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This feature enables +advertisers to simply provide high-level constraints and goals to an automated agent, which +optimizes their auction bids on their behalf. These auto-bidding intermediaries interact in +a decentralized manner in the underlying auctions, leading to new interesting practical and +theoretical questions on auction design, for example, in understanding the bidding equilibrium +properties between auto-bidder intermediaries for different auctions. In this paper, we examine +the effect of different auctions on the incentives of advertisers to report their constraints to the +auto-bidder intermediaries. More precisely, we study whether canonical auctions such as first +price auction (FPA) and second price auction (SPA) are auto-bidding incentive compatible (AIC): +whether an advertiser can gain by misreporting their constraints to the autobidder. +We consider value-maximizing advertisers in two important settings: when they have a budget +constraint and when they have a target cost-per-acquisition constraint. The main result of our +work is that for both settings, FPA and SPA are not AIC. This contrasts with FPA being AIC +when auto-bidders are constrained to bid using a (sub-optimal) uniform bidding policy. We +further extend our main result and show that any (possibly randomized) auction that is truthful +(in the classic profit-maximizing sense), scalar invariant and symmetric is not AIC. Finally, to +complement our findings, we provide sufficient market conditions for FPA and SPA to become +AIC for two advertisers. These conditions require advertisers’ valuations to be well-aligned. This +suggests that when the competition is intense for all queries, advertisers have less incentive to +misreport their constraints. +From a methodological standpoint, we develop a novel continuous model of queries. This +model provides tractability to study equilibrium with auto-bidders, which contrasts with the +standard discrete query model, which is known to be hard. Through the analysis of this model, +we uncover a surprising result: in auto-bidding with two advertisers, FPA and SPA are auction +equivalent. +†Stanford University, yeganeh@stanford.edu +∗Google, {aranyak,perlroth}@google.com +1 +arXiv:2301.13414v1 [econ.TH] 31 Jan 2023 + +1 +Introduction +Auto-bidding has become a popular tool in modern online ad auctions, allowing advertisers to +set up automated bidding strategies to optimize their goals subject to a set of constraints. By +using algorithms to adjust the bid for each query, auto-bidding offers a more efficient and effective +alternative to the traditional fine-grained bidding approach, which requires manual monitoring and +adjustment of the bids. +There are three main components in auto-bidding paradigm: 1) the advertisers who provide +high-level constraints to the auto-bidders, 2) the auto-bidder agents who bid – in a decentralized +manner – on behalf of each advertiser to maximize the advertiser’s value subject to their constraints, +and 3) the query-level auctions where queries are sold (see Figure 1). +Figure 1: The Auto-bidding Process: Advertisers submit constraints and receive query allocations +with specified costs as output. Inside the auto-bidding feature, each advertiser has an agent that +optimizes bidding profile within each advertiser’s constraints. +Current research has made important progress in studying the interactions of the second and third +components in the auto-bidding paradigm, particularly in understanding equilibrium properties (e.g., +welfare and revenue) between the auto-bidders intermediaries for different auction rules (Aggarwal +et al., 2019; Balseiro et al., 2021a; Deng et al., 2021a; Mehta, 2022; Liaw et al., 2022). There is also +work on mechanism design for this setting in more generality, i.e., between the advertisers and the +auctioneer directly abstracting out the second component (Balseiro et al., 2021c, 2022; Golrezaei +et al., 2021b). +Our work, instead, examines the relation between value-maximizing advertisers, who maximize +the value they obtain subject to a payment constraint, and the other two components of the auto- +bidding paradigm. More precisely, we study the impact of different auction rules on the incentives +of advertisers to report their constraints to the auto-bidder intermediaries. We specifically ask +whether canonical auctions such as first price auction (FPA), second price auction (SPA) and general +truthful auctions are auto-bidding incentive compatible (AIC) - in other words, can advertisers gain +by misreporting their constraints to the auto-bidder? +We consider value-maximizing advertisers in two important settings: when they have a budget +constraint and when they have a target cost-per-acquisition (tCPA) constraint1. The main result of +1The former is an upper bound on the total spend, and the latter is an upper bound on the average spend per +acquisition (sale). Our results clearly hold for more general autobidding features, such as target return on ad-spend +(tRoAS) where the constraint is an upper bound on the average spend per value generated. +1 + +Auto- +constraints +bids +Advertiser +bidder +Agent +Auto- +constraints +bids +Auction +Advertiser +bidder +per query +Agent +Auto- +constraints +bids +Advertiser +bidder +Agentour work is that for both settings, FPA and SPA are not AIC. This contrasts with FPA being AIC +when auto-bidders are constrained to bid using a (sub-optimal) uniform bidding policy. We further +generalize this surprising result and show that any (possibly randomized) truthful auction having a +scale invariance and symmetry property is also not AIC. We complement our result by providing +sufficient market conditions for FPA and SPA to become AIC for two advertisers. These conditions +require advertisers’ valuations to be well-aligned. This suggests that when the competition is intense +for all queries, advertisers have less incentive to misreport their constraints. +In our model, each advertiser strategically reports a constraint (either a tCPA or a budget) to +an auto-bidder agent which bids optimally on their behalf in each of the queries. Key in our model, +we consider a two stage game where first advertisers submit constraints to the auto-bidders and, in +the subgame, auto-bidders reach a bidding equilibrium across all query-auctions. Thus, when an +advertiser deviates and reports a different constraint to its auto-bidder, the whole bidding subgame +equilibrium can change.2 In this context, an auction rule is called auto-bidding incentive compatible +(AIC) if, for all equilibria, it is optimal for the advertiser to report their constraint to the auto-bidder. +1.1 +Main Results +We begin our results by presenting a stylized example in Section 2 that demonstrates how auto- +bidding with SPA is not AIC (Theorem 2.1). Our example consists of a simple instance with three +queries and two advertisers. This example highlights a scenario where an advertiser can benefit from +lowering their reported budget or tCPA-constraint. +We then introduce a continuous query model that departs from the standard auto-bidding model +by considering each query to be of infinitesimal size. This model provides tractability in solving +equilibrium for general auction rules like FPA which is key to study the auto-bidding incentive +compatibility properties of such auctions. Further, this continuous-query model succinctly captures +real-world scenarios where the value of a single query is negligible compared to the pool of all queries +that are sold. +Under the continuous-query model, we study the case where queries are sold using FPA and show +that in the auto-bidding paradigm, FPA is not AIC (Section 4). We first characterize the optimal +bidding strategy for each auto-bidder agent which, surprisingly, has a tractable form.3 We then +leverage this tractable form to pin down an equilibrium for the case of two auto-bidders when both +auto-bidders either face a budget or tCPA constraint. In this equilibrium, queries are divided between +the two advertisers based on the ratio of their values for each advertiser. Specifically, advertiser 1 +receives queries for which the ratio of its value to the other advertiser’s value is higher than a certain +threshold. From this point, determining the equilibrium reduces to finding a threshold that make +advertisers’ constraints tight (see Lemma 4.4 for more detail). We then show that for instances where +the threshold lacks monotonicity with the auto-bidders constraints, advertisers have an incentive +to misreport the constraint to the auto-bidder (Theorem 4.1). Conversely, when the thresholds +are monotone advertisers report constraints truthfully. We show conditions on the advertisers’ +valuations, for the two-advertisers setting, to guarantee this monotonicity (Theorem 4.10). This +condition requires a strong positive correlation of the advertisers’ valuations across the queries. As a +2This two stage model captures the idea that auto-bidding systems rapidly react to any change in the auction. +Hence, if there is any change in the bidding landscape, auto-bidders quickly converge to a new equilibrium. +3Notice that in the discrete-query model, there is no simple characterization for the auto-bidder best response in a +FPA. +2 + +practical insight, our results suggest that for settings where the competition on all queries is intense, +advertisers’ incentives to misreport is weak. +We then explore the case where, in FPA, auto-bidders are constrained to bid using a uniform +bidding strategy: the bid on each query is a constant times the advertiser’s value for the query.4 +Uniform bidding is only an optimal strategy when auctions are truthful (Aggarwal et al., 2019). +Even though for FPA these strategies are suboptimal, they have gained recent attention in the +literature due to their tractability Conitzer et al. (2022a,b); Chen et al. (2021); Gaitonde et al. (2022). +We show that in such a scenario, FPA with uniform bidding turns out to be AIC (Theorem 4.2). +However, we note that while this proves AIC in our model, the suboptimality of uniform bidding for +FPA can give rise to incentives to deviate in other ways outside our model, e.g., by splitting the +advertising campaigns into multiple campaigns with different constraints. These considerations are +important when implementing this rule in practice. +The second part of the paper pivots to the case where auctions are truthful, that is, auctions in +which it is optimal for a profit-maximizing agent to bid their value. We first study the canonical +SPA and show that, in our continuous-query model, SPA and FPA are auction equivalent. That is, +the allocation and payments among the set of reasonable equilibria (Theorem 5.5).5 As a Corollary, +the results we obtain for FPA apply to SPA as well: SPA is not AIC and; and we derive sufficient +conditions on advertisers’ valuations so that SPA is AIC for two advertisers. We then consider a +general class of randomized truthful auctions. We show that if the allocation rule satisfies these +natural conditions:6 (i) scaled invariant (if all bids are multiplied by the same factor then the +allocation doesn’t change), and (ii) is symmetric (bidders are treated equally); then the auction rule +is not AIC. +The main results of the paper are summarized in Table 1. +Per Query Auction +AIC +Second-Price Auction +Not AIC +Truthful Auctions +Not AIC +First-Price Auction +Not AIC +First-Price Auction with Uniform Bidding +AIC7 +Table 1: Main Results +1.2 +Related Work +The study of auto-bidding in ad auctions has gained significant attention in recent years. One of the +first papers to study this topic is Aggarwal et al. (2019), which presents a mathematical formulation +for the auto-bidders problem given a fixed constraints reported by advertisers. They show that +uniform bidding is an optimal strategy if and only if auctions are truthful (in the profit-maximizing +sense). They further started an important line of work to measure, using a Price of Anarchy (PoA) +approach, the welfare implications when auto-bidders are bidding in equilibrium for different auctions. +4Uniform bidding strategy is also known in the literature as pacing bidding Conitzer et al. (2022a); Chen et al. +(2021); Conitzer et al. (2022b); Gaitonde et al. (2022). +5We show the auction equivalence among uniform bidding equilibria for SPA and threshold-type equilibrium for +FPA. +6These conditions have been widely studied in the literature due to their practical use (Mehta, 2022; Liaw et al., +2022; Allouah and Besbes, 2020). +7As previously discussed, implementing FPA with the suboptimal uniform bidding policy can create other distortion +on advertisers’ incentives (e.g., splitting their campaign into multiple campaigns with different constraints). +3 + +Current results state that for SPA the PoA is 2 Aggarwal et al. (2019) and also for FPA Liaw et al. +(2022)8, and, interestingly, it can be improved if the auction uses a randomized allocation rule Mehta +(2022); Liaw et al. (2022). In a similar venue, Deng et al. (2021b); Balseiro et al. (2021b) studies +models where the auction has access to extra information and show how reserves and boosts can be +used to improve welfare and efficiency guarantees. +A second line of work, studies how to design revenue-maximizing auctions when bidders are +value-maximizing agents and may have private information about their value or their constraints +(Golrezaei et al., 2021b; Balseiro et al., 2021c,b). In all these settings, the mechanism designer is not +constrained to the presence of the auto-bidding intermediaries (Component 2 in Figure 1). Our study +has added structure by having advertisers submit their constraints first, followed by a decentralized +subgame to achieve a bidding equilibrium before allocating and determining payments. Thus, a priori +their mechanism setting can achieve broader outcomes than in our auto-bidding constraint paradigm. +Interestingly, for the one query case the authors show that FPA with a uniform bidding policy is +optimal Balseiro et al. (2021c). Our results complement theirs and show that such mechanism is +implementable in auto-bidding constraint paradigm and is AIC. +Closer to our auto-bidding paradigm, a recent line of work has started to study the incentive of +advertisers when bidding via an auto-bidder intermediary. Mehta and Perlroth (2023) show that a +profit-maximizing agent may benefit by reporting a target-based bidding strategy to the auto-bidder +when the agent has concern that the auctioneer may change (ex-post) the auction rules. Also, in +an empirical work, Li and Tang (2022) develop a new methodology to numerically approximate +auto-bidding equilibrium and show numerical examples where advertisers may benefit my reporting +their constraints on SPA. Our work complements their findings by showing under a theoretical +framework that SPA is not AIC. +Our work also connects with the literature about auction with budgeted constraint bidders. In +particular, our results are closely related to Conitzer et al. (2022a) who study FPA with uniform +bidding (a.k.a. pacing bidding). They introduce the concept of the first-price auction pacing +equilibrium (FPPE) for budget-constrained advertisers, which is the same as the equilibrium in our +auto-bidding subgame. They show that in FPPE the revenue and welfare are monotone increasing +as a function of the advertisers’ budgets. In our work, we show that in FPPE, advertisers’ values +are monotone as a function of their reported budget. In addition, they differentiate between first +and second-price by showing that FPPE is computable, unlike SPPE, where maximizing revenue +has previously been known to be NP-hard Conitzer et al. (2022b), and that the general problem +of approximating the SPPE is PPAD-complete Chen et al. (2021). In contrast, we show in the +continuous model both SPA and FPA are tractable. Interestingly, this dichotomy between FPA and +SPA (both with uniform bidding) is reflected in our work as well – the former is AIC, while the +latter is not. +Uniform bidding has been explored in a separate body of research on repeated auctions, without +the presence of auto-bidding. Balseiro and Gur (2019) investigate strategies to minimize regret in +simultaneous first-price auctions with learning. Gaitonde et al. (2022) take this concept further by +extending the approach to a wider range of auction settings. Furthermore, Golrezaei et al. (2021a) +examines how to effectively price and bid for advertising campaigns when advertisers have both ROI +and budget constraints. +8The authors show that for a general class of deterministic auctions PoA ≥ 2. +4 + +2 +Warm Up: Second Price Auction is not AIC! +To understand the implications of the auto-bidding model, we start with an example of auto-bidding +with the second-price auction. Through this example, we will demonstrate the process of determining +the equilibrium in an auto-bidding scenario and emphasize a case where the advertiser prefers to +misreport their budget leading to the following theorem. +Theorem 2.1. For the budget setting (when all advertisers are budgeted-constrained) and for the +tCPA-setting (when all advertisers are tCPA-constrained), we have that SPA is not AIC. That is, +there are some instances where an advertiser benefits by misreporting its constraint. +Proof. Consider two budget-constrained advertisers and three queries Q = {q1, q2, q3}, where the +expected value of winning query q for advertiser a is denoted by va(q), and it is publicly known (as +in Table 2). At first, each advertiser reports their budget to the auto-bidder B1 = 2, and B2 = 4. +Then the auto-bidder agents, one for each advertiser, submit the bidding profiles (to maximize their +advertisers’ value subject to the budget constraint). The next step is a second-price auction per +query, where the queries are allocated to the highest bidder. +q1 +q2 +q3 +Advertiser 1 +4 +3 +2 +Advertiser 2 +1 +1.3 +10 +Table 2: SPA with two budget constraint advertisers is not AIC: The value of each query for each +advertiser. +Finding the equilibrium bidding strategies for the auto-bidder agents is challenging, as the +auto-bidder agents have to find the best-response bids with respect to the other auto-bidder agents, +and each auto-bidder agent’s bidding profile changes the cost of queries for the rest of the agents. +To calculate such an equilibrium between auto-bidder agents, we use the result of Aggarwal et al. +(2019) to find best-response strategies. Their result states that the best response strategy in any +truthful auto-bidding auction is uniform bidding.9 In other words, each agent optimizes over one +variable, a bidding multiplier µa, and then bids on query q with respect to the scaled value µava(q). +We show that with the given budgets B1 = 2 and B2 = 4, an equilibrium exists such that +advertiser 1 only wins q1, and µ1 = 0.5 and µ2 = 1 result in such an equilibrium. To this end, +we need to check: 1) Allocation: with bidding strategies b1 = (µ1v1(q1), µ1v1(q2), µ1v1(q3)) and +b2 = (µ2v2(q1), µ2v2(q2), µ2v2(q3)), advertiser 1 wins q1 and advertiser 2 wins q2 and q3, 2) Budget +constraints are satisfied, and 3) Bidding profiles are the best response: The auto-bidder agents do +not have the incentive to increase their multiplier to get more queries. These three conditions are +checked as follows: +1. Allocation inequalities: For each query, the advertiser with the highest bid wins it. +v1(q1) +v2(q1) ≥ µ2 +µ1 += 1 +0.5 ≥ v1(q2) +v2(q2) ≥ v1(q3) +v2(q3). +2. +Budget constraints: Since the auction is second-price the cost of query q for advertiser 1 is +µ2v2(q) and for advertiser 2 is µ1v1(q). So, we must have the following inequalities to hold so +9They show uniform bidding is almost optimal, but in Appendix A we show that in this example it is exactly +optimal. +5 + +that the budget constraints are satisfied: +2 = B1 ≥ µ2v2(q1) = 1 +(Advertiser 1), +4 = B2 ≥ µ1(v1(3) + v1(q2)) = 2.5 +(Advertiser 2). +3. Best response: Does the advertiser’s agent have incentive to raise their multiplier to get more +queries? If not, they shouldn’t afford the next cheapest query. +2 < µ2(v2(q1) + v2(q2)) = 2.3 +(Advertiser 1), +4 < µ1(v1(q3) + v1(q2) + v1(q1)) = 4.5 +(Advertiser 2). +Since all the three conditions are satisfied. Thus, this profile is an equilibrium for th auto-bidders +bidding game. In this equilibrium, advertiser 1 wins q1 and advertiser 2 wins q2 and q3. +Now, consider the scenario that advertiser 1 wants to strategically report their budget B1 to the +auto-bidder. Suppose the first advertiser decreases their budget. Intuitively, the budget constraint +for the auto-bidder agent should be harder to satisfy, and hence the advertiser should not win more +queries. But, contrary to this intuition, when advertiser 1 reports a lower budget B′ +1 = 1, we show +that, given the unique auto-bidding equilibrium, advertiser 1 wins q1 and q2 (more queries than the +case where advertiser 1 reports B1 = 2). Similar to above, we can check that µ′ +1 = +1 +2.3, and µ′ +2 = 1 +results in an equilibrium (we prove the uniqueness in Appendix A): +1. Allocation: advertiser 1 wins q1 and q2 since it has a higher bid on them, +v1(q1) +v2(q1) ≥ v1(q2) +v2(q2) ≥ µ′ +2 +µ′ +1 += 1 +2.3 ≥ v1(q3) +v2(q3). +2. Budget constraints: +4 ≥ v1(q3), +and +1 = (1/2.3)(v2(q1) + v2(q2)). +3. Best response: +4 < 1(v1(q3) + v1(q2)), +and +1 < (1/2.3)(v2(q1) + v2(q2) + v2(q3)). +This surprising example leads to the first main result of the paper. In Appendix A, we will +generalize the above example to the case of tCPA-constrained advertisers with the same set of queries +as in Table 2. +Before studying other canonical auctions, in the next section we develop a tractable model of +continuous query. Under this model it turns out that the characterization of the auto-bidders bidding +equilibria when the auction is not SPA is tractable. This tractability is key for studying auto-bidding +incentive compatibility. +6 + +3 +Model +The baseline model consists of a set of A advertisers competing for q ∈ Q single-slot queries owned by +an auctioneer. We consider a continuous-query model where Q = [0, 1]. Let xa(q) be the probability +of winning query q for advertiser a. Then the expected value and payment of winning query q at price +pa(q) are xa(q)va(q)dq and pa(q)dq.10, 11 Intuitively, this continuous-query model is a first-order +approximation for instances where the size of each query relative to the whole set is small. +The auctioneer sells each query q using a query-level auction which induces the allocation and +payments (xa(q), pa(q))a∈A as a function of the bids (ba)a∈A. In this paper, we focus on the First +Price Auction (FPA), Second Price Auction (SPA) and more generally any Truthful Auction (see +Section 5.2 for details). +Auto-bidder agent: +Advertisers do not participate directly in the auctions, rather they report high-level goal constraints +to an auto-bidder agent who bids on their behalf in each of the queries. Thus, Advertiser a reports a +budget constraint Ba or a target cost-per-acquisition constraint (tCPA) Ta to the auto-bidder. Then, +the auto-bidder taking as fixed other advertiser’s bid, submits bids ba(q) to induce xa(q), pa(q) that +solves +max +� 1 +0 +xa(q)va(q)dq +(1) +s.t. +� 1 +0 +pa(q)dq ≤ Ba + Ta +� 1 +0 +xa(q)va(q)dq. +(2) +The optimal bidding policy does not have a simple characterization for a general auction. However, +when the auction is truthful (like SPA) the optimal bid take a simple form in the continuous model. +(Aggarwal et al., 2019). +Remark 3.1 (Uniform Bidding). If the per-query auction is truthful, then uniform bidding is the +optimal policy for the autobidder. Thus, ba(q) = µ · va(q) for some µ > 0. We formally prove this in +Claim 5.4. +Advertisers +Following the current paradigm in autobidding, we consider that advertisers are value-maximizers +and can be two of types: a budget-advertiser or tCPA-advertiser. Payoffs for these advertisers are as +follows. +• For a budget-advertiser with budget Ba, the payoff is +ua = +�� 1 +0 xa(q)va(q)dq +if +� 1 +0 pa(q)dq ≤ Ba +−∞ +if not. +10All functions va, xa, pa are integrable with respect to the Lebesgue measure dq. +11The set Q = [0, 1] is chosen to simplify the exposition. Our results apply to a general metric measurable space +(Q, A, λ) with atomless measure λ. +7 + +• For a tCPA-advertiser with target Ta, the payoff is +ua = +�� 1 +0 xa(q)va(q)dq +if +� 1 +0 pa(q)dq ≤ Ta · +� 1 +0 xa(q)va(q)dq +−∞ +if not. +Game, Equilibrium and Auto-bidding Incentive Compatibility (AIC) +The timing of the game is as follows. First, each advertiser depending on their type submits a +budget or target constraint to an auto-bidder agent. Then, each auto-bidder solves Problem 1 for +the respective advertiser. Finally, the per-query auctions run and allocations and payments accrue. +We consider a complete information setting and use subgame perfect equilibrium (SPE) as +solution concept. Let Va(B′ +a; Ba) the expected payoff in the subgame for a budget-advertiser with +budget Ba that reports B′ +a to the auto-bidder (likewise we define Va(T ′ +a; Ta) for the tCPA-advertiser). +Definition 3.2 (Auto-bidding Incentive Compatibility (AIC)). An auction rule is Auto-bidding +Incentive Compatible (AIC) if for every SPE we have that Va(Ba; Ba) ≥ Va(B′ +a; Ba) and Va(Ta; Ta) ≥ +Va(T ′ +a; Ta) for every Ba, B′ +a, Ta, T ′ +a. +Similar to classic notion of incentive compatibility, an auction rule satisfying AIC makes the +advertisers’ decision simpler: they simply need to report their target to the auto-bidder. However, +notice that the auto-bidder plays a subgame after advertiser’s reports. Thus, when Advertiser a +deviates and submit a different constraint, the subgame outcome may starkly change not only on +the bids of Advertiser a but also other advertisers bid may change as well. +4 +First Price Auctions +In this section, we demonstrate that the first price auction is not auto-bidding incentive compatible. +Theorem 4.1. Suppose that there are at least two budget-advertisers or two tCPA-advertisers, then +FPA is not AIC. +Later in Section 4.2, we show a complementary result by providing sufficient conditions on +advertisers’ value functions to make FPA be AIC for the case of two advertisers. We show that this +sufficient condition holds in many natural settings, suggesting that in practice FPA tends to be AIC. +Then in Section 4.3, we turn our attention to FPA where autobidders are restricted to use +uniform bidding across the queries. In this case, we extend to our continuous-query model the result +of Conitzer et al. (2022a) and show the following result. +Theorem 4.2. FPA restricted to uniform bidding is AIC. +4.1 +Proof of Theorem 4.1 +We divide the proof of Theorem 4.1 in three main steps. Step 1 characterizes the best response +bidding profile for an autobidder in the subgame. As part of our analysis, we derive a close connection +between first and second price auctions in the continuous-query model that simplifies the task of +finding the optimal bidding for each query to finding a single multiplying variable for each advertiser. +In Step 2, we leverage the tractability of our continuous-query model and pin-down the subgame +bidding equilibrium when there are either two budget-advertisers or two tCPA-advertisers in the +8 + +game (Lemma 4.4). We derive an equation that characterizes the ratio of the multipliers of each +advertiser as a function of the constraints submitted by the advertisers. This ratio defines the set of +queries that each advertiser wins, and as we will see the value accrued by each advertiser is monotone +in this ratio. So, to find a non-AIC example, one has to find scenarios where the equilibrium ratio is +not a monotone function of the input constraints which leads to the next step. +To conclude, we show in Step 3 an instance where the implicit solution for the ratio is nonmono- +tonic, demonstrating that auto-bidding in first-price auctions is not AIC. As part of our proof, we +interestingly show that AIC is harder to satisfy when advertisers face budget constraints rather than +tCPA constraints (see Corollary 4.6). +Step 1: Optimal Best Response +The following claim shows that, contrary to the discrete-query model, the best response for the +autobidder in a first price auction can be characterized as function of a single multiplier. +Claim 4.3. Taking other auto-bidders as fixed, there exists a multiplier µa ≥ 0 such that the following +bidding strategy is optimal: +ba(q) = +� +maxa′̸=a(ba′(q)) +µava(q) ≥ maxa′̸=a(ba′(q)) +0 +µava(q) ̸= maxa′(ba′(q)). +The result holds whether the advertiser is budget-constrained or tCPA-constrained12. +Proof. We show that in a first-price auction, the optimal bidding strategy is to bid on queries with +a value-to-price ratio above a certain threshold. To prove this, we assume that the bidding profile of +all advertisers is given. Since the auction is first-price, advertiser a can win each query q by fixing +small enough ϵ > 0 and paying maxa′̸=a(ba′(q)) + ϵ. So, let pa(q) = maxa′̸=a(ba′(q)), be the price of +query q. Since we have assumed that the value functions of all advertisers are integrable (i.e., there +are no measure zero sets of queries with a high value), in the optimal strategy pa is also integrable +since it is suboptimal for any advertiser to bid positive (and hence have a positive cost) on a measure +zero set of queries. +First, consider a budget-constrained advertiser. The main idea is that since the prices are +integrable, the advertiser’s problem is similar to a continuous knapsack problem. In a continuous +knapsack problem, it is well known that the optimal strategy is to choose queries with the highest +value-to-cost ratio Goodrich and Tamassia (2001). Therefore, there must exist a threshold, denoted +as µ, such that the optimal strategy is to bid on queries with a value-to-price ratio of at least µ. So +if we let µa = 1 +µ, then advertiser a bids on any query with µava(q) ≥ pa(q). +We prove it formally by contradiction. Assume to the contrary, that there exist non-zero measure +sets X, Y ⊂ Q such that for all x ∈ X and y ∈ Y , the fractional value of x is less than the fractional +value of y, i.e., +va(x) +pa(qx) < va(y) +pa(y), and in the optimal solution advertiser a gets all the queries in X and +no query in Y . However, we show that by swapping queries in X with queries in Y with the same +price, the advertiser can still satisfy its budget constraint while increasing its value. +To prove this, fix 0 < α < min( +� +X pa(q)dq, +� +Y pa(q)dq). Since the Lebesgue measure is atomless, +there exists subsets X′ ⊆ X and Y ′ ⊆ Y such that α = +� +X′ pa(q)dq = +� +Y ′ pa(q)dq. Since the value +12In FPA ties are broken in a way that is consistent with the equilibrium. This is similar to the pacing equilibrium +notion where the tie-breaking rule is endogenous to the equilibrium Conitzer et al. (2022a). +9 + +per cost of queries in Y is higher than queries in X, by swapping queries of X′ with Y ′, the value +of the new sets increases, while the cost does not change. Therefore, the initial solution cannot be +optimal. +A similar argument holds for tCPA-constrained advertisers. Swapping queries in X′ with Y ′ does +not change the cost and increases the upper bound of the tCPA constraint, resulting in a feasible +solution with a higher value. Therefore, the optimal bidding strategy for tCPA constraint is also +ba(q) as defined in the statement of the claim. +Step 2: Equilibrium Characterization +The previous step showed that the optimal bidding strategy is to bid on queries with a value-to-price +ratio above a certain threshold. Thus, we need to track one variable per auto-bidder to find the +subgame equilibrium. +In what follows, we focus on the case of finding the variables when there are only two advertisers in +the game. This characterization of equilibrium gives an implicit equation for deriving the equilibrium +bidding strategy, which makes the problem tractable in our continuous-query model.13. +From Claim 4.3 we observe that the ratio of bidding multipliers is key to determine the set of +queries that each advertiser wins. To map the space of queries to the bidding space, we introduce +the function h(q) = v1(q) +v2(q). Hence, for high values of h, the probability that advertiser 1 wins the +query increases. Also, notice that without loss of generality, we can reorder the queries on [0, 1] so +that h is non-decreasing. +In what follows, we further assume that h is increasing on [0, 1]. This implies that h is invertible +and also differentiable almost everywhere on [0, 1]. With these assumptions in place, we can now +state the following lemma to connect the subgame equilibrium to the ratio of advertisers’ values. +Lemma 4.4. [Subgame Equilibrium in FPA] Given two budget-constrained auto-bidders with budget +B1 and B2, let µ1 and µ2 be as defined in Claim 4.3 for auto-bidding with FPA. Also assume that +h(q) = v1(q) +v2(q) as defined above is strictly monotone. Then µ1 = +B2 +E[z1(z≥r)] and µ2 = µ1r, where r is +the solution of the following implicit function, +rE[1[z ≥ r)] +E[z1(z ≤ r)] = B1 +B2 +. +(3) +Here, E[·] is defined as E[P(z)] = +� ∞ +0 P(z)f(z)dz, where f(z) = v2(h−1(z)) +h′(h−1(z)) wherever h′ is defined, +and it is zero otherwise. +Also, for two tCPA auto-bidders with targets T1 and T2, we have µ1 = T1E[1(z≤r)] +E[1(z≥r)] +and µ2 = µ1r, +where r is the answer of the following implicit function, +rE[1(z ≥ r)] +E[z1(z ≥ r)] +E[1(z ≤ r)] +E[z1(z ≤ r)] = T1 +T2 +. +(4) +Remark 4.5. The function f intuitively represents the expected value of the queries that advertiser +2 can win as well as the density of the queries that advertiser 1 can win. Also, the variable r shows +the cut-off on how the queries are divided between the two advertisers. In the proof, we will see that +the advertisers’ value at equilibrium is computed with respect to f: Advertiser 1’s overall value is +� ∞ +r +zf(z)dz and advertiser 2’s overall value is +� r +0 f(z)dz. +13Notice that for the discrete-query model finding equilibrium is PPAD hard Filos-Ratsikas et al. (2021) +10 + +Proof. First, consider budget constraint auto-bidders. Given Claim 4.3, in equilibrium price of query +q is min(µ1v1(q), µ2v2(q)). Therefore, the budget constraints become: +B1 = +� 1 +0 +µ2v2(q)1(µ2v2(q) ≤ µ1v1(q))dq, +B2 = +� 1 +0 +µ1v1(q)1(µ2v2(q) ≥ µ1v1(q))dq. +With a change of variable from q to z = h(q) and letting r = µ2 +µ1 , we have: +B1 = +� ∞ +r +µ2v2(h−1(z))dh−1(z) +dz +dz +B2 = +� r +0 +µ1v1(h−1(z))dh−1(z) +dz +dz. +Observe that v1(h−1(z)) = zv2(h−1(z)), then if we let f(z) = v2(h−1)(h−1)′ = +v2(h−1(z)) +h′(h−1(z)), the +constraints become +B1 = µ2 +� ∞ +r +f(z)dz, +(5) +B2 = µ1 +� r +0 +zf(z)dz. +(6) +We obtain Equation (3) by diving both sides of Equation (5) by the respective both sides of +Equation (6). +Now, consider two tCPA constrained auto-bidders. Similar to the budget-constrained auto-bidders, +we can write +T1 +� 1 +0 +v1(q)1(µ2v2(q) ≤ µ1v1(q))dq = +� 1 +0 +µ2v2(q)1(µ2v2(q) ≤ µ1v1(q))dq +T2 +� 1 +0 +v2(q)1(µ2v2(q) ≥ µ1v1(q))dq = +� 1 +0 +µ1v1(q)1(µ2v2(q) ≥ µ1v1(q))dq +The same way of changing variables leads to the following: +T1 +T2 +� ∞ +r +xf(x)dx +� r +0 f(x) += r +� ∞ +r +f(x)dx +� r +0 xf(x)dx . +This finishes the proof of the lemma. +The previous theorem immediately implies that any example of valuation functions that is +non-AIC for budget-advertisers, it will be non-AIC for tCPA-advertisers as well. +Corollary 4.6. If auto-bidding with the first-price and two budget-advertisers is not AIC, then +auto-bidding with the same set of queries and two tCPA-advertisers is also not AIC. +11 + +Proof. Recall that advertiser 1 wins all queries with h(q) ≥ r. So, the value accrued by advertiser +1 is decreasing in r. So, if an instance of auto-bidding with tCPA-constrained advertisers is not +AIC for advertiser 1, then the corresponding function r +� ∞ +r +f(x)dx +� r +0 xf(x)dx +� r +0 f(x) +� ∞ +r +xf(x)dx (same as (4)) must be +increasing for some r′. +On the other hand, recall that r +� ∞ +r +f(x)dx +� r +0 xf(x)dx is the ratio for budget-constrained bidders equilibrium +as in (3). The additional multiplier in the equilbirum equation of tCPA constraint advertiser in +(4) is +� r +0 f(x)dx +� ∞ +r +xf(x) which is increasing in r. So, if the auto-bidding for budget-constrained bidders is +not AIC and hence the corresponding ratio is increasing for some r′, it should be increasing for the +tCPA-constrained advertisers as well, which proves the claim. +Step 3: Designing a non AIC instance +The characterization of equilibrium from Step 2 leads us to construct an instance where advertisers +have the incentive to misreport their constraints. The idea behind the proof is that the value accrued +by the advertiser 1 is decreasing in r ( as found in Lemma 4.4). Then to find a counter-example, it +will be enough to find an instance of valuation functions such that the equilibrium equation (3) is +non-monotone in r. +Proof of Theorem 4.1. We construct an instance with two budget-constrained advertisers. +By +Corollary 4.6 the same instance would work for tCPA-constrained advertisers. To prove the theorem, +we will find valuation functions v1 and v2 and budgets B1 and B2 such that the value accrued by +advertiser 1 decreases when their budget increases. +Define g(r) = +� r +0 xf(x)dx +r +� ∞ +r +f(x)dx. By Lemma 4.4, one can find the equilibrium by solving the equation +g(r) = B2 +B1 . Recall that advertiser 1 wins all queries with v1(q) +v2(q) ≥ r. So, the total value of queries +accrued by advertiser 1 is decreasing in r. Hence, to construct a non- AIC example, it is enough to +find a function f such that g is non-monotone in r. +A possible such non-monotone function g is +g(r) = (r − 1)3 + 3 +cr +− 1, +(7) +where c is chosen such that minr≥0 g(r) = 0, i.e., c = min (r−1)3+3 +r +≈ 1.95105. To see why g is +non-monotone, observe that g(r) is decreasing for r ≤ 1.8, because g′(r) = 2r3−3r2−2 +cr2 +is negative for +r ≤ 1.8, and then increasing for r ≥ 1.81. +We claim the function f defined as in, +f(r) = 3c(r − 1)2 e +� r +0 +c +(r−1)3+3 dx +((r − 1)3 + 3)2 , +(8) +would result in the function g in (7). To see why this claim is enough to finish the proof, note that +there are many ways to choose the value functions of advertisers to derive tf as in (8). One possible +way is to define v1, v2 : [0, 1] → R as v2(q) = f(tan(q))/(tan(q)2 + 1) and v1(q) = tan(q)v2(q) (see +Fig. 2). +12 + +(a) Valuation function of two advertisers. +(b) Finding the equilibrium using (3). +Figure 2: An example of two advertisers such that FPA is not AIC (proof of Theorem 4.1). When +B1 +B2 = 1200, there are three values for r (see the right panel) that lead to equilibrium, and one +(orange) leads to non-AIC equilibrium. +So it remains to prove that choosing f as in (8) would result in g as defined in (7). To derive f +from g, first we simplify g using integration by part, +g(r) = +� r +0 xf(x)dx +r +� ∞ +r +f(x)dx += r +� r +0 f(x)dx − +� r +0 +� x +0 f(y)dydx +r +� ∞ +r +f(x)dx += r +� ∞ +0 f(x)dx − +� r +0 +� x +0 f(y)dydx +r +� ∞ +r +f(x)dx +− 1, +Assuming that +� ∞ +0 f(x) is finite, the above equations lead to the following +rg(r) + r = +� r +0 +� ∞ +x f(y)dydx +� ∞ +r +f(x)dx +. +(9) +Therefore, by integrating the inverse of both sides, +log( +� r +0 +� ∞ +x +f(y)dydx) = C + +� r +0 +1 +xg(x) + xdx, +and by raising to the exponent +� r +0 +� ∞ +x +f(y)dydx = Ke +� r +0 +1 +xg(x)+x dx. +for some constants C and K > 0. Then by differentiating both sides with respect to x, +� ∞ +r +f(x)dx = +K +rg(r) + re +� r +0 +1 +xg(x)+x dx. +Note that for any choice of K ≥ 0, dividing the last two equations will result in (9). So, without loss +of generality, we can assume K = 1. By differentiating again, we can derive f as a function of g: +f(r) = (g′(r)r + g(r)) +(rg(r) + r)2 e +� r +0 +1 +xg(x)+x dx. +13 + +3.0 +- +Vi(q) +25 +V2(q) +20 +15 +LD +0.5 +0.D +0.2 +t0 +0.6 +o.B +LD +q3500 +rE[1[z ≥ r]] +E[21(z ≤ r)] +31D0 +2500 +22000 +1500 +14D0 +50D +0 +FN +4 +6 +1 +rWe need g′(r)r + g(r) ≥ 0 to ensure that f(r) ≥ 0 for all r. This holds for g as in (7). Finally, by +substituting g as in (7), we will derive f as in (8). +Remark 4.7. Note that the above proof shows that for values of r such that there exists a equilibrium +which is not AIC, there exists always a second monotone equilibrium. This follows from the fact that +the function g(r) tends to infinity as r → ∞, so, g must be increasing for some large enough r. +Before moving on to finding conditions for incentive compatibility, we also note that the above’s +characterization implies the existence of equilibrium for auto-bidding with any pairs of advertisers. +Proposition 4.8. Given auto-bidding satisfying the conditions of Lemma 4.4, the equilibrium for +all pairs of budgets or all pairs of tCPA constrained advertisers always exists. +Proof. Recall that the equilibirum exists if there exists an r such that +B2 +B1 += +� r +0 xf(x)dx +r +� ∞ +r +f(x)dx +has a solution for any value of B2 +B1 . Note that the right-hand side ( +� r +0 xf(x)dx +r +� ∞ +r +f(x)dx) is positive for any +r > 0, and it continuously grows to infinity as r → ∞. So, to make sure that every value of +B2/B1 is covered, we need to check whether at r = 0 the ratio becomes zero. By L’Hopital rule, +limz→0 +zf(z) +� ∞ +z +f(x)dx−zf(z) = 0, which is as desired. +For tCPA constrained advertiser, the second ratio +r +� r +0 f(x)dx +� ∞ +r +xf(x)dx always converges to 0, so the +equilibrium in this case always exists. +4.2 +Sufficient Conditions for Incentive Compatibility +In this section we show that the lack of ACI happens for cases where advertisers’ valuations have +unusual properties. +More precisely, the main result of the section is to characterize sufficient +conditions on the advertiser’s valuations so that FPA is AIC when there are two advertisers in the +auction. +For this goal, we recall the function f(z) = v2(h−1(z)) +h′(h−1(z)) where h(q) = v1(q) +v2(q) defined in Section 4.1. +As shown in Lemma4.4, function f behaves as a value of the queries advertiser 2 gets and the density +of queries that advertiser 1 gets. +Lemma 4.9. Consider that there are two advertisers and they can either both be budget-advertisers +or tCPA-advertisrs. Also, suppose that auto-bidder with FPA uses the optimal bidding strategy in +Claim 4.3. Then a sufficient condition for FPA to be AIC is that f has a monotone hazard rate, i.e., +f(r) +� ∞ +r +f(x)dx is non-decreasing in r. +Proof. Following the proof Theorem 4.1, if g(r) = +� r +0 +� ∞ +x +f(y)dydx +r +� ∞ +r +f(x)dx +is non-decreasing in r then the +equilibrium is AIC. The equivalent sufficient conditions obtained by imposing the inequality g′(r) ≥ 0 +is that for all r ≥ 0, +r +� � ∞ +r +f(x)dx +�2 ≥ +� � r +0 +� ∞ +x +f(y)dydx +�� � ∞ +r +f(x)dx − rf(r) +� +. +(10) +14 + +If +� ∞ +r +f(x)dx ≤ rf(r) then above’s inequality obviously holds. So, we can assume that for some +r > 0, +� ∞ +r +f(x)dx > rf(r). Since +f(z) +� ∞ +z +f(x)dx is non-decreasing in z, we must have that for all r′ ≤ r, +f(r′) +� ∞ +r′ f(x)dx ≤ +f(r) +� ∞ +r +f(x)dx ≤ 1 +r ≤ 1 +r′ . On the other hand by taking the derivative of +f(z) +� ∞ +z +f(x)dx, we must +have that f′(z) +� ∞ +z +f(x)dx + f(z)2 ≥ 0. By considering two cases on the sign of f′, for z ≤ r we +must have, f(z) +� +zf′(z) + f(z)) ≥ 0, and hence (zf(z))′ ≥ 0 for all z ≤ r. Therefore, zf(z) is +non-decreasing for z ≤ r. +On the other hand, +� r +0 +� ∞ +x +f(y)dydx = +� r +0 +� r +x +f(y)dydx + +� r +0 +� ∞ +r +f(y)dydx += r +� r +0 +f(x)dx − +� r +0 +� x +0 +f(y)dydx + r +� ∞ +r +f(x)dx += +� r +0 +xf(x)dx + r +� ∞ +r +f(x)dx, +where the second equation is by integration by part. Then by applying monotonicity of z(f)z for +z ≤ r we have +� r +0 +� ∞ +x f(y)dydx ≤ r2f(r) + r +� ∞ +r +f(x)dx. So, to prove (10) it is enough to show that +�� ∞ +r +f(x)dx +�2 +≥ +� +rf(r) + +� ∞ +r +f(x)dx +� �� ∞ +r +f(x)dx − rf(r) +� +, +which holds, since the right-hand side is equal to +� � ∞ +r +f(x)dx +�2 − (rf(r))2 strictly less than the +left-hand side. +While the condition on f has intuitive properties when seen as a density, it has the unappealing +properties to be too abstract in terms of the conditions on the advertisers’ valuation. The following +result, provides sufficient conditions on value functions v1 and v2 that makes f be monotone hazard +rate, and hence, FPA to be AIC. +Theorem 4.10. Consider two advertisers that are either budget-advertisers or tCPA-advertisers. +Assume that h(q) = v1(q) +v2(q) is increasing concave function and that v2 is non-decreasing. Then, the +equilibrium in FPA auto-bidding with bidding strategy as in Claim 4.3 is AIC. +Proof. Note that when f is non-decreasing, it also has a monotone hazard rate. Now, when h +is concave, +1 +h′ is a non-decreasing function, and since v2 is also non-decreasing, then f is also +non-decreasing. +4.3 +FPA with uniform bidding +The previous section shows that when auto-bidders have full flexibility on the bidding strategy, +FPA is not AIC. However, non-uniform bid is not simple to implement and auto-bidders may be +constrained to use simpler uniform bidding policies (aka pacing bidding). In this context, the main +result of the section is Theorem 4.2 that shows that when restricted to uniform bidding policies FPA +is AIC. Note that here, we are assuming a simple model where advertisers do not split campaigns. +So, FPA with uniform bidding is AIC but it could bring up other incentives for advertisers when it +is implemented. +15 + +Definition 4.11 (Uniform bidding equilibrium). A uniform bidding equilibrium for the auto-bidders +subgame corresponds to bid multipliers µ1, . . . , µN such that every auto-bidder a chooses the uniform +bidding policy µa that maximizes Problem (1) when restricted to uniform bidding policies with the +requirement that if advertiser a’s constraints of type 2 are not tight then µa gets its maximum possible +value.14 +The proof of Theorem 4.2 is based on the main results of Conitzer et al. (2022a). The authors +proved that the uniform-bidding equilibrium is unique and in equilibrium the multiplier of each +advertiser is the maximum multiplier over all feasible uniform bidding strategies. Their result is +for budget-constrained advertisers, and we extend it to include tCPA constrained advertisers. The +proof is deferred to Appendix B. +Lemma 4.12 (Extension of Theorem 1 in Conitzer et al. (2022a)). Given an instance of Auto- +bidding with general constraints as in (2), there is a unique uniform bidding equilibrium, and the bid +multipliers of all advertisers is maximal among all feasible uniform bidding profiles. +Now, we are ready to prove Theorem 4.2. +Proof of Theorem 4.2. Assume that advertiser 1 increases their budget or their target CPA. Then the +original uniform bidding is still feasible for all advertisers. Further, by Lemma 4.12 the equilibrium +pacing of all advertisers is maximal among all feasible pacings. So, the pacing of all advertisers +will either increase or remain the same. But the constraints of all advertisers except 1 are either +binding or their multiplier has attained its maximum value by the definition of pacing equilibrium. +Therefore, the set of queries they end up with should be a subset of their original ones since the +price of all queries will either increase or remain the same. So, it is only advertiser 1 that can win +more queries. +Remark 4.13. Conitzer et al. (2022a) show monotonicity properties of budgets in FPA with uniform +bidding equilibrium for the revenue and welfare. Instead, in our work we focus on monotonicity for +each advertiser. +5 +Truthful Auctions +This section studies auto-bidding incentive compatibility for the case where the per-query auction is +a truthful auction. +A truthful auction is an auction where the optimal bidding strategy for a profit-maximizing +agent is to bid its value. An important example of a truthful auction is Second Price Auction. As we +showed in the three-queries example of the introduction, SPA is not AIC. In this section, we show +that the previous example generalizes, in our continuous-query model, to any (randomized) truthful +auctions so long as the auction is scalar invariant and symmetric (see Assumption 5.1 below for +details). As part of our proof-technique, we obtain an auction equivalence result which is interesting +on its own: in the continuous query-model SPA and FPA have the same outcome.15 +For the remaining of the section we assume all truthful auction satisfy the following property. +14When valuations are strictly positive for all queries q ∈ [0, 1], we can easily show that bid multipliers have to be +bounded in equilibrium. When this is not the case, we set a cap sufficiently high to avoid bid multipliers going to +infinity. +15It is well-known that in the discrete-query model, FPA and SPA are not auction equivalent in the presence of +auto-bidders. +16 + +Assumption 5.1. Let (xa(b))a∈A be the allocation rule in a truthful auction given bids b = (ba)a∈A. +We assume that the allocation rule satisfies the following properties. +1. The auction always allocates: � +a∈A xa(b) = 1 +2. Scalar invariance: For any constant c > 0 and any advertiser a ∈ A, xa(b) = xa(cb). +3. Symmetry: For any pair of advertisers a, a′ ∈ A and bids b, b′, b−{a,a′} = (b)a∈A\{a,a′} we have +that +xa(ba = b, ba′ = b′, b−{a,a′}) = xa′(ba = b′, ba′ = b, b−{a,a′}). +Remark 5.2. Observe that SPA satisfies Assumption 5.1. +From the seminal result of Myerson (1981) we obtain a tractable characterization of truthful +auctions which we use in our proof. +Lemma 5.3 (Truthful auctions (Myerson, 1981)). Let (xa(b), pa(b))a∈A the allocation and pricing +rule for an auction given bids b = (ba)a∈A. The auction rule is truthful if and only if +1. Allocation rule is non-decreasing on the bid: For each bidder a ∈ A and any b′ +a ≥ ba, we have +that +xa(b′ +a, b−a) ≥ xa(ba, b−a). +2. Pricing follows Myerson’s formulae: +pa(b) = ba · xa(b) − +� ba +0 +xa(z, b−a)dz. +A second appealing property of truthful actions is that the optimal bidding strategy for auto- +bidders is simpler: in the discrete-query model uniform bidding strategy is almost optimal and can +differ from optimal by at most the value of two queries (Aggarwal et al., 2019). We revisit this result +in our continuous-query model and show that uniform bidding policy is optimal for truthful auctions. +Claim 5.4. In the continuous-query model, if the per-quuery auction is truthful then using a uniform +bidding is an optimal strategy for each auto-bidder. +Proof. We use Theorem 1 Aggarwal et al. (2019). Pick some small δ > 0 and divide the interval +[0, 1] into subintervals of length δ. Let each subinterval I be a discrete query with value functions +vj(I) = +� +I vj(q)dq. Then Theorem 1 Aggarwal et al. (2019) implies that uniform bidding differs from +optimal by at most two queries. So, the difference from optimal is bounded by 2 maxj max|I|≤δ vj(I). +Now, since the valuation functions are atomless (i.e., the value of a query is dq), by letting δ to 0, +the error of uniform bidding in the continuous case also goes to zero. +5.1 +SPA in the Continuous-Query Model +We generalize the discrete example of second price auction in Theorem 2.1 to the continuous set +of queries model showing that SPA is not AIC. The key step consists on showing that for the +continuous-query model there is an auction equivalence result between first and second price auction. +17 + +Theorem 5.5. [Auction Equivalence Result] Suppose that auto-bidder uses a uniform bid strategy +for SPA, and similarly, uses the simple bidding strategy defined in Claim 4.3 for FPA. Then, in any +subgame equilibrium the outcome of the auctions (allocations and pricing) on SPA is the same as in +FPA. +This result immediately implies that all the results for FPA in Section 4 hold for SPA as well. +Theorem 5.6. Suppose that there are at least two budget-advertisers or two tCPA-advertisers, then +even for the continuous-query model SPA is not AIC. +Similarly to FPA case, we can characterize the equilibrium for the two-advertiser case and derive +sufficient conditions on advertisers’ valuation functions so that SPA is AIC. +Theorem 5.7. Given two advertisers, let µ1 and µ2 be the bidding multipliers in equilibrium for the +subgame of the auto-bidders. Also assume that h(q) = v1(q) +v2(q) is increasing. Then +1. If the advertisers are budget-constrained with budget B1 and B2, then µ1 = +B2 +E[z1(z≥r)] and +µ2 = µ1r, where r is the answer of the following implicit function, +rE[1[z ≥ r)] +E[z1(z ≤ r)] = B1 +B2 +. +Here, E[.] is defined as E[P(z)] = +� ∞ +0 P(z)f(z)dz, where f(z) = v2(h−1(z)) +h′(h−1(z)) wherever h′ is +defined, and it is zero otherwise. +2. If the advertisers are tCPA-constrained with targets T1 and T2, we have µ1 = T1E[1(z≤r)] +E[1(z≥r)] +and +µ2 = µ1r, where r is the answer of the following implicit function, +rE[1(z ≥ r)] +E[z1(z ≥ r)] +E[1(z ≤ r)] +E[z1(z ≤ r)] = T1 +T2 +. +3. If further, v2 is non-decreasing in q, and h is concave, and advertiers are either both budget- +constrained two tCPA-constrained, then SPA is AIC. +We now demonstrate the auction equivalence between FPA and SPA. +Proof of Theorem 5.5. Note that the optimal strategy for a second-price auction is uniform bidding +with respect to the true value of the query by Claim 5.4. Also, Claim 4.3 implies that the cost obtained +by each advertiser in first-price auction in the continuous model is also depends on pacing multipliers +of the other advertiser. This claim immediately, suggests the equivalent between the optimal. bidding +strategies of first and second price auctions. So, the optimal strategy for both auctions will be the +same and therefore the resulting allocation and pricing will also be the same. Hence, it follows that +the same allocation and pricing will be a pure equilibrium under both auctions. +5.2 +Truthful Auctions Beyond Second-Price +We now present the main result of the section. We show that a general truthful auction (with +possibly random allocation) is not AIC. +18 + +Theorem 5.8. Consider a truthful auction (x, p) satisfying Assumption 5.1. If there are at least +two budget-advertisers or two tCPA-advertisers, then the truthful auction is not AIC. +The remainder of the section gives an overview of the proof of this theorem. Similar to the +FPA and SPA case, we start by characterizing the equilibrium in the continuous case when there +are two advertisers in the game. The proof relies on the observation that for auctions satisfying +Assumption 5.1, the allocation probability is a function of the bids’ ratios. So, again, similar to FPA +and SPA finding the equilibrium reduces to finding the ratio of bidding multipliers. Then to finish +the proof of Theorem 5.8 instead of providing an explicit example where auto-bidding is non-AIC, we +showed that the conditions needed for an auction’s allocation probability to satisfy are impossible. +The following theorem finds an implicit equation for the best response. We omit the proofs of +the intermediaries steps and deferred them to the Appendix C. +Theorem 5.9. Consider a truthful auction (x, p) satisfying Assumption 5.1 and assume that there +are either two budget-advertisers or two tCPA-advertisers. Let µ1 and µ2 be the bidding multipliers +used by the auto-bidders in the subgame equilibrium. Further, assume that h(q) = v1(q) +v2(q) is increasing. +Then +1. If the advertisers are budget-constrained with budget B1 and B2, then µ1 = +B1 +E[p1(rz,1)] and +µ2 = rµ1, where r is the answer of the following implicit function, +E[rp1( z +r, 1)] +E[zp1( r +z, 1)] = B1 +B2 +. +Here, E[.] is defined as E[P(z)] = +� ∞ +0 P(z)f(z)dz, where f(z) = v2(h−1(z)) +h′(h−1(z)) wherever h′ is +defined, and it is zero otherwise. +2. If the advertisers are tCPA-constrained with targets T1 and T2, we have µ1 = T1E[zg(z/r)] +E[rp1(z/r)] and +µ2 = µ1r, where r is the answer of the following implicit function, +E[x1( r +z, 1)] +E[zx1( z +r, 1)] +E[rp1( z +r, 1)] +E[zp1( r +z, 1)] = T1 +T2 +. +Because allocation probability x1 is a non-decreasing function, we can derive a similar result to +the FPA case and show if an instance is not AIC for budget-advertisers then it is also not AIC for +tCPA-advertisers. +Proposition 5.10. If for the two budget-constrained advertisers case the truthful auction is not +AIC, then for the tCPA-constrained advertisers case the same auction is also not AIC. +Using the previous results we are in position to tackle the main theorem. +Proof of Theorem 5.8. We prove Theorem 5.8 for budget constrained advertisers, since Proposi- +tion 5.10 would derive it for tCPA constraint advertisers. We use implicit function theorem to find +conditions on p1 and f to imply monotonicity in r. Let +H(x, r) = +� ∞ +0 rf(z)p1(z/r, 1)dz +� ∞ +0 f(z)zp1(r/z, 1)dz − x. +19 + +Then when advertiser 1 increases budget, the corresponding variable x increases. So, if we want to +check whether r is a non-decreasing function of x, we need dr +dx to be non-negative. By the implicit +function theorem, +dr +dx = − +∂H +∂x +∂H +∂r += +1 +∂H +∂r +. +So, assume to the contrary that r i always non-decreasing in x, then ∂H(x,r +∂r +≥ 0. +Define +p(x) = p1(x, 1). Then we have the following +E[ d +drrp(z/r)]E[zp(r/z)] ≥ E[rp(z/r)]E[ d +dr +� +zp(r/z) +� +]. +Then +d +drE[rp(z/r)] +E[rp(z/r)] +≥ +d +drE[zp(z/r)] +E[zp(z/r)] +By integrating both parts, we have that for any choice of f, +rE[p(z/r)] ≥ E[zp(r/z)]. +When the above inequality hold for any choice of v1 and v2, we claim that the following must hold +almost everywhere +p(b) ≥ bp(1/b). +(11) +To see this, assume to the contrary that there exist a measurable set B such that (11) does not hold +for it. Let qv2(q) = v1(q), therefore, f(z) = v2(z) can be any measurable function. So, we can define +f to have zero value everywhere except X, and have weight 1 over X to get a contradiction. +By substituting variable with y = 1/b in (11), p(1/b)db ≥ p(b)/bdb. Therefore, almost everywhere +p(b) = bp(1/b). By differentiating we have p′(b) = p(1/b) − p′(1/x)/x. On the other hand, as we will +see in Appendix C for any truthful auction satisfying Assumption 5.1, p′(b) = p′(1/b). Therefore, +p(b) = p′(b)(b + 1). Solving it for p, we get that the only possible AIC pricing must be of the form +p(b) = α(b + 1) for some α > 0. +Next, we will show there is no proper allocation probability satisfying the Assumption 5.1 that +would result in a pricing function p. It is not hard to see that by the Myerson’s pricing formulae, +dx1(b,1) +db += p′(b) +b . Therefore, we must have x′ +1(b, 1) = α/b, so x1(b, 1) = c log(b) + d for some constants +c > 0 and d. But x1 cannot be a valid allocation rule, since it will take negative values for small +enough b. +References +Gagan Aggarwal, Ashwinkumar Badanidiyuru Varadaraja, and Aranyak Mehta. 2019. Autobidding +with Constraints. In Web and Internet Economics 2019. +Amine Allouah and Omar Besbes. 2020. Prior-independent optimal auctions. Management Science +66, 10 (2020), 4417–4432. +20 + +Santiago Balseiro, Yuan Deng, Jieming Mao, Vahab Mirrokni, and Song Zuo. 2021a. +Robust +Auction Design in the Auto-bidding World. 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Auction Design in an Auto-Bidding Setting: Randomization Improves +Efficiency Beyond VCG. In Proceedings of the ACM Web Conference 2022 (Virtual Event, Lyon, +France) (WWW ’22). Association for Computing Machinery, New York, NY, USA, 173–181. +https://doi.org/10.1145/3485447.3512062 +Aranyak Mehta and Andres Perlroth. 2023. Auctions without commitment in the auto-bidding world. +https://doi.org/10.48550/ARXIV.2301.07312 +Roger B. Myerson. 1981. Optimal Auction Design. Mathematics of Operations Research 6, 1 (1981), +58–73. https://doi.org/10.1287/moor.6.1.58 arXiv:https://doi.org/10.1287/moor.6.1.58 +A +Second-price tCPA constrained +Proof. We continue with the proof of Theorem 2.1. We first prove the uniqueness of equilibrium in +the case of B′ +1 = 1. irst, note that there’s no equilibrium such that advertiser 1 wins all the queries. +To see this, note that the multiplier of advertiser 1 is at most 1. Hence, the price of q3 for advertiser +2 is within their budget, and they have the incentive to increase their multiplier to buy q3. Similarly, +one can see that in any equilibrium, advertiser 1 gets at least q1, since its highest price is within +their budget. +Now, assume some equilibrium exists with bidding prices ˜µ1 and ˜µ2 such that advertiser 1 gets +only q1. Then +˜µ1(v1(1) + v1(2) + v1(3)) +B2 +> 1 ≥ ˜µ2v2(1) +B1 +, +where the first inequality is because advertiser 2’s multiplier is the best response, and the second is +coming from the budget constraint for advertiser 1. Therefore, +B1 +B2 +v1(1) + v1(2) + v1(3) +v2(1) +≥ ˜µ2 +˜µ1 +, +But v1(2) +v2(2) ≥ B1 +B2 +v1(1)+v1(2)+v1(3) +v2(1) += 9 +4, and thus v1(2) +v2(2) > ˜µ2 +˜µ1 . This is in contradiction with allocation +inequalities since advertiser 2 wins q2. Therefore, we proved with B1 = 1 and B2 = 4 the equilibrium +is unique such that advertiser 1 wins q1 and q2. +22 + +Now it remains to show a non AIC example for tCPA advertisers. Again consider two advertisers, +and 3 queries, with the same values as in Table 2. Here, let the tCPA constraint of advertiser 1 be +T1 = 0.4 and for advertiser 2 be T2 = 0.7. Then again we show that there exists a unique equilibrium +in which advertiser 1 gets queries 1 and 2. +First, to prove the existence, let µ1 = 1.6 and µ2 = 1.2. Then we show this is an equilibrium +since the three following conditoins hold: +1. Allocation: advertiser 1 wins q1 and q2 since it has a higher bid on them +v1(1) +v2(1) ≥ v1(2) +v2(2) ≥ µ2 +µ1 += 1.2 +1.5 ≥ v1(3) +v2(3). +2. tCPA constraints are satisfied: +T2v2(3) ≥ µ1v1(3), +and +T1(v1(1) + v1(2)) ≥ µ2(v2(1) + v2(2)). +3. Best response: non of the advertiser can win more queries if they increase their multiplier: +T2(v2(3) + v2(2)) < µ1(v1(3) + v1(2)), +and +T1(v1(1) + v1(2) + v1(3)) < µ2(v2(1) + v2(2) + v2(3)). +Now, similar to the proof of the budget-constrained advertisers we show the equilibrium is unique. +Note that there’s no equilibrium such that advertiser 1, gets all queries since the cost of all queries +for advertiser 1 is at least v2(1) + v2(2) + v2(3) = 12.3 which is larger than T1(v1(1) + v1(2) + v1(3)). +Similarly, advertiser 2 cannot get all queries since the tCPA constraint would not hold v1(1) +v1(2) + +v1(3) > T2(v2(1) + v2(2) + v2(3)). So, to prove the uniqueness of equilibrium, it remains to show that +there’s no equilibrium that advertiser 1 only gets query 1. To contradiction, assume such equilibrium +exists with the corresponding multipliers ˜µ1 and ˜µ2. Then we must have +˜µ1(v1(1) + v1(2) + v1(3)) +T2(v2(1) + v2(2) + v2(3)) > 1 ≥ ˜µ2v2(1) +T1v1(1), +where the first inequality is because advertiser 2’s multiplier is the best response, and the second +inequality is coming from the budget constraint for advertiser 1. Therefore, +T1 +T2 +v1(1) + v1(2) + v1(3) +v2(1) + v2(2) + v2(3) +v1(1) +v2(1) ≥ ˜µ2 +˜µ1 +, +But v1(2) +v2(2) ≥ T1 +T2 +v1(1)+v1(2)+v1(3) +v2(1)+v2(2)+v2(3) +v1(1) +v2(1), and thus v1(2) +v2(2) > ˜µ2 +˜µ1 . This is in contradiction with allocation +inequalities since advertiser 2 wins q2. Therefore, we proved with T1 = 0.4 and T2 = 0.7, the +equilibrium is unique such that advertiser 1 wins q1 and q2. +Now, we show that if advertiser 1 increases their tCPA constraint to T ′ +1 = 0.6, then there exists +an equilibrium such that advertiser 1 only wins q1. Let µ′ +1 = 1 and µ2 = 2.38. Then +1. Allocation: advertiser 1 wins q1 +v1(1) +v2(1) ≥ µ′ +2 +µ′ +1 += 2.38 +1. +≥ v1(2) +v2(2) ≥ v1(3) +v2(3). +23 + +2. tCPA constraints are satisfied: +T2(v2(3) + v2(2)) ≥ µ′ +1(v1(3) + v1(2)), +and +T ′ +1v1(1) ≥ µ′ +2v2(1). +3. Best response: non of the advertiser can win more queries if they increase their multiplier: +T2(v2(1) + v2(2) + v2(3)) < µ′ +1(v1(1) + v1(2) + v1(3)), +and +T ′ +1(v1(1) + v1(2)) < µ′ +2(v2(1) + v2(2)). +B +First-price pacing equilibrium +Proof of Lemma 4.12. We follow the same steps of the proof as in Conitzer et al. (2022a) for tCPA +constrained advertisers. Consider two sets of feasible bidding multipliers µ and µ′. We will show +that µ∗ = max(µ, µ′) is also feasible, where max is the component wise maximum of the bidding +profiles for n advertisers. +Each query q is allocated to the bidder with the highest pacing bid. We need to check that +constraint (2) is satisfied. Fix advertiser a. Its multiplier in µ∗ must be also maximum in one of µ, +or µ′. without loss assume µ∗ +a = µa. Then the set of queries that a wins with bidding profile µ∗ (X∗ +a) +must be a subset of queries it wins n µ (Xa), since all other advertisers’ bids have either remained +the same or increased. On the other hand, the cost of queries a wins stays the same, since it’s a first +price auction. Since constraint (2) is feasible for bidding multipliers µ we must have +(µa − Ta) +� +q∈X +va(q) ≤ Ba. +But then since X∗ ⊆ X, we have as well +(µa − Ta) +� +q∈X∗ va(q) = (µ∗ +a − Ta) +� +q∈X∗ va(q) ≤ Ba, +which implies µ∗ is a feasible strategy. +To complete the proof we need to show the strategy that all advertisers take the maximum feasible +pace µ∗ +a = sup{µa|µ is feasible} results in an equilibrium. To see this, note that if an advertiser’s +strategy is not best-response, they have incentive to increase their pace with its constraints remaining +satisfied. But then this would result into another feasible pacing strategy and is in contradiction +with the choice of the highest pace µ∗ +a. A similar argument also shows the equilibrium is unique. +Assume there exists another pacing equilibrium where an advertiser a exists such that its pace is +less than µ∗ +a. Then by increasing their pace to µ∗ +a they will get at least as many queries as before, so +µ∗ +a is the best-response strategy. +24 + +C +Proofs for Truthful Auctions +We start by the following observation, which follows by applying Assumption 5.1 to reformulate the +allocation function in the case of two advertisers as a function of a single variable. +Claim C.1. The probability of allocating each query is a function of the ratio of bids, i.e., there +exists a non-decreasing function g : R+ → [0, 1] such that the followings hold.16 +1. x1(b1(q), b2(q)) = g( b1(q) +b2(q)), +2. g(z) + g(1/z) = 1, +3. g(0) = 0. +For example, SPA satisfies the above claim with g(z) = 1 when z = b1(q) +b2(q) ≥ 1. We are ready to +prove Theorem 5.9, which follows the similar steps of Lemma 4.4. +Proof of Theorem 5.9. By Claim 5.4, there exists µ1 and µ2 such that advertiser a bids zava(q) on +each query. Therefore, we can write the budget constraint for bidder 1 as, +B1 = +� 1 +0 +p1(b1(q), b2(q))dq = +� 1 +0 +µ1v1(q)g +�v1(q) +v2(q) +µ1 +µ2 +� +dq − +� 1 +0 +� µ1v1(q) +0 +g +� +x +v2(q)µ2 +� +dxdq +Next, with a change of variable x = v1(q)y we have +B1 = +� 1 +0 +µ1v1(q)g +�v1(q) +v2(q) +µ1 +µ2 +� +dq − +� 1 +0 +� µ1 +0 +g +� v1(q) +v2(q)µ2 +y +� +v1(q)dydq. +As before, let h(q) = v1(q) +v2(q). Then let z = h(q), we have dq = dh−1(z) = +1 +h′(h−1(z))dz. So, +B1 = +� ∞ +0 +µ1v1(h−1(z))g +�zµ1 +µ2 +� +1 +h′(h−1(z))dz − +� ∞ +0 +� µ1 +0 +g +� z +µ2 +y +� +v1(h−1(z))dy +1 +h′(h−1(z))dz. +Define f(z) = v2(h−1(z)) +h′(h−1(z)) = 1 +z +v1(h−1(z)) +h′(h−1(z)). Then we have +B1 = +� ∞ +0 +µ1zf(z)g +�zµ1 +µ2 +� +dz − +� ∞ +0 +� � µ1 +0 +g +� z +µ2 +y +� +dy +� +zf(z)dz. +Similarly, +B2 = +� ∞ +0 +µ2v2(h−1(z))(1 − g +�zµ1 +µ2 +� +) +1 +h′(h−1(z))dz − +� ∞ +0 +� µ2 +0 +g +� y +µ1z +� +v2(h−1(z))dy +1 +h′(h−1(z))dz. +B2 = +� ∞ +0 +µ2f(z)(1 − g +�zµ1 +µ2 +� +)dz − +� ∞ +0 +� µ2 +0 +g +� y +µ1z +� +dyf(z)dz. +Next, we find the implicit function to derive r = µ2 +µ1 . By change of variable we have the following +two equations: +B1 +µ1 += +� ∞ +0 +zf(z)g(z/r)dz − r +� ∞ +0 +� � z/r +0 +g(w)dw +� +f(z)dz. +16Notice that the function g is measurable since is non-decreasing. +25 + +B2 +µ2 += +� ∞ +0 +f(z)(1 − g(z/r))dz − 1 +r +� ∞ +0 +� � r/z +0 +g(w)dw +� +zf(z)dz. +The implicit function for r is the following: +B1 +B2 += +� ∞ +0 f(z) +� +zg(z/r) − r +� z/r +0 +g(w)dw +� +dz +� ∞ +0 f(z) +� +r(1 − g(z/r) − z +� r/z +0 +g(w)dw +� +dz +. +Recall the payment rule in Assumption 5.3, this can be re-written as +B1 +B2 += +� ∞ +0 rf(z)p1(z/r, 1)dz +� ∞ +0 zf(z)zp1(r/z, 1)dz , +which finishes the proof for the budget constrained advertisers. +Now, consider two tCPA constrained advertisers. Following the same argument as above, we get +the following from tightness of tCPA constraints +T1 +� ∞ +0 +zf(z)g +�zµ1 +µ2 +� +dz = +� ∞ +0 +µ1zf(z)g +�zµ1 +µ2 +� +dz − +� ∞ +0 +� � µ1 +0 +g +� z +µ2 +y +� +dy +� +zf(z)dz, +and, +T2 +� ∞ +0 +f(z)(1 − g +�zµ1 +µ2 +� +)dz = +� ∞ +0 +µ2f(z)(1 − g +�zµ1 +µ2 +� +)dz − +� ∞ +0 +� µ2 +0 +g +� y +µ1z +� +dyf(z)dz. +By dividing both sides of the equations we get the desired results. +Now, to prove the main theorem, we need to show that the values accrued by advertisers is +monotone in µ1/µ2. +Claim C.2. Let µi be the optimal bidding multiplier for advertiser i. Given the assumptions in +Theorem 5.8, the value obtained by advertiser 1 is increasing in r = µ1 +µ2 . +Proof. Following the proof of Theorem 5.9 we can write value obtained by advertiser i as +V1(B1, B2) = +� ∞ +0 +f(z)zg(rz)dz, +where r is the answer to the implicit function stated in Theorem 5.9. Monotonicity of V1(B1, B2) as +a function of r follows from the fact that g is a monotone function. +26 + diff --git a/0NFQT4oBgHgl3EQfzzas/content/tmp_files/load_file.txt b/0NFQT4oBgHgl3EQfzzas/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb0621e11232e32ef16a0a18961270c6f49ba70d --- /dev/null +++ b/0NFQT4oBgHgl3EQfzzas/content/tmp_files/load_file.txt @@ -0,0 +1,949 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf,len=948 +page_content='Incentive Compatibility in the Auto-bidding World Yeganeh Alimohammadi†, Aranyak Mehta∗ and Andres Perlroth∗ February 1, 2023 Abstract Auto-bidding has recently become a popular feature in ad auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This feature enables advertisers to simply provide high-level constraints and goals to an automated agent, which optimizes their auction bids on their behalf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' These auto-bidding intermediaries interact in a decentralized manner in the underlying auctions, leading to new interesting practical and theoretical questions on auction design, for example, in understanding the bidding equilibrium properties between auto-bidder intermediaries for different auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In this paper, we examine the effect of different auctions on the incentives of advertisers to report their constraints to the auto-bidder intermediaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' More precisely, we study whether canonical auctions such as first price auction (FPA) and second price auction (SPA) are auto-bidding incentive compatible (AIC): whether an advertiser can gain by misreporting their constraints to the autobidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We consider value-maximizing advertisers in two important settings: when they have a budget constraint and when they have a target cost-per-acquisition constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The main result of our work is that for both settings, FPA and SPA are not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This contrasts with FPA being AIC when auto-bidders are constrained to bid using a (sub-optimal) uniform bidding policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We further extend our main result and show that any (possibly randomized) auction that is truthful (in the classic profit-maximizing sense), scalar invariant and symmetric is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Finally, to complement our findings, we provide sufficient market conditions for FPA and SPA to become AIC for two advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' These conditions require advertisers’ valuations to be well-aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This suggests that when the competition is intense for all queries, advertisers have less incentive to misreport their constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' From a methodological standpoint, we develop a novel continuous model of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This model provides tractability to study equilibrium with auto-bidders, which contrasts with the standard discrete query model, which is known to be hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Through the analysis of this model, we uncover a surprising result: in auto-bidding with two advertisers, FPA and SPA are auction equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' †Stanford University, yeganeh@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='edu ∗Google, {aranyak,perlroth}@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='13414v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='TH] 31 Jan 2023 1 Introduction Auto-bidding has become a popular tool in modern online ad auctions, allowing advertisers to set up automated bidding strategies to optimize their goals subject to a set of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By using algorithms to adjust the bid for each query, auto-bidding offers a more efficient and effective alternative to the traditional fine-grained bidding approach, which requires manual monitoring and adjustment of the bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' There are three main components in auto-bidding paradigm: 1) the advertisers who provide high-level constraints to the auto-bidders, 2) the auto-bidder agents who bid – in a decentralized manner – on behalf of each advertiser to maximize the advertiser’s value subject to their constraints, and 3) the query-level auctions where queries are sold (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Figure 1: The Auto-bidding Process: Advertisers submit constraints and receive query allocations with specified costs as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Inside the auto-bidding feature, each advertiser has an agent that optimizes bidding profile within each advertiser’s constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Current research has made important progress in studying the interactions of the second and third components in the auto-bidding paradigm, particularly in understanding equilibrium properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', welfare and revenue) between the auto-bidders intermediaries for different auction rules (Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Mehta, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Liaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' There is also work on mechanism design for this setting in more generality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', between the advertisers and the auctioneer directly abstracting out the second component (Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2021c, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Golrezaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Our work, instead, examines the relation between value-maximizing advertisers, who maximize the value they obtain subject to a payment constraint, and the other two components of the auto- bidding paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' More precisely, we study the impact of different auction rules on the incentives of advertisers to report their constraints to the auto-bidder intermediaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We specifically ask whether canonical auctions such as first price auction (FPA), second price auction (SPA) and general truthful auctions are auto-bidding incentive compatible (AIC) - in other words, can advertisers gain by misreporting their constraints to the auto-bidder?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We consider value-maximizing advertisers in two important settings: when they have a budget constraint and when they have a target cost-per-acquisition (tCPA) constraint1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The main result of 1The former is an upper bound on the total spend, and the latter is an upper bound on the average spend per acquisition (sale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Our results clearly hold for more general autobidding features, such as target return on ad-spend (tRoAS) where the constraint is an upper bound on the average spend per value generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 1 Auto- constraints bids Advertiser bidder Agent Auto- constraints bids Auction Advertiser bidder per query Agent Auto- constraints bids Advertiser bidder Agentour work is that for both settings, FPA and SPA are not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This contrasts with FPA being AIC when auto-bidders are constrained to bid using a (sub-optimal) uniform bidding policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We further generalize this surprising result and show that any (possibly randomized) truthful auction having a scale invariance and symmetry property is also not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We complement our result by providing sufficient market conditions for FPA and SPA to become AIC for two advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' These conditions require advertisers’ valuations to be well-aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This suggests that when the competition is intense for all queries, advertisers have less incentive to misreport their constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In our model, each advertiser strategically reports a constraint (either a tCPA or a budget) to an auto-bidder agent which bids optimally on their behalf in each of the queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Key in our model, we consider a two stage game where first advertisers submit constraints to the auto-bidders and, in the subgame, auto-bidders reach a bidding equilibrium across all query-auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Thus, when an advertiser deviates and reports a different constraint to its auto-bidder, the whole bidding subgame equilibrium can change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 In this context, an auction rule is called auto-bidding incentive compatible (AIC) if, for all equilibria, it is optimal for the advertiser to report their constraint to the auto-bidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 Main Results We begin our results by presenting a stylized example in Section 2 that demonstrates how auto- bidding with SPA is not AIC (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Our example consists of a simple instance with three queries and two advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This example highlights a scenario where an advertiser can benefit from lowering their reported budget or tCPA-constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We then introduce a continuous query model that departs from the standard auto-bidding model by considering each query to be of infinitesimal size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This model provides tractability in solving equilibrium for general auction rules like FPA which is key to study the auto-bidding incentive compatibility properties of such auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Further, this continuous-query model succinctly captures real-world scenarios where the value of a single query is negligible compared to the pool of all queries that are sold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Under the continuous-query model, we study the case where queries are sold using FPA and show that in the auto-bidding paradigm, FPA is not AIC (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We first characterize the optimal bidding strategy for each auto-bidder agent which, surprisingly, has a tractable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 We then leverage this tractable form to pin down an equilibrium for the case of two auto-bidders when both auto-bidders either face a budget or tCPA constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In this equilibrium, queries are divided between the two advertisers based on the ratio of their values for each advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Specifically, advertiser 1 receives queries for which the ratio of its value to the other advertiser’s value is higher than a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' From this point, determining the equilibrium reduces to finding a threshold that make advertisers’ constraints tight (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4 for more detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We then show that for instances where the threshold lacks monotonicity with the auto-bidders constraints, advertisers have an incentive to misreport the constraint to the auto-bidder (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Conversely, when the thresholds are monotone advertisers report constraints truthfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We show conditions on the advertisers’ valuations, for the two-advertisers setting, to guarantee this monotonicity (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This condition requires a strong positive correlation of the advertisers’ valuations across the queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' As a 2This two stage model captures the idea that auto-bidding systems rapidly react to any change in the auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Hence, if there is any change in the bidding landscape, auto-bidders quickly converge to a new equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 3Notice that in the discrete-query model, there is no simple characterization for the auto-bidder best response in a FPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2 practical insight, our results suggest that for settings where the competition on all queries is intense, advertisers’ incentives to misreport is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We then explore the case where, in FPA, auto-bidders are constrained to bid using a uniform bidding strategy: the bid on each query is a constant times the advertiser’s value for the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4 Uniform bidding is only an optimal strategy when auctions are truthful (Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Even though for FPA these strategies are suboptimal, they have gained recent attention in the literature due to their tractability Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Gaitonde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We show that in such a scenario, FPA with uniform bidding turns out to be AIC (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' However, we note that while this proves AIC in our model, the suboptimality of uniform bidding for FPA can give rise to incentives to deviate in other ways outside our model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', by splitting the advertising campaigns into multiple campaigns with different constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' These considerations are important when implementing this rule in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The second part of the paper pivots to the case where auctions are truthful, that is, auctions in which it is optimal for a profit-maximizing agent to bid their value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We first study the canonical SPA and show that, in our continuous-query model, SPA and FPA are auction equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' That is, the allocation and payments among the set of reasonable equilibria (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5 As a Corollary, the results we obtain for FPA apply to SPA as well: SPA is not AIC and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' and we derive sufficient conditions on advertisers’ valuations so that SPA is AIC for two advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We then consider a general class of randomized truthful auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We show that if the allocation rule satisfies these natural conditions:6 (i) scaled invariant (if all bids are multiplied by the same factor then the allocation doesn’t change), and (ii) is symmetric (bidders are treated equally);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' then the auction rule is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The main results of the paper are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Per Query Auction AIC Second-Price Auction Not AIC Truthful Auctions Not AIC First-Price Auction Not AIC First-Price Auction with Uniform Bidding AIC7 Table 1: Main Results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 Related Work The study of auto-bidding in ad auctions has gained significant attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' One of the first papers to study this topic is Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2019), which presents a mathematical formulation for the auto-bidders problem given a fixed constraints reported by advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' They show that uniform bidding is an optimal strategy if and only if auctions are truthful (in the profit-maximizing sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' They further started an important line of work to measure, using a Price of Anarchy (PoA) approach, the welfare implications when auto-bidders are bidding in equilibrium for different auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 4Uniform bidding strategy is also known in the literature as pacing bidding Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Gaitonde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 5We show the auction equivalence among uniform bidding equilibria for SPA and threshold-type equilibrium for FPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 6These conditions have been widely studied in the literature due to their practical use (Mehta, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Liaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Allouah and Besbes, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 7As previously discussed, implementing FPA with the suboptimal uniform bidding policy can create other distortion on advertisers’ incentives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', splitting their campaign into multiple campaigns with different constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 3 Current results state that for SPA the PoA is 2 Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2019) and also for FPA Liaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022)8, and, interestingly, it can be improved if the auction uses a randomized allocation rule Mehta (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Liaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In a similar venue, Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2021b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2021b) studies models where the auction has access to extra information and show how reserves and boosts can be used to improve welfare and efficiency guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' A second line of work, studies how to design revenue-maximizing auctions when bidders are value-maximizing agents and may have private information about their value or their constraints (Golrezaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2021c,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In all these settings, the mechanism designer is not constrained to the presence of the auto-bidding intermediaries (Component 2 in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Our study has added structure by having advertisers submit their constraints first, followed by a decentralized subgame to achieve a bidding equilibrium before allocating and determining payments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Thus, a priori their mechanism setting can achieve broader outcomes than in our auto-bidding constraint paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Interestingly, for the one query case the authors show that FPA with a uniform bidding policy is optimal Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Our results complement theirs and show that such mechanism is implementable in auto-bidding constraint paradigm and is AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Closer to our auto-bidding paradigm, a recent line of work has started to study the incentive of advertisers when bidding via an auto-bidder intermediary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Mehta and Perlroth (2023) show that a profit-maximizing agent may benefit by reporting a target-based bidding strategy to the auto-bidder when the agent has concern that the auctioneer may change (ex-post) the auction rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Also, in an empirical work, Li and Tang (2022) develop a new methodology to numerically approximate auto-bidding equilibrium and show numerical examples where advertisers may benefit my reporting their constraints on SPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Our work complements their findings by showing under a theoretical framework that SPA is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Our work also connects with the literature about auction with budgeted constraint bidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In particular, our results are closely related to Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022a) who study FPA with uniform bidding (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' pacing bidding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' They introduce the concept of the first-price auction pacing equilibrium (FPPE) for budget-constrained advertisers, which is the same as the equilibrium in our auto-bidding subgame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' They show that in FPPE the revenue and welfare are monotone increasing as a function of the advertisers’ budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In our work, we show that in FPPE, advertisers’ values are monotone as a function of their reported budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In addition, they differentiate between first and second-price by showing that FPPE is computable, unlike SPPE, where maximizing revenue has previously been known to be NP-hard Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022b), and that the general problem of approximating the SPPE is PPAD-complete Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In contrast, we show in the continuous model both SPA and FPA are tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Interestingly, this dichotomy between FPA and SPA (both with uniform bidding) is reflected in our work as well – the former is AIC, while the latter is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Uniform bidding has been explored in a separate body of research on repeated auctions, without the presence of auto-bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Balseiro and Gur (2019) investigate strategies to minimize regret in simultaneous first-price auctions with learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Gaitonde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022) take this concept further by extending the approach to a wider range of auction settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Furthermore, Golrezaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2021a) examines how to effectively price and bid for advertising campaigns when advertisers have both ROI and budget constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 8The authors show that for a general class of deterministic auctions PoA ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 4 2 Warm Up: Second Price Auction is not AIC!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To understand the implications of the auto-bidding model, we start with an example of auto-bidding with the second-price auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Through this example, we will demonstrate the process of determining the equilibrium in an auto-bidding scenario and emphasize a case where the advertiser prefers to misreport their budget leading to the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' For the budget setting (when all advertisers are budgeted-constrained) and for the tCPA-setting (when all advertisers are tCPA-constrained), we have that SPA is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' That is, there are some instances where an advertiser benefits by misreporting its constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Consider two budget-constrained advertisers and three queries Q = {q1, q2, q3}, where the expected value of winning query q for advertiser a is denoted by va(q), and it is publicly known (as in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' At first, each advertiser reports their budget to the auto-bidder B1 = 2, and B2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then the auto-bidder agents, one for each advertiser, submit the bidding profiles (to maximize their advertisers’ value subject to the budget constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The next step is a second-price auction per query, where the queries are allocated to the highest bidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' q1 q2 q3 Advertiser 1 4 3 2 Advertiser 2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 10 Table 2: SPA with two budget constraint advertisers is not AIC: The value of each query for each advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Finding the equilibrium bidding strategies for the auto-bidder agents is challenging, as the auto-bidder agents have to find the best-response bids with respect to the other auto-bidder agents, and each auto-bidder agent’s bidding profile changes the cost of queries for the rest of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To calculate such an equilibrium between auto-bidder agents, we use the result of Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2019) to find best-response strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Their result states that the best response strategy in any truthful auto-bidding auction is uniform bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='9 In other words, each agent optimizes over one variable, a bidding multiplier µa, and then bids on query q with respect to the scaled value µava(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We show that with the given budgets B1 = 2 and B2 = 4, an equilibrium exists such that advertiser 1 only wins q1, and µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5 and µ2 = 1 result in such an equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To this end, we need to check: 1) Allocation: with bidding strategies b1 = (µ1v1(q1), µ1v1(q2), µ1v1(q3)) and b2 = (µ2v2(q1), µ2v2(q2), µ2v2(q3)), advertiser 1 wins q1 and advertiser 2 wins q2 and q3, 2) Budget constraints are satisfied, and 3) Bidding profiles are the best response: The auto-bidder agents do not have the incentive to increase their multiplier to get more queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' These three conditions are checked as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Allocation inequalities: For each query, the advertiser with the highest bid wins it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' v1(q1) v2(q1) ≥ µ2 µ1 = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5 ≥ v1(q2) v2(q2) ≥ v1(q3) v2(q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Budget constraints: Since the auction is second-price the cost of query q for advertiser 1 is µ2v2(q) and for advertiser 2 is µ1v1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, we must have the following inequalities to hold so 9They show uniform bidding is almost optimal, but in Appendix A we show that in this example it is exactly optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 5 that the budget constraints are satisfied: 2 = B1 ≥ µ2v2(q1) = 1 (Advertiser 1), 4 = B2 ≥ µ1(v1(3) + v1(q2)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5 (Advertiser 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Best response: Does the advertiser’s agent have incentive to raise their multiplier to get more queries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If not, they shouldn’t afford the next cheapest query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2 < µ2(v2(q1) + v2(q2)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 (Advertiser 1), 4 < µ1(v1(q3) + v1(q2) + v1(q1)) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5 (Advertiser 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Since all the three conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Thus, this profile is an equilibrium for th auto-bidders bidding game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In this equilibrium, advertiser 1 wins q1 and advertiser 2 wins q2 and q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, consider the scenario that advertiser 1 wants to strategically report their budget B1 to the auto-bidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Suppose the first advertiser decreases their budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Intuitively, the budget constraint for the auto-bidder agent should be harder to satisfy, and hence the advertiser should not win more queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' But, contrary to this intuition, when advertiser 1 reports a lower budget B′ 1 = 1, we show that, given the unique auto-bidding equilibrium, advertiser 1 wins q1 and q2 (more queries than the case where advertiser 1 reports B1 = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Similar to above, we can check that µ′ 1 = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3, and µ′ 2 = 1 results in an equilibrium (we prove the uniqueness in Appendix A): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Allocation: advertiser 1 wins q1 and q2 since it has a higher bid on them, v1(q1) v2(q1) ≥ v1(q2) v2(q2) ≥ µ′ 2 µ′ 1 = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 ≥ v1(q3) v2(q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Budget constraints: 4 ≥ v1(q3), and 1 = (1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3)(v2(q1) + v2(q2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Best response: 4 < 1(v1(q3) + v1(q2)), and 1 < (1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3)(v2(q1) + v2(q2) + v2(q3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This surprising example leads to the first main result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In Appendix A, we will generalize the above example to the case of tCPA-constrained advertisers with the same set of queries as in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Before studying other canonical auctions, in the next section we develop a tractable model of continuous query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Under this model it turns out that the characterization of the auto-bidders bidding equilibria when the auction is not SPA is tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This tractability is key for studying auto-bidding incentive compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 6 3 Model The baseline model consists of a set of A advertisers competing for q ∈ Q single-slot queries owned by an auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We consider a continuous-query model where Q = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let xa(q) be the probability of winning query q for advertiser a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then the expected value and payment of winning query q at price pa(q) are xa(q)va(q)dq and pa(q)dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='10, 11 Intuitively, this continuous-query model is a first-order approximation for instances where the size of each query relative to the whole set is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The auctioneer sells each query q using a query-level auction which induces the allocation and payments (xa(q), pa(q))a∈A as a function of the bids (ba)a∈A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In this paper, we focus on the First Price Auction (FPA), Second Price Auction (SPA) and more generally any Truthful Auction (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Auto-bidder agent: Advertisers do not participate directly in the auctions, rather they report high-level goal constraints to an auto-bidder agent who bids on their behalf in each of the queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Thus, Advertiser a reports a budget constraint Ba or a target cost-per-acquisition constraint (tCPA) Ta to the auto-bidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then, the auto-bidder taking as fixed other advertiser’s bid, submits bids ba(q) to induce xa(q), pa(q) that solves max � 1 0 xa(q)va(q)dq (1) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' � 1 0 pa(q)dq ≤ Ba + Ta � 1 0 xa(q)va(q)dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2) The optimal bidding policy does not have a simple characterization for a general auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' However, when the auction is truthful (like SPA) the optimal bid take a simple form in the continuous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 (Uniform Bidding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If the per-query auction is truthful, then uniform bidding is the optimal policy for the autobidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Thus, ba(q) = µ · va(q) for some µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We formally prove this in Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Advertisers Following the current paradigm in autobidding, we consider that advertisers are value-maximizers and can be two of types: a budget-advertiser or tCPA-advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Payoffs for these advertisers are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' For a budget-advertiser with budget Ba, the payoff is ua = �� 1 0 xa(q)va(q)dq if � 1 0 pa(q)dq ≤ Ba −∞ if not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 10All functions va, xa, pa are integrable with respect to the Lebesgue measure dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 11The set Q = [0, 1] is chosen to simplify the exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Our results apply to a general metric measurable space (Q, A, λ) with atomless measure λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 7 For a tCPA-advertiser with target Ta, the payoff is ua = �� 1 0 xa(q)va(q)dq if � 1 0 pa(q)dq ≤ Ta · � 1 0 xa(q)va(q)dq −∞ if not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Game, Equilibrium and Auto-bidding Incentive Compatibility (AIC) The timing of the game is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' First, each advertiser depending on their type submits a budget or target constraint to an auto-bidder agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then, each auto-bidder solves Problem 1 for the respective advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Finally, the per-query auctions run and allocations and payments accrue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We consider a complete information setting and use subgame perfect equilibrium (SPE) as solution concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let Va(B′ a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Ba) the expected payoff in the subgame for a budget-advertiser with budget Ba that reports B′ a to the auto-bidder (likewise we define Va(T ′ a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Ta) for the tCPA-advertiser).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 (Auto-bidding Incentive Compatibility (AIC)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' An auction rule is Auto-bidding Incentive Compatible (AIC) if for every SPE we have that Va(Ba;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Ba) ≥ Va(B′ a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Ba) and Va(Ta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Ta) ≥ Va(T ′ a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Ta) for every Ba, B′ a, Ta, T ′ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Similar to classic notion of incentive compatibility, an auction rule satisfying AIC makes the advertisers’ decision simpler: they simply need to report their target to the auto-bidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' However, notice that the auto-bidder plays a subgame after advertiser’s reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Thus, when Advertiser a deviates and submit a different constraint, the subgame outcome may starkly change not only on the bids of Advertiser a but also other advertisers bid may change as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 4 First Price Auctions In this section, we demonstrate that the first price auction is not auto-bidding incentive compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Suppose that there are at least two budget-advertisers or two tCPA-advertisers, then FPA is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Later in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2, we show a complementary result by providing sufficient conditions on advertisers’ value functions to make FPA be AIC for the case of two advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We show that this sufficient condition holds in many natural settings, suggesting that in practice FPA tends to be AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3, we turn our attention to FPA where autobidders are restricted to use uniform bidding across the queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In this case, we extend to our continuous-query model the result of Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022a) and show the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' FPA restricted to uniform bidding is AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 We divide the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 in three main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Step 1 characterizes the best response bidding profile for an autobidder in the subgame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' As part of our analysis, we derive a close connection between first and second price auctions in the continuous-query model that simplifies the task of finding the optimal bidding for each query to finding a single multiplying variable for each advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In Step 2, we leverage the tractability of our continuous-query model and pin-down the subgame bidding equilibrium when there are either two budget-advertisers or two tCPA-advertisers in the 8 game (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We derive an equation that characterizes the ratio of the multipliers of each advertiser as a function of the constraints submitted by the advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This ratio defines the set of queries that each advertiser wins, and as we will see the value accrued by each advertiser is monotone in this ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, to find a non-AIC example, one has to find scenarios where the equilibrium ratio is not a monotone function of the input constraints which leads to the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To conclude, we show in Step 3 an instance where the implicit solution for the ratio is nonmono- tonic, demonstrating that auto-bidding in first-price auctions is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' As part of our proof, we interestingly show that AIC is harder to satisfy when advertisers face budget constraints rather than tCPA constraints (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Step 1: Optimal Best Response The following claim shows that, contrary to the discrete-query model, the best response for the autobidder in a first price auction can be characterized as function of a single multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Taking other auto-bidders as fixed, there exists a multiplier µa ≥ 0 such that the following bidding strategy is optimal: ba(q) = � maxa′̸=a(ba′(q)) µava(q) ≥ maxa′̸=a(ba′(q)) 0 µava(q) ̸= maxa′(ba′(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The result holds whether the advertiser is budget-constrained or tCPA-constrained12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We show that in a first-price auction, the optimal bidding strategy is to bid on queries with a value-to-price ratio above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To prove this, we assume that the bidding profile of all advertisers is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Since the auction is first-price, advertiser a can win each query q by fixing small enough ϵ > 0 and paying maxa′̸=a(ba′(q)) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, let pa(q) = maxa′̸=a(ba′(q)), be the price of query q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Since we have assumed that the value functions of all advertisers are integrable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', there are no measure zero sets of queries with a high value), in the optimal strategy pa is also integrable since it is suboptimal for any advertiser to bid positive (and hence have a positive cost) on a measure zero set of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' First, consider a budget-constrained advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The main idea is that since the prices are integrable, the advertiser’s problem is similar to a continuous knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In a continuous knapsack problem, it is well known that the optimal strategy is to choose queries with the highest value-to-cost ratio Goodrich and Tamassia (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, there must exist a threshold, denoted as µ, such that the optimal strategy is to bid on queries with a value-to-price ratio of at least µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So if we let µa = 1 µ, then advertiser a bids on any query with µava(q) ≥ pa(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We prove it formally by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Assume to the contrary, that there exist non-zero measure sets X, Y ⊂ Q such that for all x ∈ X and y ∈ Y , the fractional value of x is less than the fractional value of y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', va(x) pa(qx) < va(y) pa(y), and in the optimal solution advertiser a gets all the queries in X and no query in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' However, we show that by swapping queries in X with queries in Y with the same price, the advertiser can still satisfy its budget constraint while increasing its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To prove this, fix 0 < α < min( � X pa(q)dq, � Y pa(q)dq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Since the Lebesgue measure is atomless, there exists subsets X′ ⊆ X and Y ′ ⊆ Y such that α = � X′ pa(q)dq = � Y ′ pa(q)dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Since the value 12In FPA ties are broken in a way that is consistent with the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This is similar to the pacing equilibrium notion where the tie-breaking rule is endogenous to the equilibrium Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 9 per cost of queries in Y is higher than queries in X, by swapping queries of X′ with Y ′, the value of the new sets increases, while the cost does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, the initial solution cannot be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' A similar argument holds for tCPA-constrained advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Swapping queries in X′ with Y ′ does not change the cost and increases the upper bound of the tCPA constraint, resulting in a feasible solution with a higher value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, the optimal bidding strategy for tCPA constraint is also ba(q) as defined in the statement of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Step 2: Equilibrium Characterization The previous step showed that the optimal bidding strategy is to bid on queries with a value-to-price ratio above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Thus, we need to track one variable per auto-bidder to find the subgame equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In what follows, we focus on the case of finding the variables when there are only two advertisers in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This characterization of equilibrium gives an implicit equation for deriving the equilibrium bidding strategy, which makes the problem tractable in our continuous-query model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' From Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 we observe that the ratio of bidding multipliers is key to determine the set of queries that each advertiser wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To map the space of queries to the bidding space, we introduce the function h(q) = v1(q) v2(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Hence, for high values of h, the probability that advertiser 1 wins the query increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Also, notice that without loss of generality, we can reorder the queries on [0, 1] so that h is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In what follows, we further assume that h is increasing on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This implies that h is invertible and also differentiable almost everywhere on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' With these assumptions in place, we can now state the following lemma to connect the subgame equilibrium to the ratio of advertisers’ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' [Subgame Equilibrium in FPA] Given two budget-constrained auto-bidders with budget B1 and B2, let µ1 and µ2 be as defined in Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 for auto-bidding with FPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Also assume that h(q) = v1(q) v2(q) as defined above is strictly monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then µ1 = B2 E[z1(z≥r)] and µ2 = µ1r, where r is the solution of the following implicit function, rE[1[z ≥ r)] E[z1(z ≤ r)] = B1 B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (3) Here, E[·] is defined as E[P(z)] = � ∞ 0 P(z)f(z)dz, where f(z) = v2(h−1(z)) h′(h−1(z)) wherever h′ is defined, and it is zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Also, for two tCPA auto-bidders with targets T1 and T2, we have µ1 = T1E[1(z≤r)] E[1(z≥r)] and µ2 = µ1r, where r is the answer of the following implicit function, rE[1(z ≥ r)] E[z1(z ≥ r)] E[1(z ≤ r)] E[z1(z ≤ r)] = T1 T2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (4) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The function f intuitively represents the expected value of the queries that advertiser 2 can win as well as the density of the queries that advertiser 1 can win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Also, the variable r shows the cut-off on how the queries are divided between the two advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In the proof, we will see that the advertisers’ value at equilibrium is computed with respect to f: Advertiser 1’s overall value is � ∞ r zf(z)dz and advertiser 2’s overall value is � r 0 f(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 13Notice that for the discrete-query model finding equilibrium is PPAD hard Filos-Ratsikas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2021) 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' First, consider budget constraint auto-bidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Given Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3, in equilibrium price of query q is min(µ1v1(q), µ2v2(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, the budget constraints become: B1 = � 1 0 µ2v2(q)1(µ2v2(q) ≤ µ1v1(q))dq, B2 = � 1 0 µ1v1(q)1(µ2v2(q) ≥ µ1v1(q))dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' With a change of variable from q to z = h(q) and letting r = µ2 µ1 , we have: B1 = � ∞ r µ2v2(h−1(z))dh−1(z) dz dz B2 = � r 0 µ1v1(h−1(z))dh−1(z) dz dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Observe that v1(h−1(z)) = zv2(h−1(z)), then if we let f(z) = v2(h−1)(h−1)′ = v2(h−1(z)) h′(h−1(z)), the constraints become B1 = µ2 � ∞ r f(z)dz, (5) B2 = µ1 � r 0 zf(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (6) We obtain Equation (3) by diving both sides of Equation (5) by the respective both sides of Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, consider two tCPA constrained auto-bidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Similar to the budget-constrained auto-bidders, we can write T1 � 1 0 v1(q)1(µ2v2(q) ≤ µ1v1(q))dq = � 1 0 µ2v2(q)1(µ2v2(q) ≤ µ1v1(q))dq T2 � 1 0 v2(q)1(µ2v2(q) ≥ µ1v1(q))dq = � 1 0 µ1v1(q)1(µ2v2(q) ≥ µ1v1(q))dq The same way of changing variables leads to the following: T1 T2 � ∞ r xf(x)dx � r 0 f(x) = r � ∞ r f(x)dx � r 0 xf(x)dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This finishes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The previous theorem immediately implies that any example of valuation functions that is non-AIC for budget-advertisers, it will be non-AIC for tCPA-advertisers as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If auto-bidding with the first-price and two budget-advertisers is not AIC, then auto-bidding with the same set of queries and two tCPA-advertisers is also not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Recall that advertiser 1 wins all queries with h(q) ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, the value accrued by advertiser 1 is decreasing in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, if an instance of auto-bidding with tCPA-constrained advertisers is not AIC for advertiser 1, then the corresponding function r � ∞ r f(x)dx � r 0 xf(x)dx � r 0 f(x) � ∞ r xf(x)dx (same as (4)) must be increasing for some r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' On the other hand, recall that r � ∞ r f(x)dx � r 0 xf(x)dx is the ratio for budget-constrained bidders equilibrium as in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The additional multiplier in the equilbirum equation of tCPA constraint advertiser in (4) is � r 0 f(x)dx � ∞ r xf(x) which is increasing in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, if the auto-bidding for budget-constrained bidders is not AIC and hence the corresponding ratio is increasing for some r′, it should be increasing for the tCPA-constrained advertisers as well, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Step 3: Designing a non AIC instance The characterization of equilibrium from Step 2 leads us to construct an instance where advertisers have the incentive to misreport their constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The idea behind the proof is that the value accrued by the advertiser 1 is decreasing in r ( as found in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then to find a counter-example, it will be enough to find an instance of valuation functions such that the equilibrium equation (3) is non-monotone in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We construct an instance with two budget-constrained advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='6 the same instance would work for tCPA-constrained advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To prove the theorem, we will find valuation functions v1 and v2 and budgets B1 and B2 such that the value accrued by advertiser 1 decreases when their budget increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Define g(r) = � r 0 xf(x)dx r � ∞ r f(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4, one can find the equilibrium by solving the equation g(r) = B2 B1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Recall that advertiser 1 wins all queries with v1(q) v2(q) ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, the total value of queries accrued by advertiser 1 is decreasing in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Hence, to construct a non- AIC example, it is enough to find a function f such that g is non-monotone in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' A possible such non-monotone function g is g(r) = (r − 1)3 + 3 cr − 1, (7) where c is chosen such that minr≥0 g(r) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', c = min (r−1)3+3 r ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='95105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To see why g is non-monotone, observe that g(r) is decreasing for r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='8, because g′(r) = 2r3−3r2−2 cr2 is negative for r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='8, and then increasing for r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We claim the function f defined as in, f(r) = 3c(r − 1)2 e � r 0 c (r−1)3+3 dx ((r − 1)3 + 3)2 , (8) would result in the function g in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To see why this claim is enough to finish the proof, note that there are many ways to choose the value functions of advertisers to derive tf as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' One possible way is to define v1, v2 : [0, 1] → R as v2(q) = f(tan(q))/(tan(q)2 + 1) and v1(q) = tan(q)v2(q) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 12 (a) Valuation function of two advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (b) Finding the equilibrium using (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Figure 2: An example of two advertisers such that FPA is not AIC (proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' When B1 B2 = 1200, there are three values for r (see the right panel) that lead to equilibrium, and one (orange) leads to non-AIC equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So it remains to prove that choosing f as in (8) would result in g as defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To derive f from g, first we simplify g using integration by part, g(r) = � r 0 xf(x)dx r � ∞ r f(x)dx = r � r 0 f(x)dx − � r 0 � x 0 f(y)dydx r � ∞ r f(x)dx = r � ∞ 0 f(x)dx − � r 0 � x 0 f(y)dydx r � ∞ r f(x)dx − 1, Assuming that � ∞ 0 f(x) is finite, the above equations lead to the following rg(r) + r = � r 0 � ∞ x f(y)dydx � ∞ r f(x)dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (9) Therefore, by integrating the inverse of both sides, log( � r 0 � ∞ x f(y)dydx) = C + � r 0 1 xg(x) + xdx, and by raising to the exponent � r 0 � ∞ x f(y)dydx = Ke � r 0 1 xg(x)+x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' for some constants C and K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then by differentiating both sides with respect to x, � ∞ r f(x)dx = K rg(r) + re � r 0 1 xg(x)+x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Note that for any choice of K ≥ 0, dividing the last two equations will result in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, without loss of generality, we can assume K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By differentiating again, we can derive f as a function of g: f(r) = (g′(r)r + g(r)) (rg(r) + r)2 e � r 0 1 xg(x)+x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='0 Vi(q) 25 V2(q) 20 15 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='6 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='B LD q3500 rE[1[z ≥ r]] E[21(z ≤ r)] 31D0 2500 22000 1500 14D0 50D 0 FN 4 6 1 rWe need g′(r)r + g(r) ≥ 0 to ensure that f(r) ≥ 0 for all r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This holds for g as in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Finally, by substituting g as in (7), we will derive f as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Note that the above proof shows that for values of r such that there exists a equilibrium which is not AIC, there exists always a second monotone equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This follows from the fact that the function g(r) tends to infinity as r → ∞, so, g must be increasing for some large enough r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Before moving on to finding conditions for incentive compatibility, we also note that the above’s characterization implies the existence of equilibrium for auto-bidding with any pairs of advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Given auto-bidding satisfying the conditions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4, the equilibrium for all pairs of budgets or all pairs of tCPA constrained advertisers always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Recall that the equilibirum exists if there exists an r such that B2 B1 = � r 0 xf(x)dx r � ∞ r f(x)dx has a solution for any value of B2 B1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Note that the right-hand side ( � r 0 xf(x)dx r � ∞ r f(x)dx) is positive for any r > 0, and it continuously grows to infinity as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, to make sure that every value of B2/B1 is covered, we need to check whether at r = 0 the ratio becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By L’Hopital rule, limz→0 zf(z) � ∞ z f(x)dx−zf(z) = 0, which is as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' For tCPA constrained advertiser, the second ratio r � r 0 f(x)dx � ∞ r xf(x)dx always converges to 0, so the equilibrium in this case always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 Sufficient Conditions for Incentive Compatibility In this section we show that the lack of ACI happens for cases where advertisers’ valuations have unusual properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' More precisely, the main result of the section is to characterize sufficient conditions on the advertiser’s valuations so that FPA is AIC when there are two advertisers in the auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' For this goal, we recall the function f(z) = v2(h−1(z)) h′(h−1(z)) where h(q) = v1(q) v2(q) defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' As shown in Lemma4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4, function f behaves as a value of the queries advertiser 2 gets and the density of queries that advertiser 1 gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Consider that there are two advertisers and they can either both be budget-advertisers or tCPA-advertisrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Also, suppose that auto-bidder with FPA uses the optimal bidding strategy in Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then a sufficient condition for FPA to be AIC is that f has a monotone hazard rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', f(r) � ∞ r f(x)dx is non-decreasing in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Following the proof Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1, if g(r) = � r 0 � ∞ x f(y)dydx r � ∞ r f(x)dx is non-decreasing in r then the equilibrium is AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The equivalent sufficient conditions obtained by imposing the inequality g′(r) ≥ 0 is that for all r ≥ 0, r � � ∞ r f(x)dx �2 ≥ � � r 0 � ∞ x f(y)dydx �� � ∞ r f(x)dx − rf(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (10) 14 If � ∞ r f(x)dx ≤ rf(r) then above’s inequality obviously holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, we can assume that for some r > 0, � ∞ r f(x)dx > rf(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Since f(z) � ∞ z f(x)dx is non-decreasing in z, we must have that for all r′ ≤ r, f(r′) � ∞ r′ f(x)dx ≤ f(r) � ∞ r f(x)dx ≤ 1 r ≤ 1 r′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' On the other hand by taking the derivative of f(z) � ∞ z f(x)dx, we must have that f′(z) � ∞ z f(x)dx + f(z)2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By considering two cases on the sign of f′, for z ≤ r we must have, f(z) � zf′(z) + f(z)) ≥ 0, and hence (zf(z))′ ≥ 0 for all z ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, zf(z) is non-decreasing for z ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' On the other hand, � r 0 � ∞ x f(y)dydx = � r 0 � r x f(y)dydx + � r 0 � ∞ r f(y)dydx = r � r 0 f(x)dx − � r 0 � x 0 f(y)dydx + r � ∞ r f(x)dx = � r 0 xf(x)dx + r � ∞ r f(x)dx, where the second equation is by integration by part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then by applying monotonicity of z(f)z for z ≤ r we have � r 0 � ∞ x f(y)dydx ≤ r2f(r) + r � ∞ r f(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, to prove (10) it is enough to show that �� ∞ r f(x)dx �2 ≥ � rf(r) + � ∞ r f(x)dx � �� ∞ r f(x)dx − rf(r) � , which holds, since the right-hand side is equal to � � ∞ r f(x)dx �2 − (rf(r))2 strictly less than the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' While the condition on f has intuitive properties when seen as a density, it has the unappealing properties to be too abstract in terms of the conditions on the advertisers’ valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The following result, provides sufficient conditions on value functions v1 and v2 that makes f be monotone hazard rate, and hence, FPA to be AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Consider two advertisers that are either budget-advertisers or tCPA-advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Assume that h(q) = v1(q) v2(q) is increasing concave function and that v2 is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then, the equilibrium in FPA auto-bidding with bidding strategy as in Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 is AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Note that when f is non-decreasing, it also has a monotone hazard rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, when h is concave, 1 h′ is a non-decreasing function, and since v2 is also non-decreasing, then f is also non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 FPA with uniform bidding The previous section shows that when auto-bidders have full flexibility on the bidding strategy, FPA is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' However, non-uniform bid is not simple to implement and auto-bidders may be constrained to use simpler uniform bidding policies (aka pacing bidding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In this context, the main result of the section is Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 that shows that when restricted to uniform bidding policies FPA is AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Note that here, we are assuming a simple model where advertisers do not split campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, FPA with uniform bidding is AIC but it could bring up other incentives for advertisers when it is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 15 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='11 (Uniform bidding equilibrium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' A uniform bidding equilibrium for the auto-bidders subgame corresponds to bid multipliers µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' , µN such that every auto-bidder a chooses the uniform bidding policy µa that maximizes Problem (1) when restricted to uniform bidding policies with the requirement that if advertiser a’s constraints of type 2 are not tight then µa gets its maximum possible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='14 The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 is based on the main results of Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The authors proved that the uniform-bidding equilibrium is unique and in equilibrium the multiplier of each advertiser is the maximum multiplier over all feasible uniform bidding strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Their result is for budget-constrained advertisers, and we extend it to include tCPA constrained advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The proof is deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='12 (Extension of Theorem 1 in Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Given an instance of Auto- bidding with general constraints as in (2), there is a unique uniform bidding equilibrium, and the bid multipliers of all advertisers is maximal among all feasible uniform bidding profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, we are ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Assume that advertiser 1 increases their budget or their target CPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then the original uniform bidding is still feasible for all advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Further, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='12 the equilibrium pacing of all advertisers is maximal among all feasible pacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, the pacing of all advertisers will either increase or remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' But the constraints of all advertisers except 1 are either binding or their multiplier has attained its maximum value by the definition of pacing equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, the set of queries they end up with should be a subset of their original ones since the price of all queries will either increase or remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, it is only advertiser 1 that can win more queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022a) show monotonicity properties of budgets in FPA with uniform bidding equilibrium for the revenue and welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Instead, in our work we focus on monotonicity for each advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 5 Truthful Auctions This section studies auto-bidding incentive compatibility for the case where the per-query auction is a truthful auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' A truthful auction is an auction where the optimal bidding strategy for a profit-maximizing agent is to bid its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' An important example of a truthful auction is Second Price Auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' As we showed in the three-queries example of the introduction, SPA is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In this section, we show that the previous example generalizes, in our continuous-query model, to any (randomized) truthful auctions so long as the auction is scalar invariant and symmetric (see Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 below for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' As part of our proof-technique, we obtain an auction equivalence result which is interesting on its own: in the continuous query-model SPA and FPA have the same outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='15 For the remaining of the section we assume all truthful auction satisfy the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 14When valuations are strictly positive for all queries q ∈ [0, 1], we can easily show that bid multipliers have to be bounded in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' When this is not the case, we set a cap sufficiently high to avoid bid multipliers going to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 15It is well-known that in the discrete-query model, FPA and SPA are not auction equivalent in the presence of auto-bidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 16 Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let (xa(b))a∈A be the allocation rule in a truthful auction given bids b = (ba)a∈A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We assume that the allocation rule satisfies the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The auction always allocates: � a∈A xa(b) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Scalar invariance: For any constant c > 0 and any advertiser a ∈ A, xa(b) = xa(cb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Symmetry: For any pair of advertisers a, a′ ∈ A and bids b, b′, b−{a,a′} = (b)a∈A\\{a,a′} we have that xa(ba = b, ba′ = b′, b−{a,a′}) = xa′(ba = b′, ba′ = b, b−{a,a′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Observe that SPA satisfies Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' From the seminal result of Myerson (1981) we obtain a tractable characterization of truthful auctions which we use in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 (Truthful auctions (Myerson, 1981)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let (xa(b), pa(b))a∈A the allocation and pricing rule for an auction given bids b = (ba)a∈A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The auction rule is truthful if and only if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Allocation rule is non-decreasing on the bid: For each bidder a ∈ A and any b′ a ≥ ba, we have that xa(b′ a, b−a) ≥ xa(ba, b−a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Pricing follows Myerson’s formulae: pa(b) = ba · xa(b) − � ba 0 xa(z, b−a)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' A second appealing property of truthful actions is that the optimal bidding strategy for auto- bidders is simpler: in the discrete-query model uniform bidding strategy is almost optimal and can differ from optimal by at most the value of two queries (Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We revisit this result in our continuous-query model and show that uniform bidding policy is optimal for truthful auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In the continuous-query model, if the per-quuery auction is truthful then using a uniform bidding is an optimal strategy for each auto-bidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We use Theorem 1 Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Pick some small δ > 0 and divide the interval [0, 1] into subintervals of length δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let each subinterval I be a discrete query with value functions vj(I) = � I vj(q)dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then Theorem 1 Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2019) implies that uniform bidding differs from optimal by at most two queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, the difference from optimal is bounded by 2 maxj max|I|≤δ vj(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, since the valuation functions are atomless (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', the value of a query is dq), by letting δ to 0, the error of uniform bidding in the continuous case also goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 SPA in the Continuous-Query Model We generalize the discrete example of second price auction in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 to the continuous set of queries model showing that SPA is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The key step consists on showing that for the continuous-query model there is an auction equivalence result between first and second price auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 17 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' [Auction Equivalence Result] Suppose that auto-bidder uses a uniform bid strategy for SPA, and similarly, uses the simple bidding strategy defined in Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 for FPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then, in any subgame equilibrium the outcome of the auctions (allocations and pricing) on SPA is the same as in FPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This result immediately implies that all the results for FPA in Section 4 hold for SPA as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Suppose that there are at least two budget-advertisers or two tCPA-advertisers, then even for the continuous-query model SPA is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Similarly to FPA case, we can characterize the equilibrium for the two-advertiser case and derive sufficient conditions on advertisers’ valuation functions so that SPA is AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Given two advertisers, let µ1 and µ2 be the bidding multipliers in equilibrium for the subgame of the auto-bidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Also assume that h(q) = v1(q) v2(q) is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If the advertisers are budget-constrained with budget B1 and B2, then µ1 = B2 E[z1(z≥r)] and µ2 = µ1r, where r is the answer of the following implicit function, rE[1[z ≥ r)] E[z1(z ≤ r)] = B1 B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Here, E[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='] is defined as E[P(z)] = � ∞ 0 P(z)f(z)dz, where f(z) = v2(h−1(z)) h′(h−1(z)) wherever h′ is defined, and it is zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If the advertisers are tCPA-constrained with targets T1 and T2, we have µ1 = T1E[1(z≤r)] E[1(z≥r)] and µ2 = µ1r, where r is the answer of the following implicit function, rE[1(z ≥ r)] E[z1(z ≥ r)] E[1(z ≤ r)] E[z1(z ≤ r)] = T1 T2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If further, v2 is non-decreasing in q, and h is concave, and advertiers are either both budget- constrained two tCPA-constrained, then SPA is AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We now demonstrate the auction equivalence between FPA and SPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Note that the optimal strategy for a second-price auction is uniform bidding with respect to the true value of the query by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Also, Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 implies that the cost obtained by each advertiser in first-price auction in the continuous model is also depends on pacing multipliers of the other advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This claim immediately, suggests the equivalent between the optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' bidding strategies of first and second price auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, the optimal strategy for both auctions will be the same and therefore the resulting allocation and pricing will also be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Hence, it follows that the same allocation and pricing will be a pure equilibrium under both auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 Truthful Auctions Beyond Second-Price We now present the main result of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We show that a general truthful auction (with possibly random allocation) is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 18 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Consider a truthful auction (x, p) satisfying Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If there are at least two budget-advertisers or two tCPA-advertisers, then the truthful auction is not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The remainder of the section gives an overview of the proof of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Similar to the FPA and SPA case, we start by characterizing the equilibrium in the continuous case when there are two advertisers in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The proof relies on the observation that for auctions satisfying Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1, the allocation probability is a function of the bids’ ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, again, similar to FPA and SPA finding the equilibrium reduces to finding the ratio of bidding multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then to finish the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='8 instead of providing an explicit example where auto-bidding is non-AIC, we showed that the conditions needed for an auction’s allocation probability to satisfy are impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The following theorem finds an implicit equation for the best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We omit the proofs of the intermediaries steps and deferred them to the Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Consider a truthful auction (x, p) satisfying Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 and assume that there are either two budget-advertisers or two tCPA-advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let µ1 and µ2 be the bidding multipliers used by the auto-bidders in the subgame equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Further, assume that h(q) = v1(q) v2(q) is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If the advertisers are budget-constrained with budget B1 and B2, then µ1 = B1 E[p1(rz,1)] and µ2 = rµ1, where r is the answer of the following implicit function, E[rp1( z r, 1)] E[zp1( r z, 1)] = B1 B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Here, E[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='] is defined as E[P(z)] = � ∞ 0 P(z)f(z)dz, where f(z) = v2(h−1(z)) h′(h−1(z)) wherever h′ is defined, and it is zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If the advertisers are tCPA-constrained with targets T1 and T2, we have µ1 = T1E[zg(z/r)] E[rp1(z/r)] and µ2 = µ1r, where r is the answer of the following implicit function, E[x1( r z, 1)] E[zx1( z r, 1)] E[rp1( z r, 1)] E[zp1( r z, 1)] = T1 T2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Because allocation probability x1 is a non-decreasing function, we can derive a similar result to the FPA case and show if an instance is not AIC for budget-advertisers then it is also not AIC for tCPA-advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' If for the two budget-constrained advertisers case the truthful auction is not AIC, then for the tCPA-constrained advertisers case the same auction is also not AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Using the previous results we are in position to tackle the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='8 for budget constrained advertisers, since Proposi- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='10 would derive it for tCPA constraint advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We use implicit function theorem to find conditions on p1 and f to imply monotonicity in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let H(x, r) = � ∞ 0 rf(z)p1(z/r, 1)dz � ∞ 0 f(z)zp1(r/z, 1)dz − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 19 Then when advertiser 1 increases budget, the corresponding variable x increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, if we want to check whether r is a non-decreasing function of x, we need dr dx to be non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By the implicit function theorem, dr dx = − ∂H ∂x ∂H ∂r = 1 ∂H ∂r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, assume to the contrary that r i always non-decreasing in x, then ∂H(x,r ∂r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Define p(x) = p1(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then we have the following E[ d drrp(z/r)]E[zp(r/z)] ≥ E[rp(z/r)]E[ d dr � zp(r/z) � ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then d drE[rp(z/r)] E[rp(z/r)] ≥ d drE[zp(z/r)] E[zp(z/r)] By integrating both parts, we have that for any choice of f, rE[p(z/r)] ≥ E[zp(r/z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' When the above inequality hold for any choice of v1 and v2, we claim that the following must hold almost everywhere p(b) ≥ bp(1/b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (11) To see this, assume to the contrary that there exist a measurable set B such that (11) does not hold for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let qv2(q) = v1(q), therefore, f(z) = v2(z) can be any measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, we can define f to have zero value everywhere except X, and have weight 1 over X to get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By substituting variable with y = 1/b in (11), p(1/b)db ≥ p(b)/bdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, almost everywhere p(b) = bp(1/b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By differentiating we have p′(b) = p(1/b) − p′(1/x)/x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' On the other hand, as we will see in Appendix C for any truthful auction satisfying Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1, p′(b) = p′(1/b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, p(b) = p′(b)(b + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Solving it for p, we get that the only possible AIC pricing must be of the form p(b) = α(b + 1) for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Next, we will show there is no proper allocation probability satisfying the Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 that would result in a pricing function p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' It is not hard to see that by the Myerson’s pricing formulae, dx1(b,1) db = p′(b) b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, we must have x′ 1(b, 1) = α/b, so x1(b, 1) = c log(b) + d for some constants c > 0 and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' But x1 cannot be a valid allocation rule, since it will take negative values for small enough b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' References Gagan Aggarwal, Ashwinkumar Badanidiyuru Varadaraja, and Aranyak Mehta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Autobidding with Constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In Web and Internet Economics 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Amine Allouah and Omar Besbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Prior-independent optimal auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Management Science 66, 10 (2020), 4417–4432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 20 Santiago Balseiro, Yuan Deng, Jieming Mao, Vahab Mirrokni, and Song Zuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Robust Auction Design in the Auto-bidding World.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Ranzato, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Beygelzimer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Dauphin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='S.' metadata={'source': 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+page_content=' https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='cc/paper/2021/file/ 948f847055c6bf156997ce9fb59919be-Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='pdf Santiago Balseiro, Yuan Deng, Jieming Mao, Vahab Mirrokni, and Song Zuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Robust Auction Design in the Auto-bidding World.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 34 (2021), 17777–17788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Santiago R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Balseiro, Yuan Deng, Jieming Mao, Vahab Mirrokni, and Song Zuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Optimal Mechanisms for Value Maximizers with Budget Constraints via Target Clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In Proceedings of the 23rd ACM Conference on Economics and Computation (Boulder, CO, USA) (EC ’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1145/ 3490486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3538333 Santiago R Balseiro, Yuan Deng, Jieming Mao, Vahab S Mirrokni, and Song Zuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2021c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The landscape of auto-bidding auctions: Value versus utility maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In Proceedings of the 22nd ACM Conference on Economics and Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 132–133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Santiago R Balseiro and Yonatan Gur.' metadata={'source': 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Second-Price Auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In Proceedings of the 22nd ACM Conference on Economics and Computation (Budapest, Hungary) (EC ’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Vincent Conitzer, Christian Kroer, Debmalya Panigrahi, Okke Schrijvers, Nicolas E Stier-Moses, Eric Sodomka, and Christopher A Wilkens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 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+page_content=' Auction Design in an Auto-Bidding Setting: Randomization Improves Efficiency Beyond VCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' In Proceedings of the ACM Web Conference 2022 (Virtual Event, Lyon, France) (WWW ’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 173–181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1145/3485447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3512062 Aranyak Mehta and Andres Perlroth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Auctions without commitment in the auto-bidding world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='07312 Roger B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Myerson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Optimal Auction Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Mathematics of Operations Research 6, 1 (1981), 58–73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1287/moor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='58 arXiv:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1287/moor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='58 A Second-price tCPA constrained Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We continue with the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We first prove the uniqueness of equilibrium in the case of B′ 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' irst, note that there’s no equilibrium such that advertiser 1 wins all the queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To see this, note that the multiplier of advertiser 1 is at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Hence, the price of q3 for advertiser 2 is within their budget, and they have the incentive to increase their multiplier to buy q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Similarly, one can see that in any equilibrium, advertiser 1 gets at least q1, since its highest price is within their budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, assume some equilibrium exists with bidding prices ˜µ1 and ˜µ2 such that advertiser 1 gets only q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then ˜µ1(v1(1) + v1(2) + v1(3)) B2 > 1 ≥ ˜µ2v2(1) B1 , where the first inequality is because advertiser 2’s multiplier is the best response, and the second is coming from the budget constraint for advertiser 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, B1 B2 v1(1) + v1(2) + v1(3) v2(1) ≥ ˜µ2 ˜µ1 , But v1(2) v2(2) ≥ B1 B2 v1(1)+v1(2)+v1(3) v2(1) = 9 4, and thus v1(2) v2(2) > ˜µ2 ˜µ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This is in contradiction with allocation inequalities since advertiser 2 wins q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, we proved with B1 = 1 and B2 = 4 the equilibrium is unique such that advertiser 1 wins q1 and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 22 Now it remains to show a non AIC example for tCPA advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Again consider two advertisers, and 3 queries, with the same values as in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Here, let the tCPA constraint of advertiser 1 be T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4 and for advertiser 2 be T2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then again we show that there exists a unique equilibrium in which advertiser 1 gets queries 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' First, to prove the existence, let µ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='6 and µ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then we show this is an equilibrium since the three following conditoins hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Allocation: advertiser 1 wins q1 and q2 since it has a higher bid on them v1(1) v2(1) ≥ v1(2) v2(2) ≥ µ2 µ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='5 ≥ v1(3) v2(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' tCPA constraints are satisfied: T2v2(3) ≥ µ1v1(3), and T1(v1(1) + v1(2)) ≥ µ2(v2(1) + v2(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Best response: non of the advertiser can win more queries if they increase their multiplier: T2(v2(3) + v2(2)) < µ1(v1(3) + v1(2)), and T1(v1(1) + v1(2) + v1(3)) < µ2(v2(1) + v2(2) + v2(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, similar to the proof of the budget-constrained advertisers we show the equilibrium is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Note that there’s no equilibrium such that advertiser 1, gets all queries since the cost of all queries for advertiser 1 is at least v2(1) + v2(2) + v2(3) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3 which is larger than T1(v1(1) + v1(2) + v1(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Similarly, advertiser 2 cannot get all queries since the tCPA constraint would not hold v1(1) +v1(2) + v1(3) > T2(v2(1) + v2(2) + v2(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, to prove the uniqueness of equilibrium, it remains to show that there’s no equilibrium that advertiser 1 only gets query 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To contradiction, assume such equilibrium exists with the corresponding multipliers ˜µ1 and ˜µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then we must have ˜µ1(v1(1) + v1(2) + v1(3)) T2(v2(1) + v2(2) + v2(3)) > 1 ≥ ˜µ2v2(1) T1v1(1), where the first inequality is because advertiser 2’s multiplier is the best response, and the second inequality is coming from the budget constraint for advertiser 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, T1 T2 v1(1) + v1(2) + v1(3) v2(1) + v2(2) + v2(3) v1(1) v2(1) ≥ ˜µ2 ˜µ1 , But v1(2) v2(2) ≥ T1 T2 v1(1)+v1(2)+v1(3) v2(1)+v2(2)+v2(3) v1(1) v2(1), and thus v1(2) v2(2) > ˜µ2 ˜µ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' This is in contradiction with allocation inequalities since advertiser 2 wins q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, we proved with T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4 and T2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='7, the equilibrium is unique such that advertiser 1 wins q1 and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, we show that if advertiser 1 increases their tCPA constraint to T ′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='6, then there exists an equilibrium such that advertiser 1 only wins q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let µ′ 1 = 1 and µ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Allocation: advertiser 1 wins q1 v1(1) v2(1) ≥ µ′ 2 µ′ 1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' ≥ v1(2) v2(2) ≥ v1(3) v2(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' tCPA constraints are satisfied: T2(v2(3) + v2(2)) ≥ µ′ 1(v1(3) + v1(2)), and T ′ 1v1(1) ≥ µ′ 2v2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Best response: non of the advertiser can win more queries if they increase their multiplier: T2(v2(1) + v2(2) + v2(3)) < µ′ 1(v1(1) + v1(2) + v1(3)), and T ′ 1(v1(1) + v1(2)) < µ′ 2(v2(1) + v2(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' B First-price pacing equilibrium Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We follow the same steps of the proof as in Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' (2022a) for tCPA constrained advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Consider two sets of feasible bidding multipliers µ and µ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We will show that µ∗ = max(µ, µ′) is also feasible, where max is the component wise maximum of the bidding profiles for n advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Each query q is allocated to the bidder with the highest pacing bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We need to check that constraint (2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Fix advertiser a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Its multiplier in µ∗ must be also maximum in one of µ, or µ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' without loss assume µ∗ a = µa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then the set of queries that a wins with bidding profile µ∗ (X∗ a) must be a subset of queries it wins n µ (Xa), since all other advertisers’ bids have either remained the same or increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' On the other hand, the cost of queries a wins stays the same, since it’s a first price auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Since constraint (2) is feasible for bidding multipliers µ we must have (µa − Ta) � q∈X va(q) ≤ Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' But then since X∗ ⊆ X, we have as well (µa − Ta) � q∈X∗ va(q) = (µ∗ a − Ta) � q∈X∗ va(q) ≤ Ba, which implies µ∗ is a feasible strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To complete the proof we need to show the strategy that all advertisers take the maximum feasible pace µ∗ a = sup{µa|µ is feasible} results in an equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' To see this, note that if an advertiser’s strategy is not best-response, they have incentive to increase their pace with its constraints remaining satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' But then this would result into another feasible pacing strategy and is in contradiction with the choice of the highest pace µ∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' A similar argument also shows the equilibrium is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Assume there exists another pacing equilibrium where an advertiser a exists such that its pace is less than µ∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then by increasing their pace to µ∗ a they will get at least as many queries as before, so µ∗ a is the best-response strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 24 C Proofs for Truthful Auctions We start by the following observation, which follows by applying Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1 to reformulate the allocation function in the case of two advertisers as a function of a single variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Claim C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The probability of allocating each query is a function of the ratio of bids, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=', there exists a non-decreasing function g : R+ → [0, 1] such that the followings hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' x1(b1(q), b2(q)) = g( b1(q) b2(q)), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' g(z) + g(1/z) = 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' g(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' For example, SPA satisfies the above claim with g(z) = 1 when z = b1(q) b2(q) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' We are ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='9, which follows the similar steps of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='4, there exists µ1 and µ2 such that advertiser a bids zava(q) on each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Therefore, we can write the budget constraint for bidder 1 as, B1 = � 1 0 p1(b1(q), b2(q))dq = � 1 0 µ1v1(q)g �v1(q) v2(q) µ1 µ2 � dq − � 1 0 � µ1v1(q) 0 g � x v2(q)µ2 � dxdq Next, with a change of variable x = v1(q)y we have B1 = � 1 0 µ1v1(q)g �v1(q) v2(q) µ1 µ2 � dq − � 1 0 � µ1 0 g � v1(q) v2(q)µ2 y � v1(q)dydq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' As before, let h(q) = v1(q) v2(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then let z = h(q), we have dq = dh−1(z) = 1 h′(h−1(z))dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' So, B1 = � ∞ 0 µ1v1(h−1(z))g �zµ1 µ2 � 1 h′(h−1(z))dz − � ∞ 0 � µ1 0 g � z µ2 y � v1(h−1(z))dy 1 h′(h−1(z))dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Define f(z) = v2(h−1(z)) h′(h−1(z)) = 1 z v1(h−1(z)) h′(h−1(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Then we have B1 = � ∞ 0 µ1zf(z)g �zµ1 µ2 � dz − � ∞ 0 � � µ1 0 g � z µ2 y � dy � zf(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Similarly, B2 = � ∞ 0 µ2v2(h−1(z))(1 − g �zµ1 µ2 � ) 1 h′(h−1(z))dz − � ∞ 0 � µ2 0 g � y µ1z � v2(h−1(z))dy 1 h′(h−1(z))dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' B2 = � ∞ 0 µ2f(z)(1 − g �zµ1 µ2 � )dz − � ∞ 0 � µ2 0 g � y µ1z � dyf(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Next, we find the implicit function to derive r = µ2 µ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By change of variable we have the following two equations: B1 µ1 = � ∞ 0 zf(z)g(z/r)dz − r � ∞ 0 � � z/r 0 g(w)dw � f(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 16Notice that the function g is measurable since is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 25 B2 µ2 = � ∞ 0 f(z)(1 − g(z/r))dz − 1 r � ∞ 0 � � r/z 0 g(w)dw � zf(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' The implicit function for r is the following: B1 B2 = � ∞ 0 f(z) � zg(z/r) − r � z/r 0 g(w)dw � dz � ∞ 0 f(z) � r(1 − g(z/r) − z � r/z 0 g(w)dw � dz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Recall the payment rule in Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='3, this can be re-written as B1 B2 = � ∞ 0 rf(z)p1(z/r, 1)dz � ∞ 0 zf(z)zp1(r/z, 1)dz , which finishes the proof for the budget constrained advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, consider two tCPA constrained advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Following the same argument as above, we get the following from tightness of tCPA constraints T1 � ∞ 0 zf(z)g �zµ1 µ2 � dz = � ∞ 0 µ1zf(z)g �zµ1 µ2 � dz − � ∞ 0 � � µ1 0 g � z µ2 y � dy � zf(z)dz, and, T2 � ∞ 0 f(z)(1 − g �zµ1 µ2 � )dz = � ∞ 0 µ2f(z)(1 − g �zµ1 µ2 � )dz − � ∞ 0 � µ2 0 g � y µ1z � dyf(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' By dividing both sides of the equations we get the desired results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Now, to prove the main theorem, we need to show that the values accrued by advertisers is monotone in µ1/µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Claim C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Let µi be the optimal bidding multiplier for advertiser i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Given the assumptions in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='8, the value obtained by advertiser 1 is increasing in r = µ1 µ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Following the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='9 we can write value obtained by advertiser i as V1(B1, B2) = � ∞ 0 f(z)zg(rz)dz, where r is the answer to the implicit function stated in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' Monotonicity of V1(B1, B2) as a function of r follows from the fact that g is a monotone function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFQT4oBgHgl3EQfzzas/content/2301.13414v1.pdf'} diff --git a/0dFKT4oBgHgl3EQfNy2J/content/2301.11756v1.pdf 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a/39FAT4oBgHgl3EQflh2J/content/tmp_files/2301.08618v1.pdf.txt b/39FAT4oBgHgl3EQflh2J/content/tmp_files/2301.08618v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bda91e2967a715c7dd7a4f526830704fb550262f --- /dev/null +++ b/39FAT4oBgHgl3EQflh2J/content/tmp_files/2301.08618v1.pdf.txt @@ -0,0 +1,1102 @@ +arXiv:2301.08618v1 [cs.LG] 20 Jan 2023 +1 +Coupled Physics-informed Neural Networks for +Inferring Solutions of Partial Differential Equations +with Unknown Source Terms +Aina Wang, Pan Qin, Xi-Ming Sun, Senior Member, IEEE, +Abstract—Physics-informed neural networks (PINNs) provide +a transformative development for approximating the solutions +to partial differential equations (PDEs). This work proposes +a coupled physics-informed neural network (C-PINN) for the +nonhomogeneous PDEs with unknown dynamical source terms, +which is used to describe the systems with external forces and +cannot be well approximated by the existing PINNs. In our +method, two neural networks, NetU and NetG, are proposed. +NetU is constructed to generate a quasi-solution satisfying PDEs +under study. NetG is used to regularize the training of NetU. +Then, the two networks are integrated into a data-physics-hybrid +cost function. Finally, we propose a hierarchical training strategy +to optimize and couple the two networks. The performance of +C-PINN is proved by approximating several classical PDEs. +Index Terms—Coupled physics-informed neural network, hier- +archical training strategy, partial differential equations, unknown +source term +I. Introduction +P +ARTIAL differential equations (PDEs) are one of the +general representations for describing spatio-temporal +dependence in physics [1], medicine [2], engineering [3], +finance [4], and weather [5], [6]. Numerical approaches, like +the finite difference method (FDM) [7] and finite element +(FEM) [8], [9], have been widely investigated and applied. +FDM used a topologically square lines network to construct +PDEs’ discretization. Thus, complex geometries in multiple +dimensions challenge FDM [10]. On the other hand, compli- +cated geometries can be treated with FEM [11]. The greatest +difficulty of classical numerical approaches is balancing the +accuracy and efficiency of forming meshes. +Among the numerical methods for solving PDEs, the +Galerkin method is a famous computation method in which the +linear combination of basis functions was employed to approx- +imate the solutions to PDEs [12]. Motivated by this, several +works have used machine learning models to replace the linear +combination of basis functions to construct data-efficient and +physics-informed learning methods for solving PDEs [13]– +[15]. Successful applications of deep learning methods to +various fields, like image [16], text [17], and speech recogni- +tion [18], ensure that they are excellent replacers of the linear +combination of basis functions for solving PDEs [4]. Conse- +quently, leveraging the well-known approximation capability +The authors are with the Key Laboratory of Intelligent Control and Opti- +mization for Industrial Equipment of Ministry of Education and the School +of Control Science and Engineering, Dalian University of Technology, Dalian +116024, China e-mail: WangAn@mail.dlut.edu.cn, qp112cn@dlut.edu.cn, +sunxm@dlut.edu.cn (Corresponding author: Pan Qin) +of neural networks to solve PDEs is a natural idea and has +been investigated in various forms previously [19]–[21]. The +framework of physics-informed neural networks (PINNs) [22] +was introduced to solve the forward problems while respecting +any given physical laws governed by PDEs, including the +nonlinear operator, initial, and boundary conditions. Within +the PINNs framework, both the sparse measurements and +the physical knowledge were fully integrated into cost func- +tion [23], [24]. The solution with respect to spatio-temporal +dependence was obtained by training the cost function. Note +that the approximation obtained by machine learning and deep +learning is meshfree, which has no problem on balancing +accuracy and efficiency of forming meshes. +Meanwhile, the potential of using PINNs to solve the inverse +problem is promising [25]. A hybrid PINN was proposed +to solve PDEs in [26], in which a local fitting method was +combined with neural networks to solve PDEs. The hybrid +PINN was used to identify unknown constant parameters +in PDEs. The generative adversarial network (GAN) [27] +was also physics-informed to solve the inverse problems. +The stochastic physics-informed GAN was investigated for +estimating the distributions of unknown parameters in PDEs. +The recent work [28] encoded the physical laws governed +by PDEs into the architecture of GANs to solve the inverse +problems for stochastic PDEs. PINNs were also combined with +the Bayesian method to solve inverse problems from noisy data +[29]. +PDEs can be classified into homogeneous and nonhomoge- +neous types. Systems without external forces can be described +by the homogeneous PDEs. The nonhomogeneous PDEs can +be applied to reveal the continuous energy propagation be- +havior of the source and hereby are effective for describing +practical systems driven by external forces. The function forms +of the solution and the source term were both assumed to be +unknown in [30], in which the measurements of the source +term should be obtained separately from the measurements of +the solution. However, the independent measurements of the +external forces cannot always be easily obtained from practical +situations. The recent work [31] can directly solve the steady- +state PDEs’ forward and inverse problems, where the source +terms were assumed to be constant. Thus, [31] was not feasible +for systems with unsteady external forces, which should be +described by dynamical functions. +Although the aforementioned methods have made great +progress on unknown parameters, prior information or mea- +surements on external forces cannot always be easily obtained + +2 +from practical situations. For example, the real distribution of +the seismic wave field underground is unknown [32]; the vast +of signals internal engine, indicating the operation state of +the engine, cannot be isolated [33]. Furthermore, the existing +methods with the assumption of the constant source term +cannot be readily extended to describe the spatio-temporal +dependence of complex dynamical systems. The determination +of dynamical source terms with less prior information or even +without any prior information is an under-investigated issue. +To this end, this paper proposes a coupled-PINN (C-PINN), +using the sparse measurements and limited prior information +of PDEs, to solve PDEs with unknown source terms. In our +method, two neural networks, NetU and NetG, are proposed. +NetU is applied to generate a quasi-solution satisfying PDEs +under study; NetG is used to regularize the training of NetU. +Then, the two networks are integrated into a data-physics- +hybrid cost function. Furthermore, we propose a hierarchical +training strategy to optimize and couple the two networks. +Finally, the proposed C-PINN is applied to solve several +classical PDEs to demonstrate its performance. +The rest of the paper is organized as follows. The classical +PINNs is briefly reviewed in Section II. A C-PINN using the +sparse measurements and limited prior knowledge to solve +PDEs with unknown source terms is proposed in Section III. +Meanwhile, the two neural networks, NetU and NetG, are +proposed in our method. Furthermore, a hierarchical training +strategy is proposed to optimize and couple the two networks. +In Section IV, our proposed C-PINN is validated with four +case studies. In Section V, the concluding remarks and the +future work are presented. +II. Brief Review of PINNs +In this section, we briefly review the basic idea of PINNs +for data-driven solutions to PDEs and data-driven discovery +of PDEs [22]. +Data-driven solutions to PDEs describe that solve PDEs of +the generalized form +ut(x, t) + N[u(x, t)] = 0, x ∈ Ω ⊆ Rd, t ∈ [0, T] ⊂ R +(1) +with known parameters. Here, x is the spatial variable, t is +the temporal variable with t = 0 being at the initial state, +u : Rd × R → R denotes the hidden solution, N[·] is a +series of partial differential operators, the domain Ω ⊆ Rd is +a spatial bounded open set with the boundary ∂Ω. Analytical +or numerical methods have been widely investigated to find +proper solution ψ(x, t) satisfying (1) [34]. The left-hand-side +of (1) can be used to define a residual function as the following +f(x, t) := ut(x, t) + N[u(x, t)], +(2) +where a neural network is used to approximate the solution +ψ(x, t) to PDEs. The inverse problem is focused on the data- +driven discovery of PDEs of the generalized form (1), where +unknown parameters of PDEs here turn into parameters of +PINNs. +PINNs for both problems can be trained by minimizing the +cost function +MSE = MSED + MSEPH. +(3) +Here, MSED is formulated as the following +MSED = +� +(x,t,u)∈D +� +ˆu +� +x, t; ˆΘU +� +− u (x, t) +�2 , +(4) +where ˆu +� +x, t; ˆΘU +� +is the function of neural network with ˆΘU +being its trained parameter set. Let D denote the training +dataset. This mean squared error term can be considered as +the data-driven loss. MSEPH is as the following +MSEPH = +� +(x,t)∈E +ˆf (x, t)2 , +(5) +which regularizes ˆu +� +x, t; ˆΘU +� +to satisfy (1). Let E denote +the set of collocation points. This regularization term can be +considered as the physics-informed loss for the homogeneous +PDEs. Here, ˆf (x, t) is defined as +ˆf (x, t) := ˆut +� +x, t; ˆΘU +� ++ N +� +ˆu +� +x, t; ˆΘU +�� +, +(6) +where ˆut +� +x, t; ˆΘU +� +and N +� +ˆu +� +x, t; ˆΘU +�� +can be obtained using +automatic differential [35]. +III. Constructing C-PINN +C-PINN for solving PDEs with unknown source terms is +presented in this section. The nonhomogeneous PDEs are of +the following generalized form +ut(x, t)+N[u(x, t)] = g(x, t), x ∈ Ω ⊆ Rd, t ∈ [0, T] ⊂ R, (7) +where x and t are the spatial and temporal variable, respec- +tively, u : Rd ×R → R is similar to (1), g : Rd ×R → R denotes +the general types of source terms including linear, nonlinear, +state-steady, or dynamical, Ω is a spatial bounded open set with +the boundary ∂Ω. Without loss of generality, the spatial set of +(7) is subjected to Dirichlet boundary, Neumann boundary, or +the hybrid of Dirichlet and Neumann boundary conditions. In +general, g(x, t) is used as source terms to describe the external +forces for dynamical systems and cannot always be separately +measured, as mentioned in Section I. +Different from (6), the residual function is defined for the +nonhomogeneous case as the following +fN(x, t) := f(x, t)−g(x, t) = ut(x, t)+N[u(x, t)]−g(x, t). (8) +When g(x, t) is exactly known, ˆfN(x, t), obtained with auto- +matic differential from (8), can be directly used to regularize +the approximation of u(x, t). However, the unknown g(x, t) +will lead to unknown fN(x, t), which makes the aforemen- +tioned regularization infeasible. +Therefore, the goal of C-PINN is to approximate the solu- +tion to PDEs with unknown source terms described by (7). To +this end, there are two neural networks included in C-PINN: +(a) NetU for approximating the solution satisfying (7); (b) +NetG for regularizing the training of NetU. +1) Cost function: +To train C-PINN, the training dataset +is uniformly sampled from the system governed by (7). The +training dataset D divided into D = DB∪DI with DB∩DI = ∅, +where DB denotes the boundary and initial training dataset and +DI is the training dataset of interior of Ω. Collocation points + +3 +(x, t) ∈ E correspond to those of (x, t, u) ∈ DI. Then, we adopt +the following data-physics-hybrid cost function +MSE = MSED + MSEPN +(9) +to train our proposed C-PINN. MSED and MSEPN in (9) +are the data-driven loss and physics-informed loss for the +nonhomogeneous PDEs, respectively. MSED adopts the same +form of (4). MSEPN is as the following +MSEPN = +� +(x,t)∈E +� ˆf (x, t) − ˆg +� +x, t; ˆΘG +��2 , +where ˆg +� +x, t; ˆΘG +� +is the function of NetG with ˆΘG being +its trained parameter set, +ˆf(x, t) has been defined by (2). +MSEPN corresponds to the physics-informed loss for the +nonhomogeneous PDEs obtained from (8) imposed at a finite +set of collocation points (x, t) ∈ E, which is used to regularize +ˆu +� +x, t; ˆΘU +� +of NetU to satisfy (7). +2) Hierarchical training strategy: Considering the relation +between NetU and NetG in (3), a hierarchical training strategy +is proposed. In many cases, the exact formulation or even +sparse measurements of g(x, t) are not available, while the +sparse measurements DI can be obtained to enforce the +structure of (7) to achieve ˆΘG. Thus, ΘU and ΘG should be +iteratively estimated with mutual dependence. Assume k is the +present iteration step, the core issue of the hierarchical train- +ing strategy is described by the following two optimization +problems +ˆΘ(k+1) +G += arg min +ΘG +� +MSED +� ˆΘ(k) +U +� ++ MSEPN +� +ΘG; ˆΘ(k) +U +�� += arg min +ΘG +MSEPN +� +ΘG; ˆΘ(k) +U +� +(10) +and +ˆΘ(k+1) +U += arg min +ΘU +� +MSED (ΘU) + MSEPN +� +ΘU; ˆΘ(k+1) +G +�� +, (11) +where ˆΘ(k) +U is the estimated parameter set of NetU at kth step, +ˆΘ(k+1) +G +is the estimated parameter set of NetG at (k + 1)th step, +Θ(k+1) +U +is the estimated parameter set of NetU at (k + 1)th +step, which is used to describe the function ˆu +� +x, t; ˆΘ(k+1) +U +� +. +The details of the hierarchical training strategy are obtained +by Algorithm 1. +Note that Θ(0) +U and Θ(0) +G are used as a given parameter set +for NetU and the initialization of the parameter set for NetG +at Step 0, respectively. Furthermore, the iterative transmission +of parameter sets of NetG and NetU happens in the algorithm. +IV. Numerical experiments +In this section, our proposed C-PINN is applied to solve +several classical PDEs to demonstrate its performance. All +the examples are implemented with TensorFlow. The fully +connected structure with a hyperbolic tangent activation func- +tion is applied, which is initialized by Xavier. These training +dataset (x, t, u) ∈ D and collocation points (x, t) ∈ E are +then input into NetU and NetG. L-BFGS [36] is used to +hierarchically solve the optimization problems (10) and (11) +to couple the two networks. +Algorithm 1 The hierarchical strategy of optimizing and +coupling for C-PINN. +-Initialize: Randomly sampled training dataset (x, t, u) ∈ D +and collocation points (x, t) ∈ E. Randomly generate initial +parameter sets Θ(0) +U and Θ(0) +G . +- Step 0: Assume the kth iteration has achieved ˆΘ(k) +U and +ˆΘ(k) +G . +Repeat: +- Step k-1: Training for NetG by solving the optimization +problem (10) to obtain ˆΘ(k+1) +G +, where the estimations of +ˆut +� +x, t; ˆΘ(k) +U +� ++ N +� +ˆu(x, t; ˆΘ(k) +U +� +in MSEPN is obtained from +the former iteration result ˆΘ(k) +U . +- Step k-2: Training for NetU by solving the optimization +problem (11) to obtain ˆΘ(k+1) +U +, which is used to estimate +ˆg +� +x, t; Θ(k+1) +G +� +in MSEPN. +-Until the stop criterion is satisfied. +-Return the solution function ˆΘU → ˆu +� +x, t; ˆΘU +� +, which can +predict the solution (8) with any point (x, t) in Ω. +We evaluate the performance of our proposed C-PINN by +means of root mean squared error (RMSE) +RMSE = +� +1 +|T| +� +(x,t)∈T +(u (x, t) − ˆu (x, t))2, +where |T| is the cardinality with respect to the collocation +points (x, t) ∈ T, T is the set of testing collocation points. +u (x, t) and ˆu (x, t) denote the ground truth and the cor- +responding predictions, respectively. To further validate the +performance of C-PINN, the Pearson’s correlation coefficient +(CC) +CC = +cov (u (x, t) , ˆu (x, t)) +√Var u (x, t) √Var ˆu (x, t) +is also used to measure the similarity between ground truth and +prediction, where CC is the correlation coefficient of u(x, t) +and ˆu(x, t), cov (u (x, t) , ˆu (x, t)) is the covariance between +u(x, t) and ˆu(x, t), and Var u (x, t) and Var ˆu (x, t) are variance +of u(x, t) and ˆu(x, t), respectively. +A. Case 1: 1-D Heat Equation +C-PINN is first applied to solve the heat equation with +unknown external forces, where both Dirichlet and Neumann +boundary conditions are conducted to demonstrate its perfor- +mance. +1) Dirichlet Boundary Condition +Here, we consider the heat equation with Dirichlet boundary +condition as the following +∂u +∂t = a2 ∂2u +∂x2 + g(x, t), +0 < x < L, t > 0 +u|t=0 = φ(x), +0 ⩽ x ⩽ L +u|x=0 = 0, +u|x=L = 0, +t > 0, +(12) +where thermal diffusivity a += +1, u(x, t) is the primary +variable and means the temperature at (x, t), L = π is the +length of bounded rod, φ(x) = 0 is initial temperature, and + +4 +g(x, t) = xe−t denotes the unknown external heat source at +(x, t). The analytical solution u (x, t) to (12) is obtained with +respect to [37]. In this experiment, the setting-ups of C-PINN +are as follows. There are eight hidden layers with 20 units +in each of them for both NetU and NetG. A total of 110 +training data (x, t, u(x, t)) in D with t ∈ [0, 6], including 10 +training data in DI and 100 training data in DB, are randomly +sampled, 10 sparse collocation points are randomly sampled to +enforce the structure of (12). Fig. 1 shows the sparse training +dataset and the prediction results. Specifically, the magnitude +of the predictions ˆu(x, t) using the training dataset is shown in +Fig. 1(a) with a heat map. In this case, RMSE is 4.225390e−02 +and the correlation coefficient is 9.785444e − 01. Moreover, +we compare the ground truths and the predictions at fixed- +time t= 1.5, 3, and 4.5 in Fig. 1(b) to (d), respectively. The +evaluation criteria in Table I are applied to further quantify +the performance of our proposed C-PINN. +0 +1 +2 +3 +4 +5 +6 +t +0 +1 +2 +3 +x +��� ������������������uˆ(x, t) +����training data ����� +���training data ������ +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0 +2 +x +0 +1 +u(x, t) +��� t = 1.5 +0 +2 +x +0 +1 +u(x, t) +��� t = 3 +0 +2 +x +0 +1 +u(x, t) +��� t = 4.5 +������������ +���������� +Fig. 1. +(a) Predictions ˆu (x, t) for the 1-D heat equation with Dirichlet +boundary condition; (b), (c), and (d) Comparisons of the ground truths and +predictions corresponding to the fixed-time t= 1.5, 3, and 4.5 snapshots +depicted by the dashed vertical lines in (a), respectively. +TABLE I +Evaluation criteria for the three temporal snapshots depicted by the dashed +vertical lines in Fig. 1-(a) +Criteria +1.5 +3 +4.5 +RMSE +4.600305e-02 1.342719e-02 +2.991229e-02 +CC +9.753408e-01 9.912983e-01 +9.805664e-01 +Subsequently, the experiment for PDE with Neumann +boundary condition will be further explored to show the +general performance of C-PINN. +2) Neumann Boundary Condition +Heat equation with Neumann boundary condition is defined +as +∂u +∂t = a2 ∂2u +∂x2 + g(x, t), +0 < x < L, t > 0 +u|t=0 = φ(x), +0 ⩽ x ⩽ L +u|x=0 = 0, +∂u +∂x +�����x=L += 0, +t > 0, +(13) +with the thermal diffusivity a = 1, the length of bounded +rod L = π, the initial temperature φ(x) = sin (x/2), and +the external heat source is g(x, t) = sin (x/2). The analytical +solution u(x, t) to (13) is obtained according to [37]. In this +example, NetU is of three hidden layers consisting of 30 +neurons individually. NetG is of eight hidden layers consisting +of 20 units individually. (x, t, u(x, t)) in D are considered with +t ∈ [0, 10]. A total of 130 training data in DB, including 10 +initial training data, 60 left boundary training data, and 60 right +boundary training data are randomly sampled. Moreover, the +20 sparse collocation points are randomly sampled to enforce +the structure of (13). The magnitude of the predictions ˆu(x, t) +using the training dataset is shown in Fig. 2(a). RMSE is +5.748950e−02 and the correlation coefficient is 9.988286e−01. +Moreover, we compare the ground truths and the predictions at +fixed-time t= 3, 6, and 9 in Fig. 2(b) to (d), respectively. The +evaluation criteria in Table II are applied to further evaluate +the performance of our proposed C-PINN. +0 +2 +4 +6 +8 +10 +t +0 +1 +2 +3 +x +��� ������������������ uˆ(x, t) +130 training data������ +20 training data ������ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0 +2 +x +0 +1 +2 +3 +u(x, t) +��� t = 3 +0 +2 +x +0 +1 +2 +3 +u(x, t) +��� t = 6 +0 +2 +x +0 +1 +2 +3 +u(x, t) +��� t = 9 +������������� +Prediction +Fig. 2. +(a) Predictions ˆu (x, t) for the 1-D heat equation with Neumann +boundary condition. (b), (c), and (d) Comparisons of the ground truths and +predictions correspond to the fixed-time t= 3, 6, and 9 snapshots depicted by +the dashed vertical lines in (a), respectively. +TABLE II +Evaluation criteria for the three temporal snapshots depicted by the dashed +vertical lines in Fig. 2-(a). +Criteria +3 +6 +9 +RMSE +5.343142e-02 +5.884118e-02 +7.064205e-02 +CC +9.982448e-01 +9.990231e-01 +9.984719e-01 +B. Case 2: 1-D Wave Equation +The wave equation is as the following +∂2u +∂t2 = a2 ∂2u +∂x2 + g(x, t), +0 < x < L, t > 0 +u|x=0 = 0, +u|x=L = 0, +t > 0 +u|t=0 = 0, +∂u +∂t +�����t=0 += 0, +0 ⩽ x ⩽ L, +(14) + +5 +where the wave speed a is 1, the length of bounded string L +is π, the time of wave propagation t is 6, the external force is +g(x, t) = sin 2πx +L sin 2aπt +L +at(x, t) and displacement u(x, t) at (x, t) according to [37] is +further investigated. +In this experiment, NetU is of three hidden layers consisting +of 30 neurons individually. NetG is of eight hidden layers +consisting of 20 units individually. A total of 210 training +data (x, t, u (x, t)) in D, including 50 initial training data, 120 +boundary training data, and 40 collation points are randomly +sampled. Fig. 3(a) shows the sparse training dataset and the +magnitude of displacement ˆu(x, t) at (x, t). Fig. 3(b) to (d) +show the comparisons of ground truths and predictions corre- +sponding to the three fixed-time t=1.5, 3, and 4.5, which are +depicted by the dashed vertical lines in Fig. 3(a), respectively. +RMSE is 7.068626e − 02 and the correlation coefficient is +9.864411e− 01. The evaluation criteria for the three temporal +snapshots are listed in Table III. +0 +1 +2 +3 +4 +5 +6 +� +0 +1 +2 +3 +x +(�� ����������������� ˆu(x, t) +220 training data ����� +40 training data ����� +−1.0 +−0.5 +0.0 +0.5 +1.0 +0 +2 +x +−1 +0 +1 +u(x, t) +�b� ������� +0 +2 +x +−1 +0 +1 +u(x, t) +��� ����� +0 +2 +x +−1 +0 +1 +u(x, t) +��� ������� +������������� +���������� +Fig. 3. +(a) Predictions ˆu (x, t) for 1-D wave equation. (b), (c), and (d) +Comparisons of the ground truths and predictions corresponding to the fixed- +time t=1.5, 3, and 4.5 snapshots depicted by the dashed vertical lines in (a), +respectively. +TABLE III +Evaluation criteria for the three temporal snapshots depicted by the dashed +vertical lines in Fig. 3-(a). +Criteria +1.5 +3. +4.5 +RMSE +1.424030e-01 +3.305190e-02 +5.201132e-02 +CC +9.6238994e-01 +9.985312e-01 +9.983170e-01 +C. Case 3: 2-D Poisson Equation +We further consider the following 2-D Poisson equation +∂2u +∂x2 + ∂2u +∂y2 = T0, +0 < x < a, 0 < y < b +u(x, 0) = 0, +u(x, b) = T, +0 ⩽ x ⩽ a +u(0, y) = 0, +u(a, y) = 0, +0 ⩽ y ⩽ b, +(15) +where T is 1, the constant source term T0 = 1 is unknown, and +a = b = 1. The analytical solution u(x, y) to (15) is obtained +according to [37]. In this experiment, the setting-ups of C- +PINN are as follows. There are eight hidden layers with 20 +units in each of them for both NetU and NetG. Thirty training +data in DB and 3 collocation points in DI are used. Fig. 4(a) +shows the sparse training dataset and the predictions ˆu(x, y). +Fig. 4(b) to (d) show the prediction performance of fixed- +location y=0.2, 0.4, and 0.6 snapshots depicted in Fig. 4(a), +respectively. RMSE is 1.594000e − 02 and the correlation +coefficient is 9.997390e − 01. The corresponding evaluation +criteria are listed in Table IV. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.00 +0.25 +0.50 +0.75 +1.00 +x +��� ���������������������uˆ(x, y) +30 training data ������ +3 training data ����� +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +x +0.0 +0.5 +1.0 +1.5 +u(x, y) +0 +1 +x +0.0 +0.5 +1.0 +1.5 +u(x, y) +0 +1 +x +0.0 +0.5 +1.0 +1.5 +u(x, y) +(��� ����� +������������� +���������� +y +y +(c�� ����4 +y +(b�� ����2 +y +Fig. 4. +(a) Predictions ˆu (x, y) for the 2-D Poisson equation. (b), (c), and, +(d) Comparisons of the ground truths and predictions corresponding to the +fixed-location y = 0.2, 0.4, and 0.6 snapshots depicted by the dashed vertical +lines in (a), respectively. +TABLE IV +Evaluation criteria for the three fixed-location snapshots depicted by the +dashed vertical lines in Fig. 4-(a). +Criteria +0.2 +0.4 +0.6 +RMSE +1.763408e-02 +1.139888e-02 +7.696680e-03 +CC +9.986055e-01 +9.999703e-01 +9.999656e-01 +D. Case 4: 3-D Helmholtz Equation +C-PINN is also applied to solve 3-D Helmholtz equation +with an unknown source term. In particular, we consider the +same test PDEs that were previously suggested in [26] +∆u(x) + p2u(x) = g(x) in Ω ⊂ R3 +u(x) = u0(x) on ∂Ω, +(16) +where ∆ = +∂ +∂x2 + ∂ +∂y2 + ∂ +∂z2 is Laplacian operator, x = (x, y, z)⊤ +is coordinates with x, y, z ∈ (0, 1/4] , p = 5 is the wavenumber, + +6 +a suitable g(x) is the right-hand side of (16) so that +u(x) = (0.1 sin (2πx) + tanh (10x)) sin (2πy)sin (2πz) +is the analytical solution of (16) [26]. In this experiment, +NetU is of three hidden layers consisting of 100, 50, and +50 neurons individually. NetG is of eight hidden layers con- +sisting of 20 units individually. Sixty training data and 120 +collocation points are sampled. Fig. 5(a) shows the solution +of (x, y, z = 0.12) snapshot. Furthermore, Fig. 5(b) to (d) +show the comparisons of ground truths and predictions, which +are extracted at (x = 0.05, z = 0.12), (x = 0.15, z = 0.12), +and (x = 0.2, z = 0.12), respectively. The evaluation criteria +for this extractions are listed in Table V. In this experiment, +RMSE is 1.192859e − 01, and the correlation coefficient is +9.057524e − 01. +0.05 +0.10 +0.15 +0.20 +0.25 +x +0.1 +0.2 +y +(a) 3−D Helmholtz Equation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.10.2 +y +0.0 +0.2 +0.4 +0.6 +u(x, y) +�b� x = 0.05 +0.10.2 +y +0.0 +0.2 +0.4 +0.6 +u(x, y) +��� x = 0.15 +0.10.2 +y +0.0 +0.2 +0.4 +0.6 +u(x, y) +��� x =0.2 +������������� +Prediction +�uˆ(x,�y,�z�=�25 +3�) +Fig. 5. (a) Predictions ˆu (x, y, z = 0.12) for 3-D Helmholtz equation. (b), (c) +and, (d) Comparisons of the ground truths and predictions corresponding to +the (x = 0.05, z = 0.12), (x = 0.15, z = 0.12), and (x = 0.20, z = 0.12) +snapshots depicted by the dashed vertical lines in (a), respectively. +TABLE V +Evaluation criteria for the three snapshots depicted by the dashed vertical +lines in Fig. 5-(a). +Criteria +0.05 +0.15 +0.2 +RMSE +7.043735e-02 +7.548533e-02 +5.179414e-02 +CC +9.604538e-01 +9.998589e-01 +9.964517e-01 +V. Conclusion +This paper proposes a novel PINN, called C-PINN, to solve +PDEs with less prior information or even without any prior +information for source terms. In our approach, two neural net- +works, NetU and NetG, are proposed with a fully-connected +structure. NetU for approximating the solution satisfying +PDEs under study; NetG for regularizing the training of NetU. +Then, the two networks are integrated into a data-physics- +hybrid cost function. Furthermore, the two networks are op- +timized and coupled by the proposed hierarchical training +strategy. Finally, C-PINN is applied to solve several classical +PDEs to testify to its performance. Note that C-PINN inherits +the advantages of PINN, such as sparse property and automatic +differential. C-PINN is proposed to solve such a dilemma as +the governing equation of dynamical systems with unknown +forces. Thus, C-PINN can be further applied to infer the +unknown source terms. Meanwhile, C-PINN can be extended +to identify the operators from the sparse measurements. +In the future, we will continue to use our C-PINN in +various scenarios, like solving PDEs with unknown struc- +ture parameters and high-dimension PDEs. For the case, the +structures of PDE are totally unknown, regularization method +will be combined with C-PINN to select operators from +the sparse measurements. Our proposed C-PINN has been +shown to solve several classical PDEs successfully. 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Westwig, Mathematical physics: applied +mathematics for scientists and engineers. +John Wiley & Sons, 2010. + diff --git a/39FAT4oBgHgl3EQflh2J/content/tmp_files/load_file.txt b/39FAT4oBgHgl3EQflh2J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c43367df3db3233db73ab1b3fb2d114c3b7c33f1 --- /dev/null +++ b/39FAT4oBgHgl3EQflh2J/content/tmp_files/load_file.txt @@ -0,0 +1,706 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf,len=705 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='08618v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='LG] 20 Jan 2023 1 Coupled Physics-informed Neural Networks for Inferring Solutions of Partial Differential Equations with Unknown Source Terms Aina Wang, Pan Qin, Xi-Ming Sun, Senior Member, IEEE, Abstract—Physics-informed neural networks (PINNs) provide a transformative development for approximating the solutions to partial differential equations (PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' This work proposes a coupled physics-informed neural network (C-PINN) for the nonhomogeneous PDEs with unknown dynamical source terms, which is used to describe the systems with external forces and cannot be well approximated by the existing PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In our method, two neural networks, NetU and NetG, are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' NetU is constructed to generate a quasi-solution satisfying PDEs under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' NetG is used to regularize the training of NetU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Then, the two networks are integrated into a data-physics-hybrid cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Finally, we propose a hierarchical training strategy to optimize and couple the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The performance of C-PINN is proved by approximating several classical PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Index Terms—Coupled physics-informed neural network, hier- archical training strategy, partial differential equations, unknown source term I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Introduction P ARTIAL differential equations (PDEs) are one of the general representations for describing spatio-temporal dependence in physics [1], medicine [2], engineering [3], finance [4], and weather [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Numerical approaches, like the finite difference method (FDM) [7] and finite element (FEM) [8], [9], have been widely investigated and applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' FDM used a topologically square lines network to construct PDEs’ discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Thus, complex geometries in multiple dimensions challenge FDM [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' On the other hand, compli- cated geometries can be treated with FEM [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The greatest difficulty of classical numerical approaches is balancing the accuracy and efficiency of forming meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Among the numerical methods for solving PDEs, the Galerkin method is a famous computation method in which the linear combination of basis functions was employed to approx- imate the solutions to PDEs [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Motivated by this, several works have used machine learning models to replace the linear combination of basis functions to construct data-efficient and physics-informed learning methods for solving PDEs [13]– [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Successful applications of deep learning methods to various fields, like image [16], text [17], and speech recogni- tion [18], ensure that they are excellent replacers of the linear combination of basis functions for solving PDEs [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Conse- quently, leveraging the well-known approximation capability The authors are with the Key Laboratory of Intelligent Control and Opti- mization for Industrial Equipment of Ministry of Education and the School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China e-mail: WangAn@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='dlut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='cn, qp112cn@dlut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='cn, sunxm@dlut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='cn (Corresponding author: Pan Qin) of neural networks to solve PDEs is a natural idea and has been investigated in various forms previously [19]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The framework of physics-informed neural networks (PINNs) [22] was introduced to solve the forward problems while respecting any given physical laws governed by PDEs, including the nonlinear operator, initial, and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Within the PINNs framework, both the sparse measurements and the physical knowledge were fully integrated into cost func- tion [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The solution with respect to spatio-temporal dependence was obtained by training the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Note that the approximation obtained by machine learning and deep learning is meshfree, which has no problem on balancing accuracy and efficiency of forming meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Meanwhile, the potential of using PINNs to solve the inverse problem is promising [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' A hybrid PINN was proposed to solve PDEs in [26], in which a local fitting method was combined with neural networks to solve PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The hybrid PINN was used to identify unknown constant parameters in PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The generative adversarial network (GAN) [27] was also physics-informed to solve the inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The stochastic physics-informed GAN was investigated for estimating the distributions of unknown parameters in PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The recent work [28] encoded the physical laws governed by PDEs into the architecture of GANs to solve the inverse problems for stochastic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' PINNs were also combined with the Bayesian method to solve inverse problems from noisy data [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' PDEs can be classified into homogeneous and nonhomoge- neous types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Systems without external forces can be described by the homogeneous PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The nonhomogeneous PDEs can be applied to reveal the continuous energy propagation be- havior of the source and hereby are effective for describing practical systems driven by external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The function forms of the solution and the source term were both assumed to be unknown in [30], in which the measurements of the source term should be obtained separately from the measurements of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' However, the independent measurements of the external forces cannot always be easily obtained from practical situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The recent work [31] can directly solve the steady- state PDEs’ forward and inverse problems, where the source terms were assumed to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Thus, [31] was not feasible for systems with unsteady external forces, which should be described by dynamical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Although the aforementioned methods have made great progress on unknown parameters, prior information or mea- surements on external forces cannot always be easily obtained 2 from practical situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' For example, the real distribution of the seismic wave field underground is unknown [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' the vast of signals internal engine, indicating the operation state of the engine, cannot be isolated [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Furthermore, the existing methods with the assumption of the constant source term cannot be readily extended to describe the spatio-temporal dependence of complex dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The determination of dynamical source terms with less prior information or even without any prior information is an under-investigated issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' To this end, this paper proposes a coupled-PINN (C-PINN), using the sparse measurements and limited prior information of PDEs, to solve PDEs with unknown source terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In our method, two neural networks, NetU and NetG, are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' NetU is applied to generate a quasi-solution satisfying PDEs under study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' NetG is used to regularize the training of NetU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Then, the two networks are integrated into a data-physics- hybrid cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Furthermore, we propose a hierarchical training strategy to optimize and couple the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Finally, the proposed C-PINN is applied to solve several classical PDEs to demonstrate its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The classical PINNs is briefly reviewed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' A C-PINN using the sparse measurements and limited prior knowledge to solve PDEs with unknown source terms is proposed in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Meanwhile, the two neural networks, NetU and NetG, are proposed in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Furthermore, a hierarchical training strategy is proposed to optimize and couple the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In Section IV, our proposed C-PINN is validated with four case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In Section V, the concluding remarks and the future work are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Brief Review of PINNs In this section, we briefly review the basic idea of PINNs for data-driven solutions to PDEs and data-driven discovery of PDEs [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Data-driven solutions to PDEs describe that solve PDEs of the generalized form ut(x, t) + N[u(x, t)] = 0, x ∈ Ω ⊆ Rd, t ∈ [0, T] ⊂ R (1) with known parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Here, x is the spatial variable, t is the temporal variable with t = 0 being at the initial state, u : Rd × R → R denotes the hidden solution, N[·] is a series of partial differential operators, the domain Ω ⊆ Rd is a spatial bounded open set with the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Analytical or numerical methods have been widely investigated to find proper solution ψ(x, t) satisfying (1) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The left-hand-side of (1) can be used to define a residual function as the following f(x, t) := ut(x, t) + N[u(x, t)], (2) where a neural network is used to approximate the solution ψ(x, t) to PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The inverse problem is focused on the data- driven discovery of PDEs of the generalized form (1), where unknown parameters of PDEs here turn into parameters of PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' PINNs for both problems can be trained by minimizing the cost function MSE = MSED + MSEPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (3) Here, MSED is formulated as the following MSED = � (x,t,u)∈D � ˆu � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘU � − u (x, t) �2 , (4) where ˆu � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘU � is the function of neural network with ˆΘU being its trained parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Let D denote the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' This mean squared error term can be considered as the data-driven loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' MSEPH is as the following MSEPH = � (x,t)∈E ˆf (x, t)2 , (5) which regularizes ˆu � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘU � to satisfy (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Let E denote the set of collocation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' This regularization term can be considered as the physics-informed loss for the homogeneous PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Here, ˆf (x, t) is defined as ˆf (x, t) := ˆut � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘU � + N � ˆu � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘU �� , (6) where ˆut � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘU � and N � ˆu � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘU �� can be obtained using automatic differential [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Constructing C-PINN C-PINN for solving PDEs with unknown source terms is presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The nonhomogeneous PDEs are of the following generalized form ut(x, t)+N[u(x, t)] = g(x, t), x ∈ Ω ⊆ Rd, t ∈ [0, T] ⊂ R, (7) where x and t are the spatial and temporal variable, respec- tively, u : Rd ×R → R is similar to (1), g : Rd ×R → R denotes the general types of source terms including linear, nonlinear, state-steady, or dynamical, Ω is a spatial bounded open set with the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Without loss of generality, the spatial set of (7) is subjected to Dirichlet boundary, Neumann boundary, or the hybrid of Dirichlet and Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In general, g(x, t) is used as source terms to describe the external forces for dynamical systems and cannot always be separately measured, as mentioned in Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Different from (6), the residual function is defined for the nonhomogeneous case as the following fN(x, t) := f(x, t)−g(x, t) = ut(x, t)+N[u(x, t)]−g(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (8) When g(x, t) is exactly known, ˆfN(x, t), obtained with auto- matic differential from (8), can be directly used to regularize the approximation of u(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' However, the unknown g(x, t) will lead to unknown fN(x, t), which makes the aforemen- tioned regularization infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Therefore, the goal of C-PINN is to approximate the solu- tion to PDEs with unknown source terms described by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' To this end, there are two neural networks included in C-PINN: (a) NetU for approximating the solution satisfying (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (b) NetG for regularizing the training of NetU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 1) Cost function: To train C-PINN, the training dataset is uniformly sampled from the system governed by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The training dataset D divided into D = DB∪DI with DB∩DI = ∅, where DB denotes the boundary and initial training dataset and DI is the training dataset of interior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Collocation points 3 (x, t) ∈ E correspond to those of (x, t, u) ∈ DI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Then, we adopt the following data-physics-hybrid cost function MSE = MSED + MSEPN (9) to train our proposed C-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' MSED and MSEPN in (9) are the data-driven loss and physics-informed loss for the nonhomogeneous PDEs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' MSED adopts the same form of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' MSEPN is as the following MSEPN = � (x,t)∈E � ˆf (x, t) − ˆg � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘG ��2 , where ˆg � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘG � is the function of NetG with ˆΘG being its trained parameter set, ˆf(x, t) has been defined by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' MSEPN corresponds to the physics-informed loss for the nonhomogeneous PDEs obtained from (8) imposed at a finite set of collocation points (x, t) ∈ E, which is used to regularize ˆu � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘU � of NetU to satisfy (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 2) Hierarchical training strategy: Considering the relation between NetU and NetG in (3), a hierarchical training strategy is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In many cases, the exact formulation or even sparse measurements of g(x, t) are not available, while the sparse measurements DI can be obtained to enforce the structure of (7) to achieve ˆΘG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Thus, ΘU and ΘG should be iteratively estimated with mutual dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Assume k is the present iteration step, the core issue of the hierarchical train- ing strategy is described by the following two optimization problems ˆΘ(k+1) G = arg min ΘG � MSED � ˆΘ(k) U � + MSEPN � ΘG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘ(k) U �� = arg min ΘG MSEPN � ΘG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘ(k) U � (10) and ˆΘ(k+1) U = arg min ΘU � MSED (ΘU) + MSEPN � ΘU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘ(k+1) G �� , (11) where ˆΘ(k) U is the estimated parameter set of NetU at kth step, ˆΘ(k+1) G is the estimated parameter set of NetG at (k + 1)th step, Θ(k+1) U is the estimated parameter set of NetU at (k + 1)th step, which is used to describe the function ˆu � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘ(k+1) U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The details of the hierarchical training strategy are obtained by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Note that Θ(0) U and Θ(0) G are used as a given parameter set for NetU and the initialization of the parameter set for NetG at Step 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Furthermore, the iterative transmission of parameter sets of NetG and NetU happens in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Numerical experiments In this section, our proposed C-PINN is applied to solve several classical PDEs to demonstrate its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' All the examples are implemented with TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The fully connected structure with a hyperbolic tangent activation func- tion is applied, which is initialized by Xavier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' These training dataset (x, t, u) ∈ D and collocation points (x, t) ∈ E are then input into NetU and NetG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' L-BFGS [36] is used to hierarchically solve the optimization problems (10) and (11) to couple the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Algorithm 1 The hierarchical strategy of optimizing and coupling for C-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Initialize: Randomly sampled training dataset (x, t, u) ∈ D and collocation points (x, t) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Randomly generate initial parameter sets Θ(0) U and Θ(0) G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Step 0: Assume the kth iteration has achieved ˆΘ(k) U and ˆΘ(k) G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Repeat: Step k-1: Training for NetG by solving the optimization problem (10) to obtain ˆΘ(k+1) G , where the estimations of ˆut � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘ(k) U � + N � ˆu(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘ(k) U � in MSEPN is obtained from the former iteration result ˆΘ(k) U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Step k-2: Training for NetU by solving the optimization problem (11) to obtain ˆΘ(k+1) U , which is used to estimate ˆg � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Θ(k+1) G � in MSEPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Until the stop criterion is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Return the solution function ˆΘU → ˆu � x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' ˆΘU � , which can predict the solution (8) with any point (x, t) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' We evaluate the performance of our proposed C-PINN by means of root mean squared error (RMSE) RMSE = � 1 |T| � (x,t)∈T (u (x, t) − ˆu (x, t))2, where |T| is the cardinality with respect to the collocation points (x, t) ∈ T, T is the set of testing collocation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' u (x, t) and ˆu (x, t) denote the ground truth and the cor- responding predictions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' To further validate the performance of C-PINN, the Pearson’s correlation coefficient (CC) CC = cov (u (x, t) , ˆu (x, t)) √Var u (x, t) √Var ˆu (x, t) is also used to measure the similarity between ground truth and prediction, where CC is the correlation coefficient of u(x, t) and ˆu(x, t), cov (u (x, t) , ˆu (x, t)) is the covariance between u(x, t) and ˆu(x, t), and Var u (x, t) and Var ˆu (x, t) are variance of u(x, t) and ˆu(x, t), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Case 1: 1-D Heat Equation C-PINN is first applied to solve the heat equation with unknown external forces, where both Dirichlet and Neumann boundary conditions are conducted to demonstrate its perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 1) Dirichlet Boundary Condition Here, we consider the heat equation with Dirichlet boundary condition as the following ∂u ∂t = a2 ∂2u ∂x2 + g(x, t), 0 < x < L, t > 0 u|t=0 = φ(x), 0 ⩽ x ⩽ L u|x=0 = 0, u|x=L = 0, t > 0, (12) where thermal diffusivity a = 1, u(x, t) is the primary variable and means the temperature at (x, t), L = π is the length of bounded rod, φ(x) = 0 is initial temperature, and 4 g(x, t) = xe−t denotes the unknown external heat source at (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The analytical solution u (x, t) to (12) is obtained with respect to [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In this experiment, the setting-ups of C-PINN are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' There are eight hidden layers with 20 units in each of them for both NetU and NetG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' A total of 110 training data (x, t, u(x, t)) in D with t ∈ [0, 6], including 10 training data in DI and 100 training data in DB, are randomly sampled, 10 sparse collocation points are randomly sampled to enforce the structure of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 1 shows the sparse training dataset and the prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Specifically, the magnitude of the predictions ˆu(x, t) using the training dataset is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 1(a) with a heat map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In this case, RMSE is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='225390e−02 and the correlation coefficient is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='785444e − 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Moreover, we compare the ground truths and the predictions at fixed- time t= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 1(b) to (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The evaluation criteria in Table I are applied to further quantify the performance of our proposed C-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 0 1 2 3 4 5 6 t 0 1 2 3 x ��� ������������������uˆ(x, t) ����training data ����� ���training data ������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='7 0 2 x 0 1 u(x, t) ��� t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 0 2 x 0 1 u(x, t) ��� t = 3 0 2 x 0 1 u(x, t) ��� t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 ������������ ���������� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (a) Predictions ˆu (x, t) for the 1-D heat equation with Dirichlet boundary condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (b), (c), and (d) Comparisons of the ground truths and predictions corresponding to the fixed-time t= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 snapshots depicted by the dashed vertical lines in (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' TABLE I Evaluation criteria for the three temporal snapshots depicted by the dashed vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 1-(a) Criteria 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 RMSE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='600305e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='342719e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='991229e-02 CC 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='753408e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='912983e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='805664e-01 Subsequently, the experiment for PDE with Neumann boundary condition will be further explored to show the general performance of C-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 2) Neumann Boundary Condition Heat equation with Neumann boundary condition is defined as ∂u ∂t = a2 ∂2u ∂x2 + g(x, t), 0 < x < L, t > 0 u|t=0 = φ(x), 0 ⩽ x ⩽ L u|x=0 = 0, ∂u ∂x �����x=L = 0, t > 0, (13) with the thermal diffusivity a = 1, the length of bounded rod L = π, the initial temperature φ(x) = sin (x/2), and the external heat source is g(x, t) = sin (x/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The analytical solution u(x, t) to (13) is obtained according to [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In this example, NetU is of three hidden layers consisting of 30 neurons individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' NetG is of eight hidden layers consisting of 20 units individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (x, t, u(x, t)) in D are considered with t ∈ [0, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' A total of 130 training data in DB, including 10 initial training data, 60 left boundary training data, and 60 right boundary training data are randomly sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Moreover, the 20 sparse collocation points are randomly sampled to enforce the structure of (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The magnitude of the predictions ˆu(x, t) using the training dataset is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' RMSE is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='748950e−02 and the correlation coefficient is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='988286e−01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Moreover, we compare the ground truths and the predictions at fixed-time t= 3, 6, and 9 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 2(b) to (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The evaluation criteria in Table II are applied to further evaluate the performance of our proposed C-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 0 2 4 6 8 10 t 0 1 2 3 x ��� ������������������ uˆ(x, t) 130 training data������ 20 training data ������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 0 2 x 0 1 2 3 u(x, t) ��� t = 3 0 2 x 0 1 2 3 u(x, t) ��� t = 6 0 2 x 0 1 2 3 u(x, t) ��� t = 9 ������������� Prediction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (a) Predictions ˆu (x, t) for the 1-D heat equation with Neumann boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (b), (c), and (d) Comparisons of the ground truths and predictions correspond to the fixed-time t= 3, 6, and 9 snapshots depicted by the dashed vertical lines in (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' TABLE II Evaluation criteria for the three temporal snapshots depicted by the dashed vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 2-(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Criteria 3 6 9 RMSE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='343142e-02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='884118e-02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='064205e-02 CC 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='982448e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='990231e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='984719e-01 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Case 2: 1-D Wave Equation The wave equation is as the following ∂2u ∂t2 = a2 ∂2u ∂x2 + g(x, t), 0 < x < L, t > 0 u|x=0 = 0, u|x=L = 0, t > 0 u|t=0 = 0, ∂u ∂t �����t=0 = 0, 0 ⩽ x ⩽ L, (14) 5 where the wave speed a is 1, the length of bounded string L is π, the time of wave propagation t is 6, the external force is g(x, t) = sin 2πx L sin 2aπt L at(x, t) and displacement u(x, t) at (x, t) according to [37] is further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In this experiment, NetU is of three hidden layers consisting of 30 neurons individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' NetG is of eight hidden layers consisting of 20 units individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' A total of 210 training data (x, t, u (x, t)) in D, including 50 initial training data, 120 boundary training data, and 40 collation points are randomly sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 3(a) shows the sparse training dataset and the magnitude of displacement ˆu(x, t) at (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 3(b) to (d) show the comparisons of ground truths and predictions corre- sponding to the three fixed-time t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5, which are depicted by the dashed vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 3(a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' RMSE is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='068626e − 02 and the correlation coefficient is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='864411e− 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The evaluation criteria for the three temporal snapshots are listed in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 0 1 2 3 4 5 6 � 0 1 2 3 x (�� ����������������� ˆu(x, t) 220 training data ����� 40 training data ����� −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0 2 x −1 0 1 u(x, t) �b� ������� 0 2 x −1 0 1 u(x, t) ��� ����� 0 2 x −1 0 1 u(x, t) ��� ������� ������������� ���������� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (a) Predictions ˆu (x, t) for 1-D wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (b), (c), and (d) Comparisons of the ground truths and predictions corresponding to the fixed- time t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 snapshots depicted by the dashed vertical lines in (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' TABLE III Evaluation criteria for the three temporal snapshots depicted by the dashed vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 3-(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Criteria 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 RMSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='424030e-01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='305190e-02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='201132e-02 CC 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6238994e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='985312e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='983170e-01 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Case 3: 2-D Poisson Equation We further consider the following 2-D Poisson equation ∂2u ∂x2 + ∂2u ∂y2 = T0, 0 < x < a, 0 < y < b u(x, 0) = 0, u(x, b) = T, 0 ⩽ x ⩽ a u(0, y) = 0, u(a, y) = 0, 0 ⩽ y ⩽ b, (15) where T is 1, the constant source term T0 = 1 is unknown, and a = b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The analytical solution u(x, y) to (15) is obtained according to [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In this experiment, the setting-ups of C- PINN are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' There are eight hidden layers with 20 units in each of them for both NetU and NetG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Thirty training data in DB and 3 collocation points in DI are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 4(a) shows the sparse training dataset and the predictions ˆu(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 4(b) to (d) show the prediction performance of fixed- location y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='4, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6 snapshots depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 4(a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' RMSE is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='594000e − 02 and the correlation coefficient is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='997390e − 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The corresponding evaluation criteria are listed in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='00 x ��� ���������������������uˆ(x, y) 30 training data ������ 3 training data ����� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 0 1 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 u(x, y) 0 1 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 u(x, y) 0 1 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 u(x, y) (��� ����� ������������� ���������� y y (c�� ����4 y (b�� ����2 y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (a) Predictions ˆu (x, y) for the 2-D Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (b), (c), and, (d) Comparisons of the ground truths and predictions corresponding to the fixed-location y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='4, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6 snapshots depicted by the dashed vertical lines in (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' TABLE IV Evaluation criteria for the three fixed-location snapshots depicted by the dashed vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 4-(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Criteria 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6 RMSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='763408e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='139888e-02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='696680e-03 CC 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='986055e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='999703e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='999656e-01 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Case 4: 3-D Helmholtz Equation C-PINN is also applied to solve 3-D Helmholtz equation with an unknown source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In particular, we consider the same test PDEs that were previously suggested in [26] ∆u(x) + p2u(x) = g(x) in Ω ⊂ R3 u(x) = u0(x) on ∂Ω, (16) where ∆ = ∂ ∂x2 + ∂ ∂y2 + ∂ ∂z2 is Laplacian operator, x = (x, y, z)⊤ is coordinates with x, y, z ∈ (0, 1/4] , p = 5 is the wavenumber, 6 a suitable g(x) is the right-hand side of (16) so that u(x) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='1 sin (2πx) + tanh (10x)) sin (2πy)sin (2πz) is the analytical solution of (16) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In this experiment, NetU is of three hidden layers consisting of 100, 50, and 50 neurons individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' NetG is of eight hidden layers con- sisting of 20 units individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Sixty training data and 120 collocation points are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 5(a) shows the solution of (x, y, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='12) snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 5(b) to (d) show the comparisons of ground truths and predictions, which are extracted at (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='05, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='12), (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='15, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='12), and (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='12), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' The evaluation criteria for this extractions are listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In this experiment, RMSE is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='192859e − 01, and the correlation coefficient is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='057524e − 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 y (a) 3−D Helmholtz Equation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6 u(x, y) �b� x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6 u(x, y) ��� x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='6 u(x, y) ��� x =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 ������������� Prediction �uˆ(x,�y,�z�=�25 3�) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (a) Predictions ˆu (x, y, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='12) for 3-D Helmholtz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' (b), (c) and, (d) Comparisons of the ground truths and predictions corresponding to the (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='05, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='12), (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='15, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='12), and (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='20, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='12) snapshots depicted by the dashed vertical lines in (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' TABLE V Evaluation criteria for the three snapshots depicted by the dashed vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' 5-(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Criteria 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='2 RMSE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='043735e-02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='548533e-02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='179414e-02 CC 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='604538e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='998589e-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content='964517e-01 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Conclusion This paper proposes a novel PINN, called C-PINN, to solve PDEs with less prior information or even without any prior information for source terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In our approach, two neural net- works, NetU and NetG, are proposed with a fully-connected structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' NetU for approximating the solution satisfying PDEs under study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' NetG for regularizing the training of NetU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Then, the two networks are integrated into a data-physics- hybrid cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Furthermore, the two networks are op- timized and coupled by the proposed hierarchical training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Finally, C-PINN is applied to solve several classical PDEs to testify to its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Note that C-PINN inherits the advantages of PINN, such as sparse property and automatic differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' C-PINN is proposed to solve such a dilemma as the governing equation of dynamical systems with unknown forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Thus, C-PINN can be further applied to infer the unknown source terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Meanwhile, C-PINN can be extended to identify the operators from the sparse measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' In the future, we will continue to use our C-PINN in various scenarios, like solving PDEs with unknown struc- ture parameters and high-dimension PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' For the case, the structures of PDE are totally unknown, regularization method will be combined with C-PINN to select operators from the sparse measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' Our proposed C-PINN has been shown to solve several classical PDEs successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' For more complex situations, the features extraction, like convolution and pooling, will be added to C-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FAT4oBgHgl3EQflh2J/content/2301.08618v1.pdf'} +page_content=' References [1] H.' metadata={'source': 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b/3dFST4oBgHgl3EQfYzh6/content/tmp_files/2301.13789v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4ff85bde1c0da3177fc2a486869676cc677c00c --- /dev/null +++ b/3dFST4oBgHgl3EQfYzh6/content/tmp_files/2301.13789v1.pdf.txt @@ -0,0 +1,1063 @@ +arXiv:2301.13789v1 [math.CO] 31 Jan 2023 +The Minimum Degree Removal Lemma Thresholds +Lior Gishboliner∗ +Zhihan Jin∗ +Benny Sudakov∗ +Abstract +The graph removal lemma is a fundamental result in extremal graph theory which says that for every +fixed graph H and ε > 0, if an n-vertex graph G contains εn2 edge-disjoint copies of H then G contains +δnv(H) copies of H for some δ = δ(ε, H) > 0. The current proofs of the removal lemma give only very +weak bounds on δ(ε, H), and it is also known that δ(ε, H) is not polynomial in ε unless H is bipartite. +Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that δ(ε, H) +depends polynomially or linearly on ε. In this paper we answer several questions of Fox and Wigderson +on this topic. +1 +Introduction +The graph removal lemma, first proved by Ruzsa and Szemerédi [23], is a fundamental result in extremal +graph theory. It also have important applications to additive combinatorics and property testing. The lemma +states that for every fixed graph H and ε > 0, if an n-vertex graph G contains εn2 edge-disjoint copies of H +then G it contains δnv(H) copies of H, where δ = δ(ε, H) > 0. Unfortunately, the current proofs of the graph +removal lemma give only very weak bounds on δ = δ(ε, H) and it is a very important problem to understand +the dependence of δ on ε. The best known result, due to Fox [11], proves that 1/δ is at most a tower of +exponents of height logarithmic in 1/ε. Ideally, one would like to have better bounds on 1/δ, where an +optimal bound would be that δ is polynomial in ε. However, it is known [2] that δ(ε, H) is only polynomial +in ε if H is bipartite. This situation led Fox and Wigderson [12] to initiate the study of minimum degree +conditions which guarantee that δ(ε, H) depends polynomially or linearly on ε. Formally, let δ(ε, H; γ) be +the maximum δ ∈ [0, 1] such that if G is an n-vertex graph with minimum degree at least γn and with εn2 +edge-disjoint copies of H, then G contains δnv(H) copies of H. +Definition 1.1. Let H be a graph. +1. The linear removal threshold of H, denoted δlin-rem(H), is the infimum γ such that δ(ε, H; γ) depends +linearly on ε, i.e. δ(ε, H; γ) ≥ µε for some µ = µ(γ) > 0 and all ε > 0. +2. The polynomial removal threshold of H, denoted δpoly-rem(H), is the infimum γ such that δ(ε, H; γ) +depends polynomially on ε, i.e. δ(ε, H; γ) ≥ µε1/µ for some µ = µ(γ) > 0 and all ε > 0. +Trivially, δlin-rem(H) ≥ δpoly-rem(H). +Fox and Wigderson [12] initiated the study of δlin-rem(H) and +δpoly-rem(H), and proved that δlin-rem(Kr) = δpoly-rem(Kr) = 2r−5 +2r−3 for every r ≥ 3, where Kr is the clique on +r vertices. They further asked to determine the removal lemma thresholds of odd cycles. Here we completely +resolve this question. The following theorem handles the polynomial removal threshold. +Theorem 1.2. δpoly-rem(C2k+1) = +1 +2k+1. +Theorem 1.2 also answers another question of Fox and Wigderson [12], of whether δlin-rem(H) and +δpoly-rem(H) can only obtain finitely many values on r-chromatic graphs H for a given r ≥ 3. Theorem 1.2 +shows that δpoly-rem(H) obtains infinitely many values for 3-chromatic graphs. In contrast, δlin-rem(H) ob- +tains only three possible values for 3-chromatic graphs. Indeed, the following theorem determines δlin-rem(H) +for every 3-chromatic H. An edge xy of H is called critical if χ(H − xy) < χ(H). +∗Department of Mathematics, ETH, Zürich, Switzerland. Research supported in part by SNSF grant 200021_196965. Email: +{lior.gishboliner, zhihan.jin, benjamin.sudakov}@math.ethz.ch. +1 + +Theorem 1.3. For a graph H with χ(H) = 3, it holds that +δlin-rem(H) = + + + + + +1 +2 +H has no critical edge, +1 +3 +H has a critical edge and contains a triangle, +1 +4 +H has a critical edge and odd-girth(H) ≥ 5. +Theorems 1.2 and 1.3 show a separation between the polynomial and linear removal thresholds, giving a +sequence of graphs (i.e. C5, C7, . . . ) where the polynomial threshold tends to 0 while the linear threshold is +constant 1 +4. +The parameters δpoly-rem and δlin-rem are related to two other well-studied minimum degree thresholds: +the chromatic threshold and the homomorphism threshold. The chromatic threshold of a graph H is the +infimum γ such that every n-vertex H-free graph G with δ(G) ≥ γn has bounded cromatic number, i.e., +there exists C = C(γ) such that χ(G) ≤ C. The study of the chromatic threshold originates in the work of +Erdős and Simonovits [10] from the ’70s. Following multiple works [4, 15, 16, 7, 5, 25, 26, 19, 6, 14, 20], the +chromatic threshold of every graph was determined by Allen et al. [1]. +Moving on to the homomorphism threshold, we define it more generally for families of graphs. The +homomorphism threshold of a graph-family H, denoted δhom(H), is the infimum γ for which there exists an +H-free graph F = F(γ) such that every n-vertex H-free graph G with δ(G) ≥ γn is homomorphic to F. +When H = {H}, we write δhom(H). This parameter was widely studied in recent years [18, 22, 17, 8, 24]. +It turns out that δhom is closely related to δpoly-rem(H), as the following theorem shows. For a graph H, let +IH denote the set of all minimal (with respect to inclusion) graphs H′ such that H is homomorphic to H′. +Theorem 1.4. For every graph H, δpoly-rem(H) ≤ δhom(IH). +Note that IC2k+1 = {C3, . . . , C2k+1}. Using this, the upper bound in Theorem 1.2 follows immediately +by combining Theorem 1.4 with the result of Ebsen and Schacht [8] that δhom({C3, . . . , C2k+1}) = +1 +2k+1. +The lower bound in Theorem 1.2 was established in [12]; for completeness, we sketch the proof in Section 3. +The rest of this short paper is organized as follows. Section 2 contains some preliminary lemmas. In +Section 3 we prove the lower bounds in Theorems 1.2 and 1.3. Section 4 gives the proof of Theorem 1.4, and +Section 5 gives the proof of the upper bounds in Theorem 1.3. In the last section we discuss further related +problems. +2 +Preliminaries +Throughout this paper, we always consider labeled copies of some fixed graph H and write copy of H for +simplicity. We use δ(G) for the minimum degree of G, and write H → F to denote that there is a homo- +morphism from H to F. For a graph H on [h] and integers s1, s2, . . . , sh > 0, we denote by H[s1, . . . , sh] +the blow-up of H where each vertex i ∈ V (H) is replaced by a set Si of size si (and edges are replaced with +complete bipartite graphs). The following lemma is standard. +Lemma 2.1. Let H be a fixed graph on vertex set [h] and let s1, s2, . . . , sh ∈ N. There exists a constant +c = c(H, s1, . . . , sh) > 0 such that the following holds. Let G be an n-vertex graph and V1, . . . , Vh ⊆ V (G). +Suppose that G contains at least ρnh copies of H mapping i to Vi for all i ∈ [h]. Then G contains at least +cρ +1 +c · ns1+···+sh copies of H[s1, . . . , sh] mapping Si to Vi for all i ∈ [h]. +Note that the sets V1, . . . , Vh in Lemma 2.1 do not have to be disjoint. The proof of Lemma 2.1 works +by defining an auxiliary h-uniform hypergraph G whose hyperedges correspond to the copies of H in which +vertex i is mapped to Vi. By assumption, G has at least ρnh edges. By the hypergraph generalization of the +Koväri-Sós-Turán theorem, see [9], G contains poly(ρ)ns1+···+sh copies of K(h) +s1,...,sh, the complete h-partite +hypergraph with parts of size s1, . . . , sh. Each copy of K(h) +s1,...,sh gives a copy of H[s1, . . . , sh] mapping Si to Vi. +Fox and Wigderson [12, Proposition 4.1] proved the following useful fact. +Lemma 2.2. If H → F and F is a subgraph of H, then δpoly-rem(H) = δpoly-rem(F). +2 + +The following lemma is an asymmetric removal-type statement for odd cycles, which gives polynomial +bounds. It may be of independent interest. A similar result has appeared very recently in [13]. +Lemma 2.3. For 1 ≤ ℓ < k, there exists a constant c = c(k) > 0 such that if an n-vertex graph G has εn2 +edge-disjoint copies of C2ℓ+1, then it has at least cε1/cn2k+1 copies of C2k+1. +Proof. Let C be a collection of εn2 edge-disjoint copies of C2ℓ+1 in G. There exists a collection C′ ⊆ C such +that |C′| ≥ εn2/2 and each vertex v ∈ V (G) belongs to either 0 or at least εn/2 of the cycles in C′. Indeed, +to obtain C′, we repeatedly delete from C all cycles containing a vertex v which belongs to at least one but +less than εn/2 of the cycles in C (without changing the graph). The set of cycles left at the end is C′. In +this process, we delete at most εn2/2 cycles altogether (because the process lasts for at most n steps); hence +|C′| ≥ εn2/2. Let V be the set of vertices contained in at least εn/2 cycles from C′, so |V | ≥ εn/2. With +a slight abuse of notation, we may replace G with G[V ], C with C′ and ε/2 with ε, and denote |V | by n. +Hence, from now on, we assume that each vertex v ∈ V (G) is contained in at least εn of the cycles in C. +This implies that |N(v)| ≥ 2εn for every v ∈ V (G). +Fix any v0 ∈ V (G) and let C(v0) be the set of cycles C ∈ C such that C ∩ N(v0) ̸= ∅ and v0 /∈ C. +The number of cycles C ∈ C intersecting N(v0) is at least |N(v0)| · εn/(2ℓ + 1) ≥ 2ε2n2/(2ℓ + 1), and the +number of cycles containing v0 is at most n. Hence, |C(v0)| ≥ 2ε2n2/(2ℓ + 1) − n ≥ ε2n2/(ℓ + 1). Take +a random partition V0, V1, . . . , Vℓ of V (G) \ {v0}, where each vertex is put in one of the parts uniformly +and independently. For a cycle (x1, . . . , x2ℓ+1) ∈ C(v0) with xℓ+1 ∈ N(v0), say that (x1, . . . , x2ℓ+1) is good +if xℓ+1 ∈ V0 and xℓ+1−i, xℓ+1+i ∈ Vi for 1 ≤ i ≤ ℓ (so in particular x1, x2ℓ+1 ∈ Vℓ). +The probability +that (x1, . . . , x2ℓ+1) is good is 1/(ℓ + 1)2ℓ+1, so there is a collection of good cycles C′(v0) ⊆ C0 of size +|C′(v0)| ≥ |C(v0)|/(ℓ + 1)2ℓ+1 ≥ ε2n2/(ℓ + 1)2ℓ+2. +Put γ := ε2/(ℓ + 1)2ℓ+2. +By the same argument as +above, there is a collection C′′(v0) ⊆ C′(v0) with |C′′(v0)| ≥ γn2/2 such that each vertex is contained in +either 0 or at least γn/2 cycles from C′′(v0). Let W be the set of vertices contained in at least γn/2 cycles +from C′′(v0). Note that W ∩ V0 ⊆ N(v0) by definition. Also, each vertex in W ∩ Vℓ has at least γn/2 +neighbors in W ∩ Vℓ, and for each 1 ≤ i ≤ ℓ, each vertex in W ∩ Vi has at least γn/2 neighbors in W ∩ Vi−1. +It follows that W ∩ Vℓ contains at least 1 +2|W ∩ Vℓ| · �2k−2ℓ−2 +i=0 +(γn/2 − i) = poly(γ)n2k−2ℓ paths of length +2k − 2ℓ − 1. We now construct a collection of copies of C2k+1 as follows. Choose a path yℓ+1, yℓ+2, . . . , y2k−ℓ +of length 2k − 2ℓ − 1 in W ∩ Vℓ. +For each i = ℓ, . . . , 1, take a neighbor yi ∈ W ∩ Vi−1 of yi+1 and a +neighbor y2k−i+1 ∈ W ∩ Vi−1 of y2k−i, such that the vertices y1, . . . , y2k are all different. Then y1, . . . , y2k +is a path and y1, y2k ∈ W ∩ V0 ⊆ N(v0), so v0, y1, . . . , y2k is a copy of C2ℓ+1. The number of choices for +the path yℓ+1, yℓ+2, . . . , y2k−ℓ is poly(γ)n2k−2ℓ and the number of choices for each vertex yi, y2k−i+1 ∈ Vi−1 +(i = ℓ, . . . , 1) is at least γn/2. Hence, the total number of choices for y1, . . . , y2k is poly(γ)n2k. As there are +n choices for v0, we get a total of poly(γ)n2k+1 = polyk(ε)n2k+1 copies of C2k+1, as required. +3 +Lower bounds +Here we prove the lower bounds in Theorems 1.2 and 1.3. The lower bound in Theorem 1.2 was proved in +[12, Theorem 4.3]. For completeness, we include a sketch of the proof: +Lemma 3.1. δpoly-rem(C2k+1) ≥ +1 +2k+1. +Proof. Fix an arbitrary α > 0. In [2] it was proved that for every ε, there exists a (2k + 1)-partite graph +with parts V1, . . . , V2k+1 of size αn/(2k + 1) each, with εn2 edge-disjoint copies of C2k+1, but with only +εω(1)n2k+1 copies of C2k+1 in total (where the ω(1) term may depend on α). Add sets U1, . . . , U2k+1 of size +(1 − α)n/(2k + 1) each, and add the complete bipartite graphs (Ui, Vi), 1 ≤ i ≤ 2k + 1, and (Ui, Ui+1), +1 ≤ i ≤ 2k. See Figure 1. It is easy to see that this graph has minimum degree (1 − α)n/(2k + 1), and every +copy of C2k+1 is contained in V1 ∪ · · · ∪ V2k+1. Letting α → 0, we get that δpoly-rem(C2k+1) ≥ +1 +2k+1. +By combining the fact that δpoly-rem(C3) = 1 +3 with Lemma 2.2 (with F = C3), we get that δlin-rem(H) ≥ +δpoly-rem(H) = 1 +3 for every 3-chromatic graph H containing a triangle. This proves the lower bound in the +second case of Theorem 1.3. Now we prove the lower bounds in the other two cases. We prove a more general +statement for r-chromatic graphs. +3 + +V2 +V3 +V4 +V5 +V1 +U2 +U3 +U4 +U5 +U1 +Figure 1: Proof of Lemma 3.1 for C5. Heavy edges indicate complete bipartite graphs while dashed edges +form the Ruzsa–Szemerédi construction for C5 (see [2]). +Lemma 3.2. Let H be a graph with χ(H) = r ≥ 3. +Then, +3r−8 +3r−5 ≤ δlin-rem(H) ≤ +r−2 +r−1. +Moreover, +δlin-rem(H) = r−2 +r−1 if H contains no critical edge. +Proof. Denote h = |V (H)|. The bound δlin-rem(H) ≤ r−2 +r−1 holds for every r-chromatic graph H; this follows +from the Erdős-Simonovits supersaturation theorem, see by [12, Section 4.1] for the details. +Suppose now that H contains no critical edge, and let us show that δlin-rem(H) ≥ r−2 +r−1. To this end, we +construct, for every small enough ε and infinitely many n, an n-vertex graph G with δ(G) ≥ r−2 +r−1n, such that +G has at most O(ε2nh) copies of H, but Ω(εn2) edges must be deleted to turn G into an H-free graph. Let +T (n, r − 1) be the Turán graph, i.e. the complete (r − 1)-partite graph with balanced parts V1, . . . , Vr−1. +Add an εn-regular graph inside V1 and let the resulting graph be G. We first claim that G contains O(ε2nh) +copies of H. As H contains no critical edge and χ(H) = r, every copy of H in G contains two edges e and +e′ inside V1. If e and e′ are disjoint, then there are at most n2(εn)2 = ε2n4 choices for e and e′ and then at +most nh−4 choices for the other h− 4 vertices of H. Therefore, there are at most ε2nh such H-copies. And if +e and e′ intersect, then there are at most n(εn)2 = ε2n3 choices for e and e′ and then at most nh−3 choices +for the remaining vertices, again giving at most ε2nh such H-copies. So G indeed has O(ε2nh) copies of H. +On the other hand, we claim that one must delete Ω(εn2) edges to destroy all H-copies in G. Observe +that G has at least 1 +2 |V1|·εn·|V2|·· · ··|Vr−1| = Ωr(εnr) copies of Kr, and every edge participates in at most +nr−2 of these copies. Thus, deleting cεn2 edges can destroy at most cεnr copies of Kr. If c is a small enough +constant (depending on r), then after deleting any cεn2 edges, there are still Ω(εnr) copies of Kr. Then, +by Lemma 2.1, the remaining graph contains Kr[h], the h-blowup of Kr, and hence H. This completes the +proof that δlin-rem(H) ≥ r−2 +r−1. +We now prove that δlin-rem(H) ≥ 3r−8 +3r−5 for every r-chromatic graph H. It suffices to construct, for every +small enough ε and infinitely many n, an n-vertex graph G with δ(G) ≥ 3r−8 +3r−5n, such that G has at most +O(ε2nh) copies of H but at least Ω(εn2) edges must be deleted to turn G into an H-free graph. The vertex +set of G consists of r + 1 disjoint sets V0, V1, V2, . . . , Vr, where |Vi| = +n +3r−5 for i = 0, 1, 2, 3 and |Vi| = +3n +3r−5 +for i = 4, 5, . . ., r. Put complete bipartite graphs between V0 and V1, between V0 ∪ V1 and V4 ∪ · · · ∪ Vr, and +between Vi to Vj for all 2 ≤ i < j ≤ r. Put εn-regular bipartite graphs between V1 and V2, and between V1 +and V3. The resulting graph is G (see Figure 2). It is easy check that δ(G) ≥ 3r−8 +3r−5n. Indeed, let 0 ≤ i ≤ r +and v ∈ Vi. If 4 ≤ i ≤ r then v is connected to all vertices except for Vi; if i ∈ {2, 3} then v is connected to +all vertices except V0 ∪ V1 ∪ Vi; and if i ∈ {0, 1} then v is connected to all vertices except V2 ∪ V3 ∪ Vi. In +any case, the neighborhood of v misses at most +3n +3r−5 vertices. +We claim that G has at most O(ε2nh) copies of H. Indeed, observe that if we delete all edges between V1 +and V2 then the remaining graph is (r − 1)-colorable with coloring V1 ∪ V2, V0 ∪ V3, V4, . . . , Vr. Hence, every +copy of H must contain an edge e between V1 and V2. Similarly, every copy of H must contain an edge e′ +between V1 and V3. If e, e′ are disjoint then there are at most n2(εn)2 = ε2n4 ways to choose e, e′ and then +at most nh−4 ways to choose the remaining vertices of H. And if e and e′ intersect then there are at most +n(εn)2 = ε2n3 ways to choose e, e′ and at most nh−3 for the remaining h − 3 vertices of H. In both cases, +the number of H-copies is at most ε2nh, as required. +Now we show that one must delete Ω(εn2) edges to destroy all copies of H in G. Observe that G has +|V1| · (εn)2 · |V4| · · · · · |Vr| = Ω(ε2nr) copies of Kr between the sets V1, . . . , Vr. We claim that every edge f +4 + +V1 +V2 +V3 +V0 +V1 +V2 +V3 +V4 +V0 +Figure 2: Proof of Lemma 3.2, r = 3 (left) and r = 4 (right). Heavy edges indicate complete bipartite graphs +while dashed edges indicate εn-regular bipartite graphs. +participates in at most εnr−2 of these r-cliques. Indeed, by the same argument as above, every copy of Kr +containing f must contain an edge e from E(V1, V2) and an edge e′ from E(V1, V3). Suppose without loss of +generality that e ̸= f (the case e′ ̸= f is symmetric). In the case f ∩ e = ∅, there are at most n · εn = εn2 +choices for e and at most nr−4 choices for the remaining vertices of Kr, giving at most εnr−2 copies of Kr +containing f. And if f, e intersect, then there are at most εn choices for e and at most nr−3 for the remaining +r − 3 vertices, giving again εnr−2. +We see that deleting cεn2 edges of G can destroy at most cε2nr copies of Kr. Hence, if c is a small +enough constant, then after deleting any cεn2 edges there are still Ω(ε2nr) copies of Kr left. By Lemma 2.1, +the remaining graph contains a copy of Kr[h] and hence H. This completes the proof. +4 +Polynomial removal thresholds: Proof of Theorem 1.4 +We say that an n-vertex graph G is ε-far from a graph property P (e.g. being H-free for a given graph H, or +being homomorphic to a given graph F) if one must delete at least εn2 edges to make G satisfy P. Trivially, +if G has εn2 edge-disjoint copies of H, then it is ε-far from being H-free. We need the following result from +[21]. +Theorem 4.1. For every graph F on f vertices and for every ε > 0, there is q = qF (ε) = poly(f/ε), such +that the following holds. If a graph G is ε-far from being homomorphic to F, then for a sample of q vertices +x1, . . . , xq ∈ V (G), taken uniformly with repetitions, it holds that G[{x1, . . . , xq}] is not homomorphic to F +with probability at least 2 +3. +Theorem 4.1 is proved in Section 2 of [21]. +In fact, [21] proves a more general result on property +testing of the so-called 0/1-partition properties. Such a property is given by an integer f and a function +d : [f]2 → {0, 1, ⊥}, and a graph G satisfies the property if it has a partition V (G) = V1 ∪ · · · ∪ Vf such +that for every 1 ≤ i, j ≤ f (possibly i = j), it holds that (Vi, Vj) is complete if d(i, j) = 1 and (Vi, Vj) is +empty if d(i, j) = 0 (if d(i, j) =⊥ then there are no restrictions). One can express the property of having a +homomorphism into F in this language, simply by setting d(i, j) = 0 for i = j and ij /∈ E(F). In [21], the +class of these partition properties is denoted GPP0,1, and every such property is shown to be testable by +sampling poly(f/ε) vertices. This implies Theorem 4.1. +Proof of Theorem 1.4. Recall that IH is the set of minimal graphs H′ (with respect to inclusion) such that +H is homomorphic to H′. For convenience, put δ := δhom(IH). Our goal is to show that δpoly-rem(H) ≤ δ+α +for every α > 0. So fix α > 0 and let G be a graph with minimum degree δ(G) ≥ (δ + α)n and with +εn2 edge-disjoint copies of H. By the definition of the homomorphism threshold, there is an IH-free graph +F (depending only on IH and α) such that if a graph G0 is IH-free and has minimum degree at least +(δ + α +2 ) · |V (G0)|, then G0 is homomorphic to F. Observe that if a graph G0 is homomorphic to F then +G0 is H-free, because F is free of any homomorphic image of H. It follows that G is ε-far from being +homomorphic to F, because G is ε-far from being H-free. Now we apply Theorem 4.1. Let q = qF (ε) be +given by Theorem 4.1. We assume that q ≫ log(1/α) +α2 +and n ≫ q2 without loss of generality. Sample q vertices +x1, . . . , xq ∈ V (G) with repetition and let X = {x1, . . . , xq}. By Theorem 4.1, G[X] is not homomorphic to +F with probability at least 2/3. As n ≫ q2, the vertices x1, . . . , xq are pairwise-distinct with probability at +least 0.99. Also, for every i ∈ [q], the number of indices j ∈ [q] \ {i} with xixj ∈ E(G) dominates a binomial +5 + +distribution B(q − 1, δ(G) +n ). By the Chernoff bound (see e.g. [3, Appendix A]) and as δ(G) ≥ (δ + α)n, +the number of such indices is at least (δ + α +2 )q with probability 1 − e−Ω(qα2). Taking the union bound over +i ∈ [q], we get that δ(G[X]) ≥ (δ + α +2 )|X| with probability at least 1 − qe−Ω(qα2) ≥ 0.9, as q ≫ log(1/α) +α2 +. +Hence, with probability at least 1 +2 it holds that δ(G[X]) ≥ (δ + α +2 )|X| and G[X] is not homomorphic to F. If +this happens, then G[X] is not IH-free (by the choice of F), hence G[X] contains a copy of some H′ ∈ IH. +By averaging, there is H′ ∈ IH such that G[X] contains a copy of H′ with probability at least +1 +2|IH|. Put +k = |V (H′)| and let M be the number of copies of H′ in G. The probability that G[X] contains a copy of H′ +is at most M( q +n)k. Using the fact that q = polyH,α( 1 +ε), we conclude that M ≥ +1 +2|IH| · ( n +q )k ≥ polyH,α(ε)nk. +As H → H′, there exists H′′, a blow-up of H′, such that H′′ have the same number of vertices as H, and +that H ⊂ H′′. By Lemma 2.1 for H′ with Vi = V (G) for all i, there exist polyH,α(ε)nv(H′′) copies of H′′ in +G, and thus polyH,α(ε)nv(H) copies of H. This completes the proof. +5 +Linear removal thresholds: Proof of Theorem 1.3 +Here we prove the upper bounds in Theorem 1.3; the lower bounds were proved in Section 3. The first case +of Theorem 1.3 follows from Lemma 3.2, so it remains to prove the other two cases. We begin with some +preparation. For disjoint sets A1, . . . , Am, we write � +i∈[m] Ai × Ai+1 to denote all pairs of vertices which +have one endpoint in Ai and one in Ai+1 for some 1 ≤ i ≤ m, with subscripts always taken modulo m. So a +graph G has a homomorphism to the cycle Cm if and only if there is a partition V (G) = A1 ∪ · · · ∪ Am with +E(G) ⊆ � +i∈[m] Ai × Ai+1. +Lemma 5.1. Suppose H is a graph such that χ(H) = 3, H contains a critical edge xy, and odd-girth(H) ≥ +2k + 1. Then, +• There is a partition V (H) = A1 ·∪ A2 ·∪ A3 ·∪ B such that A1 = {x}, A2 = {y} and E(H) ⊆ (A3 × B) ∪ +(� +i∈[3] Ai × Ai+1); +• if k ≥ 2, there is a partition V (H) = A1 ·∪ A2 ·∪ · · · ·∪ A2k+1 such that A1 = {x}, A2 = {y} and +E(H) ⊆ � +i∈[2k+1] Ai × Ai+1. In particular, H is homomorphic to C2k+1. +Proof. Write H′ = H − xy, so H′ is bipartite. Let V (H) = V (H′) = L ·∪ R be a bipartition of H′. As +χ(H) = 3, x and y must both lie in the same side of the bipartition. Without loss of generality, assume that +x, y ∈ L. For the first item, set A1 = {x}, A2 = {y}, A3 = R and B = L\{x, y}. Then every edge of G goes +between B and A3 or between two of the sets A1, A2, A3, as required. +Suppose now that k ≥ 2, i.e. odd-girth(H) = 2k + 1 ≥ 5. For 1 ≤ i ≤ k, let Xi be the set of vertices at +distance (i − 1) from x in H′, and let Yi be the set of vertices at distance (i − 1) from y in H′. Note that +X1 = {x} and Y1 = {y}. Also, Xi, Yi lie in L if i is odd and in R if i is even. Write +L′ := L\ +k� +i=1 +(Xi ∪ Yi), +R′ := R\ +k� +i=1 +(Xi ∪ Yi), +We first claim that {X1, . . . , Xk, Y1, . . . , Yk, L′, R′} forms a partition of V (H). +The sets X1, . . . , Xk are +clearly pairwise-disjoint, and so are Y1, . . . , Yk. Also, all of these sets are disjoint from L′, R′ by definition. +So we only need to check Xi and Yj are disjoint for every pair 1 ≤ i, j ≤ k. Suppose for contradiction that +there exists u ∈ Xi ∩ Yj for some 1 ≤ i, j ≤ k. Then i ≡ j (mod 2), because otherwise Xi, Yj are contained +in different parts of the bipartition L, R. By the definition of Xi and Yj, H′ has a path x = x1, x2, . . . , xi = u +and a path y = y1, y2, . . . , yj = u. Then, x = x1, x2, . . . , xi = u = yj, yj−1, . . . , y1, y, x forms a closed walk of +length i+j −1, which is odd as i ≡ j (mod 2). Hence, odd-girth(H) ≤ 2k−1, contradicting our assumption. +By definition, there are no edges between Xi and Xj for j − i ≥ 2, and similarly for Yi, Yj. Also, there +are no edges between L′ ∪ R′ and �k−1 +i=1 (Xi ∪ Yi) because the vertices in L′ ∪ R′ are at distance more than k +to x, y. Moreover, if k is even then there are no edges between Xk ∪ Yk and R′, and if k is odd then there +are no edges between Xk ∪ Yk and L′. Next, we show that there are no edges between Xi and Yj for any +1 ≤ i, j ≤ k except (i, j) = (1, 1). Indeed, if i = j then e(Xi, Yj) = 0 because Xi, Yj are on the same side +6 + +x +y +X2 +Y2 +L′ +R′ +x +y +X2 +Y2 +X3 +Y3 +R′ +L′ +Figure 3: Proof of Lemma 5.1, k = 2 (left) and k = 3 (right). Edges indicate bipartite graphs where edges +can be present. +of the bipartition L, R. So suppose that i ̸= j, say i < j, and assume by contradiction that there is an +edge uv with u ∈ Xi, v ∈ Yj. Then v is at distance at most i + 1 ≤ k from x, implying that Yj intersects +X1 ∪ · · · ∪ Xi+1, a contradiction. +Finally, we define the partition A1, . . . , A2k+1 that satisfies the assertion of the second item. If k is even +then take A1, . . . , A2k+1 to be X1, Y1, . . . , Yk−1, Yk∪R′, L′, Xk, . . . , X2, and if k is odd then take A1, . . . , A2k+1 +to be X1, Y1, . . . , Yk−1, Yk ∪ L′, R′, Xk, . . . , X2. See Figure 3 for an illustration. By the above, in both cases +it holds that E(H) ⊆ � +i∈[2k+1] Ai × Ai+1, as required. +For vertex u ∈ V (G), denote by NG(u) the neighborhood of u and let degG(u) = |NG(u)|. For vertices +u, v ∈ V (G), denote by NG(u, v) the common neighborhood of u, v and let degG(u, v) = |NG(u, v)|. +Lemma 5.2. Let H be a graph on h vertices such that χ(H) = 3 and H contains a critical edge xy. Let G +be a graph on n vertices with δ(G) ≥ αn. Let ab ∈ E(G) such that degG(a, b) ≥ αn. Then, there are at least +poly(α)nh−2 copies of H in G mapping xy ∈ E(H) to ab ∈ E(G). +Proof. By the first item of Lemma 5.1, there is a partition V (H) = A1 ·∪ A2 ·∪ A3 ·∪ B such that A1 = +{x}, A2 = {y} and E(H) ⊆ (A3 × B) ∪ � +i∈[3] Ai × Ai+1. Let s = |A3| and t = |B|. Each u ∈ NG(a, b) has at +least αn − 2 ≥ αn +2 neighbors not equal to a, b. Hence, there are at least 1 +2 · |NG(a, b)| · αn +2 ≥ α2n2 +4 +edges uv +with u ∈ NG(a, b) and v /∈ {a, b}. Applying Lemma 2.1 with H = K2, V1 = NG(a, b) and V2 = V (G)\{a, b}, +we see that there are poly(α)ns+t pairs of disjoint sets (S, T ) such that |S| = s, |T | = t, S ⊆ NG(a, b), +a, b /∈ T , and S, T form a complete bipartite graph in G. Given any such pair, it is safe to map x to a, y to +b, A3 to S and B to T to obtain an H-copy. Hence, G contains at least poly(α)ns+t = poly(α)nh−2 copies +of H mapping xy to ab. +Lemma 5.3. Let H be a graph on h vertices such that χ(H) = 3, H contains a critical edge xy, and +odd-girth(H) ≥ 5. Let G be a graph on n vertices, let ab ∈ E(G), and suppose that there exists A ⊂ NG(a) +and B ⊂ NG(b) such that |A| , |B| ≥ αn and |NG(a′, b′)| ≥ αn for all distinct a′ ∈ A and b′ ∈ B. Then there +are at least poly(α)nh−2 copies of H in G mapping xy ∈ E(H) to ab ∈ E(G). +Proof. By Lemma 5.1 (using odd-girth(H) ≥ 5), there exists a partition V (H) = A1 ·∪ · · · ·∪ A5 such that +A1 = {x}, A2 = {y}, and E(H) ⊆ � +i∈[5] Ai × Ai+1. Put si = |Ai| for i ∈ [5]. +There are at least (|A||B| − |A|)/2 ≥ α2n2/3 pairs {a′, b′} of distinct vertices with a′ ∈ A, b′ ∈ B +(the factor of 2 is due to the fact that each pair in A ∩ B is counted twice). +Each such pair a′, b′ has +at least αn − 2 ≥ αn/2 common neighbors c′ /∈ {a, b}, by assumption. +Therefore, there are at least +α2n2 +3 +· αn +2 += α3n3 +6 +triples (a′, b′, c′) such that a′ ∈ A, b′ ∈ B, and c′ ̸= a, b is a common neighbor of a′, b′. +By Lemma 2.1 with H = K2,1 and V1 = A, V2 = B, V3 = V (G)\{a, b}, there are at least poly(α)ns3+s4+s5 +corresponding copies of K2,1[s3, s5, s4], i.e., triples of disjoint sets (R, S, T ) such that R ⊆ A, S ⊆ B, a, b /∈ T , +|R| = s5, |S| = s3, |T | = s4, and (R, T ) and (S, T ) form complete bipartite graphs in G. Given any such +7 + +triple, we can safely map A1 = {x} to a, A2 = {y} to b, A5 to R, A3 to S, and A4 to T to obtain a copy of +H. Thus, there are at least poly(α)ns3+s4+s5 = poly(α)nh−2 copies of H mapping xy to ab. +In the following theorem we prove the upper bound in the second case of Theorem 1.3. +Theorem 5.4. Let H be a graph such that χ(H) = 3, H has a critical edge xy, and H contains a triangle. +Then, δlin-rem(H) ≤ 1 +3. +Proof. Write h = v(H). Fix an arbitrary α > 0, and let G be an n-vertex graph with minimum degree +δ(G) ≥ ( 1 +3 + α)n and with a collection C = {H1, . . . , Hm} of m := εn2 edge-disjoint copies of H. +For +each i = 1, . . . , m, there exist u, v, w ∈ V (Hi) forming a triangle (because H contains a triangle). +As +degG(u) + degG(v) + degG(w) ≥ 3δ(G) ≥ (1 + 3α)n, two of u, v, w have at least αn common neighbors. We +denote these two vertices by ai and bi. By Lemma 5.2, G has at least poly(α)nh−2 copies of H which map +xy to aibi. The edges a1b1, . . . , ambm are distinct because H1, . . . , Hm are edge-disjoint. Hence, summing +over all i = 1, . . . , m, we see that G contains at least εn2 · poly(α)nh−2 = poly(α)εnh copies of H. This +proves that δlin-rem(H) ≤ 1 +3 + α, and taking α → 0 gives δlin-rem(H) ≤ 1 +3. +In what follows, we need the following very well-known observation, originating in the work of Andrásfai, +Erdős and Sós, see [4, Remark 1.6]. +Lemma 5.5. If δ(G) > +2 +2k+1n and odd-girth(G) ≥ 2k + 1 for k ≥ 2, then G is bipartite. +Proof. Suppose by contradiction that G is not bipartite and take a shortest odd cycle C in G, so |C| ≥ 2k+1. +As � +x∈C deg(x) ≥ (2k+1)δ(G) > 2n, there exists a vertex v /∈ C with at least 3 neighbors on C. Then there +are two neighbors x, y ∈ C of v such that the distance of x, y along C is not equal to 2. Then by taking the +odd path between x, y along C and adding the edges vx, vy, we get a shorter odd cycle, a contradiction. +We will also use the following result of Letzter and Snyder, see [17, Corollary 32]. +Theorem 5.6 ([17]). Let G be a {C3, C5}-free graph on n vertices with δ(G) > n +4 . Then G is homomorphic +to C7. +We can now finally prove the upper bound in the last case of Theorem 1.3. +Theorem 5.7. Let H be a graph such that χ(H) = 3, H contains critical edge xy, and odd-girth(H) ≥ 5. +Then δlin-rem(H) ≤ 1 +4. +Proof. Denote h = |V (H)|. Write odd-girth(G) = 2k + 1 ≥ 5. By the second item of Lemma 5.1, there +is a partition V (H) = A1 ·∪ A2 ·∪ · · · ·∪ A2k+1 such that |A1| = |A2| = 1, and E(H) ⊆ � +i∈[2k+1] Ai × Ai+1. +Denote si = |Ai| for each i ∈ [2k + 1], so H is a subgraph of the blow-up C2k+1[s1, . . . , s2k+1] of C2k+1. Let +c1 = c1(C2k+1, s1, . . . , s2k+1) > 0 and c2 = c2(k) > 0 be the constants given by Lemma 2.1 and Lemma 2.3, +respectively. According to Theorem 1.2, δpoly-rem(C2k+1) = +1 +2k+1 < 1 +4, and hence there exists a constant +c3 = c3(k) > 0 such that if G is a graph on n vertices with δ(G) ≥ +n +4 and at least εn2 edge-disjoint +C2k+1-copies, then G contains at least c3ε +1 +c3 n2k+1 copies of C2k+1. Set c := c1 · min(c2, c3). +Let α > 0 and ε be small enough; it suffices to assume that ε < +� +α2 +200k(k+2) +�1/c +. Let G be a graph on +n vertices with δ(G) ≥ ( 1 +4 + α)n which contains at least εn2 edge-disjoint copies of H. Our goal is to show +that G contains ΩH,α(εnh) copies of H. Suppose first that G contains at least εcn2 edge-disjoint copies of +C2ℓ+1 for some 1 ≤ ℓ ≤ k. If ℓ < k, then G contains Ωk(εc/c2n2k+1) = Ωk(εc1n2k+1) copies of C2k+1 by +Lemma 2.3 and the choice of c2. And if ℓ = k, then G contains Ωk(εc/c3n2k+1) = Ωk(εc1n2k+1) copies of +C2k+1 by Theorem 1.2 and the choice of c3. In either case, G contains Ωk(εc1n2k+1) copies of C2k+1. But +then, by Lemma 2.1 (with V1 = · · · = V2k+1 = V (G)), G contains at least ΩH(εc1/c1nh) = ΩH(εnh) copies +of C2k+1[s1, . . . , s2k+1], and hence ΩH,α(εnh) copies of H. This concludes the proof of this case. +From now on, assume that G contains at most εcn2 edge-disjoint C2ℓ+1-copies for every ℓ ∈ [k]. Let Cℓ +be a maximal collection of edge-disjoint C2ℓ+1-copies in G, so |Cℓ| ≤ εcn2. Let Ec be the set of edges which +8 + +are contained in one of the cycles in C1 ∪ · · · ∪ Ck. Let S be the set of vertices which are incident with at +least αn +10 edges from Ec. Then +|Ec| ≤ +k +� +ℓ=1 +(2ℓ + 1)εcn2 = k(k + 2)εcn2 and |S| ≤ 2 |Ec| +αn/10 ≤ 20k(k + 2)εc +α +n < αn +10 , +(1) +where the last inequality holds by our assumed bound on ε. Let G′ be the subgraph of G obtained by deleting +the edges in Ec and the vertices in S. Note that G′ ⊆ G − Ec is {C3, C5, . . . , C2k+1}-free because for every +1 ≤ ℓ ≤ k, we removed all edges from a maximal collection of edge-disjoint C2ℓ+1-copies. +Claim 5.8. |V (G′)| > (1 − α +10)n and δ(G′) > ( 1 +4 + 4α +5 )n. +Proof. The first inequality follows from (1) as |V (G′)| = n − |S|. Each v ∈ V (G)\S has at most αn +10 incident +edges from Ec, and at most |S| < αn +10 neighbors in S, thereby degG′(v) > degG(v) − αn +5 ≥ ( 1 +4 + 4α +5 )n. Hence, +δ(G′) > ( 1 +4 + 4α +5 )n. +Claim 5.9. G′ is homomorphic to C7. Moreover, G′ is bipartite unless k = 2. +Proof. Recall that G′ is {C3, C5, . . . , C2k+1}-free. As k ≥ 2, G′ is {C3, C5}-free. Also, δ(G′) > n +4 ≥ |V (G′)| +4 +by Claim 5.8. +So G′ is homomorphic to C7 by Theorem 5.6. +If k ≥ 3, i.e. +odd-girth(H) ≥ 7, then +odd-girth(G′) ≥ 2k + 3 ≥ 9. As δ(G′) > n +4 , G′ is bipartite by Lemma 5.5. +The rest of the proof is divided into two cases based whether or not G′ is bipartite. These cases are handled +by Propositions 5.10 and 5.11, respectively. +Proposition 5.10. Suppose that G′ is bipartite. Then G has ΩH,α(εnh) copies of H. +Proof. Let (L′, R′) be a bipartition of G′, so V (G) = L′ ·∪ R′ ·∪ S. Let L1 ⊆ S (resp. R1 ⊆ S) be the set of +vertices of S having at most αn +5 neighbors in L′ (resp. R′). Let G′′ be the bipartite subgraph of G induced +by the bipartition (L′′, R′′) := (L′ ·∪ L1, R′ ·∪ R1). Let S′′ = V (G)\(L′′ ·∪ R′′), so V (G) = L′′ ·∪ R′′ ·∪ S′′. +We claim that δ(G′′) ≥ ( 1 +4 + α +2 )n. First, as G′ is a subgraph of G′′, we have degG′′(v) > ( 1 +4 + 4α +5 )n for +each v ∈ V (G′) ⊆ V (G′′) by Claim 5.8. Now we consider vertices in V (G′′) \ V (G′) = L1 ∪ R1. Each v ∈ L1 +has at most |S| ≤ αn +10 neighbors in S and at most αn +5 +neighbors in L′, by the definition of L1. Hence, v +has at least degG(v) − 3α +10 n ≥ ( 1 +4 + α +2 )n neighbors in R′ ⊆ V (G′′). By the symmetric argument for vertices +v ∈ R1, we get that δ(G′′) ≥ ( 1 +4 + α +2 )n, as required. +For an edge uv ∈ E(G)\E(G′′), we say uv is of type I if u, v ∈ L′′ or u, v ∈ R′′, and we say that uv is +of type II if u ∈ S′′ or v ∈ S′′. Every edge in E(G)\E(G′′) is of type I or II. Since χ(H) = 3 and G′′ is +bipartite, each copy of H in G must contain an edge of type I or an edge of type II (or both). As G has +εn2 edge-disjoint H-copies, G contains at least εn2 +2 +edges of type I or at least εn2 +2 +edges of type II. We now +consider these two cases separately. See Fig. 4 for an illustration. Recall that xy ∈ E(H) denotes a critical +edge of H. +Case 1: +G contains εn2 +2 +edges of type I. +Fix any edge ab ∈ E(G) of type I. Without loss of generality, +assume a, b ∈ L′′ (the case a, b ∈ R′′ is symmetric). We claim that G has poly(α)nh−2 copies of H mapping +xy ∈ E(H) to ab ∈ E(G). If degG(a, b) ≥ αn +2 then this holds by Lemma 5.2. Otherwise, degG(a, b) < αn +2 , +and thus +|R′′| ≥ |NG′′(a) ∪ NG′′(b)| ≥ degG′′(a) + degG′′(b) − degG(a, b) > 2δ(G′′) − αn +2 > n +2 , +using that δ(G′′) ≥ ( 1 +4 + α +2 )n. Thus, |L′′| < n +2 . This implies that for all a′ ∈ NG′′(a), b′ ∈ NG′′(b), +degG′′(a′, b′) ≥ 2δ(G′′) − |L′′| ≥ αn. +Now, by Lemma 5.3 (with A = NG′′(a) and B = NG′′(b)), there are poly(α)nh−2 copies of H mapping xy +to ab, as claimed. Summing over all edges ab of type I, we get εn2 +2 · poly(α)nh−2 = poly(α)εnh copies of H. +This completes the proof in Case 1. +9 + +L′′ +R′′ +a +b +L′′ +R′′ +a +b +a′ +b′ +L′′ +R′′ +S′′ +a +b +a′ +b′ +Figure 4: Proof of Proposition 5.10: Case 1 with degG(a, b) ≥ +αn +2 +(left), Case 1 with degG(a, b) < +αn +2 +(middle) and Case 2 (right). The red part is the common neighborhood of a and b (or a′ and b′). +Case 2: +G contains εn2 +2 +edges of type II. +Note that the number of edges of type II is trivially at most +|S′′| n. +Thus, |S′′| ≥ +εn +2 . +Fix some a ∈ S′′. +By the definition of L′′, R′′ and S′′, v has at least +αn +5 +neighbors in L′ ⊆ L′′ and at least αn +5 neighbors in R′ ⊆ R′′. Without loss of generality, assume |L′′| ≤ |R′′|, +thereby |L′′| ≤ +n +2 . +Now fix any b ∈ L′′ adjacent to a; there are at least +αn +5 +choices for b. +We have +|NG(a) ∩ R′′| ≥ αn +5 and |NG′′(b)| ≥ δ(G′′) > n +4 , and for all a′ ∈ NG(a) ∩ R′′, b′ ∈ NG′′(b) ⊆ R′′ it holds that +degG′′(a′, b′) ≥ 2δ(G′′) − |L′′| ≥ αn. Therefore, by Lemma 5.3, G has poly(α)nh−2 copies of H mapping xy +to ab. Enumerating over all a ∈ S′′ and b ∈ NG(a) ∩ L′′, we again get ΩH,α(εnh) copies of H in G. This +completes the proof of Proposition 5.10. +Proposition 5.11. Suppose G′ is non-bipartite but homomorphic to C7. Then G has ΩH,α(εnh) copies of H. +Proof. By Claim 5.9 we must have k = 2 , so odd-girth(H) = 5. The proof is similar to that of Proposi- +tion 5.10, but instead of a bipartition of G′, we use a partition corresponding to a homomorphism into C7. +Let V (G)\S = V (G′) = V ′ +1 ·∪ V ′ +2 ·∪ · · · ·∪ V ′ +7 be a partition of V (G′) such that E(G′) ⊆ � +i∈[7] V ′ +i × V ′ +i+1. +Here and later, all subscripts are modulo 7. We have V ′ +i ̸= ∅ for all i ∈ [7], because otherwise G′ would be +bipartite. For i ∈ [7], let Si be the set of vertices in S having at most 2αn +5 +neighbors in V (G′)\ (V ′ +i−1 ∪V ′ +i+1). +In case v lies in multiple Si’s, we put v arbitrarily in one of them. Set V ′′ +i +:= V ′ +i ∪ Si. Let G′′ be the +7-partite subgraph of G with parts V ′′ +1 , . . . , V ′′ +7 and with all edges of G between V ′′ +i +and V ′′ +i+1, i = 1, . . . , 7. +By definition, G′ is a subgraph of G′′, and G′′ is homomorphic to C7 via the homomorphism V ′′ +i �→ i. Put +S′′ := V (G)\V (G′′) = S \ �7 +i=1 Si. We now collect the following useful properties. +Claim 5.12. The following holds: +(i) δ(G′′) ≥ ( 1 +4 + α +2 )n. +(ii) For every i ∈ [7] and for every u, v ∈ V ′′ +i +or u ∈ V ′′ +i , v ∈ V ′′ +i+2, it holds that degG′′(u, v) ≥ αn. +(iii) For every i ∈ [7], every v ∈ V ′′ +i +has at least αn neighbors in V ′′ +i−1 and at least αn neighbors in V ′′ +i+1. +(iv) For every a ∈ S′′, there are i, j with j − i ≡ 1, 3 (mod 7) and |NG(a) ∩ V ′′ +i | , +��NG(a) ∩ V ′′ +j +�� > 2αn +25 . +Proof. fds +(i) Let i ∈ [7] and v ∈ V ′′ +i . If v ∈ V (G′), then degG′′(v) ≥ degG′(v) ≥ δ(G′) > ( 1 +4 + α +2 )n, using Claim 5.8. +Otherwise, v ∈ Si. By definition, v has at most 2αn +5 +neighbours in V (G′)\(V ′ +i−1 ∪V ′ +i+1). Also, v has at +most |S| ≤ αn +10 neighbours in S. It follows that v has at least degG(v)− 2αn +5 − αn +10 ≥ ( 1 +4 + α +2 )n neighbors +in V ′′ +i−1 ∪ V ′′ +i+1. Hence, degG′′(v) > ( 1 +4 + α +2 )n. +(ii) First, observe that +|V ′′ +i | + +��V ′′ +i+2 +�� ≥ +�1 +4 + α +2 +� +n +(2) +10 + +for all i ∈ [7]. Indeed, V ′′ +i+1 is non-empty, and fixing any v ∈ V ′′ +i+1, we have |V ′′ +i | + +��V ′′ +i+2 +�� ≥ degG′′(v) ≥ +δ(G′′) ≥ ( 1 +4 + α +2 )n. By applying (2) to the pairs (i + 2, i + 4) and (i − 2, i), we get +��V ′′ +i−1 +�� + +��V ′′ +i+1 +�� + +��V ′′ +i+3 +�� ≤ n − ( +��V ′′ +i+2 +�� + +��V ′′ +i+4 +��) − ( +��V ′′ +i−2 +�� + |V ′′ +i |) ≤ n − 2 +�1 +4 + α +2 +� +n < n +2 . +(3) +Now let i ∈ [7]. For u, v ∈ V ′′ +i +we have NG′′(u) ∪ NG′′(v) ⊆ V ′′ +i−1 ∪ V ′′ +i+1, and for u ∈ V ′′ +i , v ∈ V ′′ +i+2 +we have NG′′(u) ∪ NG′′(v) ⊆ V ′′ +i−1 ∪ V ′′ +i+1 ∪ V ′′ +i+3. In both cases, |NG′′(u) ∪ NG′′(v)| < n +2 by (3). As +degG′′(u) + degG′′(v) ≥ 2δ(G′′) ≥ ( 1 +2 + α)n, we have degG′′(u, v) > αn, as required. +(iii) We first argue that |V ′′ +i | ≤ ( 1 +4 − 3α +2 )n for each i ∈ [7]. Indeed, by applying (2) to the pairs (i − 1, i + 1), +(i + 2, i + 4), (i + 3, i + 5), we get +|V ′′ +i | ≤ n − ( +��V ′′ +i−1 +�� + +��V ′′ +i+1 +��) − ( +��V ′′ +i+2 +�� + +��V ′′ +i+4 +��) − ( +��V ′′ +i+3 +�� + +��V ′′ +i+5 +��) ≤ n − 3 +�1 +4 + α +2 +� +n = +�1 +4 − 3α +2 +� +n. +Now, for every v ∈ V ′′ +i , we have NG′′(v) ⊆ V ′′ +i−1 ∪ V ′′ +i+1 and +��V ′′ +i−1 +�� , +��V ′′ +i+1 +�� < ( 1 +4 − 3α +2 )n. Hence, v has +at least degG′′(v) − ( 1 +4 − 3α +2 )n ≥ αn neighbors in each of V ′′ +i−1, V ′′ +i+1. +(iv) Let I be the set of i with |NG(a) ∩ V ′′ +i | ≥ 2αn +25 . If I is empty, then a has less than 5 · 2αn +25 += 2αn +5 +neighbors in every V (G′)\(V ′ +i−1 ∪V ′ +i+1) and therefore can not be in S′′. Suppose for contradiction that +there exist no i, j ∈ I with j − i ≡ 1, 3 (mod 7). We claim that there is j ∈ [7] such that I ⊆ {j, j + 2}. +Fix an arbitrary i ∈ I. Then, i ± 1, i ± 3 /∈ I by assumption. Also, at most one of i + 2, i − 2 is +in I, because (i − 2) − (i + 2) ≡ 3 (mod 7). So I ⊆ {i, i + 2} or I ⊆ {i − 2, i}, proving our claim +that I ⊆ {j, j + 2} for some j. By the definition of I, a has at most 5 · 2αn +25 += +2αn +5 +neighbors in +V (G′)\(V ′ +j ∪ V ′ +j+2). Hence, a ∈ Sj+1. This contradicts the fact that a ∈ S′′, as S′′ ∩ Si+1 = ∅. +We continue with the proof of Proposition 5.11. Recall that the edges in E(G) \ E(G′′) are precisely +the edges of G not belonging to � +i∈[7] V ′′ +i × V ′′ +i+1. For an edge ab ∈ E(G)\E(G′′), we say ab is of type I if +a, b ∈ V (G′′), and of type II if a ∈ S′′ or b ∈ S′′. Clearly, every edge in E(G)\E(G′′) is either of type I +or of type II. Since odd-girth(H) = 5 and C5 is not homomorphic to C7, every H-copy in G must contain +some edge of type I or of type II (or both). As G has εn2 edge-disjoint H-copies, G must have at least εn2 +2 +edges of type I or at least εn2 +2 +edges of type II. We consider these two cases separately. See Fig. 5 for an +illustration. Recall that xy ∈ E(H) denotes a critical edge of H. +Case 1: +G contains εn2 +2 +edges of type I. Fix any edge ab of type I, where a ∈ V ′′ +i and b ∈ V ′′ +j for i, j ∈ [7]. +We now show that G has poly(α)nh−2 copies of H mapping xy ∈ E(H) to ab. As ab /∈ E(G′′), we have +i−j ≡ 0, ±2, ±3 (mod 7). When j−i ≡ 0, ±2 (mod 7), we have degG(a, b) ≥ degG′′(a, b) > αn by Claim 5.12 +(ii). Then, by Lemma 5.2, G has poly(α)nh−2 copies of H mapping xy to ab, as required. Now suppose that +j−i ≡ ±3 (mod 7), say j ≡ i+3 (mod 7). Denote A := NG(a)∩V ′′ +i−1 and B := NG(b)∩V ′′ +j+1 = NG(b)∩V ′′ +i−3. +We have that |A| , |B| ≥ αn by Claim 5.12 (iii), and |NG(a′, b′)| > αn for all a′ ∈ A, b′ ∈ B by Claim 5.12 +(ii). Now, by Lemma 5.3, G has poly(α)nh−2 copies of H mapping xy to ab, proving our claim. Summing +over all edges ab of type I, we get εn2 +2 · poly(α)nh−2 = ΩH,α(εnh) copies of H in G, finishing this case. +Case 2: +G contains εn2 +2 +edges of type II. Notice that the number edges incident to S′′ is at most |S′′| n, +meaning that |S′′| ≥ εn +2 . Fix any a ∈ S′′. By Claim 5.12 (iv), there exist i, j ∈ [7] with j − i ≡ 1, 3 (mod 7) +and |NG(a) ∩ V ′′ +i | , +��NG(a) ∩ V ′′ +j +�� > 2αn +25 . Fix any b ∈ NG(a) ∩ V ′′ +i (there are at least 2αn +25 choices for b). Take +A = NG(a)∩V ′′ +j and B = NG(b)∩V ′′ +i+1. We have that |A| ≥ 2αn +25 , and |B| ≥ αn by Claim 5.12 (iii). Further, +as j − (i + 1) ≡ 0, 2 (mod 7), Claim 5.12 (ii) implies that |NG(a′, b′)| > αn for all a′ ∈ A, b′ ∈ B. Now, +by Lemma 5.3, G has poly(α)nh−2 copies of H mapping xy to ab. Summing over all choices of a ∈ S′′ and +b ∈ V ′′ +i , we acquire |S′′| · 2αn +25 · poly(α)nh−2 = ΩH,α(εnh) copies of H in G. This completes the proof of Case +2, and hence the proposition. +Propositions 5.10 and 5.11 imply the theorem. +11 + +V ′′ +1 +V ′′ +2 +V ′′ +3 +V ′′ +4 +V ′′ +5 +V ′′ +6 +V ′′ +7 +a +b +V ′′ +1 +V ′′ +2 +V ′′ +3 +V ′′ +4 +V ′′ +5 +V ′′ +6 +V ′′ +7 +a +b +a′ +b′ +V ′′ +1 +V ′′ +2 +V ′′ +3 +V ′′ +4 +V ′′ +5 +V ′′ +6 +V ′′ +7 +S′′ a +b +a′ +b′ +Figure 5: Proof of Proposition 5.11: Case 1 for j = i + 2 (left), Case 1 for j = i + 3 (middle) and Case 2 for +j = i + 3 (right). The red part is the common neighborhood of a and b (or a′ and b′). +6 +Concluding remarks and open questions +It would be interesting to determine the possible values of δpoly-rem(H) for 3-chromatic graphs H. So far we +know that +1 +2k+1 is a value for each k ≥ 1. Is there a graph H with 1 +5 < δpoly-rem(H) < 1 +3? 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Combina- +torica, 27(2):241–243, 2007. 1 +13 + diff --git a/3dFST4oBgHgl3EQfYzh6/content/tmp_files/load_file.txt b/3dFST4oBgHgl3EQfYzh6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..493457fd612c117e4c6a2bd5b803a851b1eb5743 --- /dev/null +++ b/3dFST4oBgHgl3EQfYzh6/content/tmp_files/load_file.txt @@ -0,0 +1,992 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf,len=991 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='13789v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='CO] 31 Jan 2023 The Minimum Degree Removal Lemma Thresholds Lior Gishboliner∗ Zhihan Jin∗ Benny Sudakov∗ Abstract The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph H and ε > 0, if an n-vertex graph G contains εn2 edge-disjoint copies of H then G contains δnv(H) copies of H for some δ = δ(ε, H) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The current proofs of the removal lemma give only very weak bounds on δ(ε, H), and it is also known that δ(ε, H) is not polynomial in ε unless H is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that δ(ε, H) depends polynomially or linearly on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In this paper we answer several questions of Fox and Wigderson on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 1 Introduction The graph removal lemma, first proved by Ruzsa and Szemerédi [23], is a fundamental result in extremal graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It also have important applications to additive combinatorics and property testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The lemma states that for every fixed graph H and ε > 0, if an n-vertex graph G contains εn2 edge-disjoint copies of H then G it contains δnv(H) copies of H, where δ = δ(ε, H) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Unfortunately, the current proofs of the graph removal lemma give only very weak bounds on δ = δ(ε, H) and it is a very important problem to understand the dependence of δ on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The best known result, due to Fox [11], proves that 1/δ is at most a tower of exponents of height logarithmic in 1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Ideally, one would like to have better bounds on 1/δ, where an optimal bound would be that δ is polynomial in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' However, it is known [2] that δ(ε, H) is only polynomial in ε if H is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This situation led Fox and Wigderson [12] to initiate the study of minimum degree conditions which guarantee that δ(ε, H) depends polynomially or linearly on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Formally, let δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) be the maximum δ ∈ [0, 1] such that if G is an n-vertex graph with minimum degree at least γn and with εn2 edge-disjoint copies of H, then G contains δnv(H) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The linear removal threshold of H, denoted δlin-rem(H), is the infimum γ such that δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) depends linearly on ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) ≥ µε for some µ = µ(γ) > 0 and all ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The polynomial removal threshold of H, denoted δpoly-rem(H), is the infimum γ such that δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) depends polynomially on ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) ≥ µε1/µ for some µ = µ(γ) > 0 and all ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Trivially, δlin-rem(H) ≥ δpoly-rem(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fox and Wigderson [12] initiated the study of δlin-rem(H) and δpoly-rem(H), and proved that δlin-rem(Kr) = δpoly-rem(Kr) = 2r−5 2r−3 for every r ≥ 3, where Kr is the clique on r vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' They further asked to determine the removal lemma thresholds of odd cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Here we completely resolve this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The following theorem handles the polynomial removal threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' δpoly-rem(C2k+1) = 1 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 also answers another question of Fox and Wigderson [12], of whether δlin-rem(H) and δpoly-rem(H) can only obtain finitely many values on r-chromatic graphs H for a given r ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 shows that δpoly-rem(H) obtains infinitely many values for 3-chromatic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In contrast, δlin-rem(H) ob- tains only three possible values for 3-chromatic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, the following theorem determines δlin-rem(H) for every 3-chromatic H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' An edge xy of H is called critical if χ(H − xy) < χ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' ∗Department of Mathematics, ETH, Zürich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Research supported in part by SNSF grant 200021_196965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Email: {lior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='gishboliner, zhihan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='jin, benjamin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='sudakov}@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 1 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For a graph H with χ(H) = 3, it holds that δlin-rem(H) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 2 H has no critical edge, 1 3 H has a critical edge and contains a triangle, 1 4 H has a critical edge and odd-girth(H) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 show a separation between the polynomial and linear removal thresholds, giving a sequence of graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' C5, C7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' ) where the polynomial threshold tends to 0 while the linear threshold is constant 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The parameters δpoly-rem and δlin-rem are related to two other well-studied minimum degree thresholds: the chromatic threshold and the homomorphism threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The chromatic threshold of a graph H is the infimum γ such that every n-vertex H-free graph G with δ(G) ≥ γn has bounded cromatic number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=', there exists C = C(γ) such that χ(G) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The study of the chromatic threshold originates in the work of Erdős and Simonovits [10] from the ’70s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Following multiple works [4, 15, 16, 7, 5, 25, 26, 19, 6, 14, 20], the chromatic threshold of every graph was determined by Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Moving on to the homomorphism threshold, we define it more generally for families of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The homomorphism threshold of a graph-family H, denoted δhom(H), is the infimum γ for which there exists an H-free graph F = F(γ) such that every n-vertex H-free graph G with δ(G) ≥ γn is homomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' When H = {H}, we write δhom(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This parameter was widely studied in recent years [18, 22, 17, 8, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It turns out that δhom is closely related to δpoly-rem(H), as the following theorem shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For a graph H, let IH denote the set of all minimal (with respect to inclusion) graphs H′ such that H is homomorphic to H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For every graph H, δpoly-rem(H) ≤ δhom(IH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that IC2k+1 = {C3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , C2k+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Using this, the upper bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 follows immediately by combining Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4 with the result of Ebsen and Schacht [8] that δhom({C3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , C2k+1}) = 1 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The lower bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 was established in [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' for completeness, we sketch the proof in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The rest of this short paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Section 2 contains some preliminary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In Section 3 we prove the lower bounds in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Section 4 gives the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4, and Section 5 gives the proof of the upper bounds in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In the last section we discuss further related problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 2 Preliminaries Throughout this paper, we always consider labeled copies of some fixed graph H and write copy of H for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We use δ(G) for the minimum degree of G, and write H → F to denote that there is a homo- morphism from H to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For a graph H on [h] and integers s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh > 0, we denote by H[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh] the blow-up of H where each vertex i ∈ V (H) is replaced by a set Si of size si (and edges are replaced with complete bipartite graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The following lemma is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a fixed graph on vertex set [h] and let s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' There exists a constant c = c(H, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh) > 0 such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be an n-vertex graph and V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vh ⊆ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose that G contains at least ρnh copies of H mapping i to Vi for all i ∈ [h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then G contains at least cρ 1 c · ns1+···+sh copies of H[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh] mapping Si to Vi for all i ∈ [h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that the sets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vh in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 do not have to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 works by defining an auxiliary h-uniform hypergraph G whose hyperedges correspond to the copies of H in which vertex i is mapped to Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By assumption, G has at least ρnh edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the hypergraph generalization of the Koväri-Sós-Turán theorem, see [9], G contains poly(ρ)ns1+···+sh copies of K(h) s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=',sh, the complete h-partite hypergraph with parts of size s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each copy of K(h) s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=',sh gives a copy of H[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh] mapping Si to Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fox and Wigderson [12, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1] proved the following useful fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If H → F and F is a subgraph of H, then δpoly-rem(H) = δpoly-rem(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 2 The following lemma is an asymmetric removal-type statement for odd cycles, which gives polynomial bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' A similar result has appeared very recently in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For 1 ≤ ℓ < k, there exists a constant c = c(k) > 0 such that if an n-vertex graph G has εn2 edge-disjoint copies of C2ℓ+1, then it has at least cε1/cn2k+1 copies of C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let C be a collection of εn2 edge-disjoint copies of C2ℓ+1 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' There exists a collection C′ ⊆ C such that |C′| ≥ εn2/2 and each vertex v ∈ V (G) belongs to either 0 or at least εn/2 of the cycles in C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, to obtain C′, we repeatedly delete from C all cycles containing a vertex v which belongs to at least one but less than εn/2 of the cycles in C (without changing the graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The set of cycles left at the end is C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In this process, we delete at most εn2/2 cycles altogether (because the process lasts for at most n steps);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' hence |C′| ≥ εn2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let V be the set of vertices contained in at least εn/2 cycles from C′, so |V | ≥ εn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' With a slight abuse of notation, we may replace G with G[V ], C with C′ and ε/2 with ε, and denote |V | by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, from now on, we assume that each vertex v ∈ V (G) is contained in at least εn of the cycles in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This implies that |N(v)| ≥ 2εn for every v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any v0 ∈ V (G) and let C(v0) be the set of cycles C ∈ C such that C ∩ N(v0) ̸= ∅ and v0 /∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The number of cycles C ∈ C intersecting N(v0) is at least |N(v0)| · εn/(2ℓ + 1) ≥ 2ε2n2/(2ℓ + 1), and the number of cycles containing v0 is at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, |C(v0)| ≥ 2ε2n2/(2ℓ + 1) − n ≥ ε2n2/(ℓ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Take a random partition V0, V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vℓ of V (G) \\ {v0}, where each vertex is put in one of the parts uniformly and independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For a cycle (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , x2ℓ+1) ∈ C(v0) with xℓ+1 ∈ N(v0), say that (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , x2ℓ+1) is good if xℓ+1 ∈ V0 and xℓ+1−i, xℓ+1+i ∈ Vi for 1 ≤ i ≤ ℓ (so in particular x1, x2ℓ+1 ∈ Vℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The probability that (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , x2ℓ+1) is good is 1/(ℓ + 1)2ℓ+1, so there is a collection of good cycles C′(v0) ⊆ C0 of size |C′(v0)| ≥ |C(v0)|/(ℓ + 1)2ℓ+1 ≥ ε2n2/(ℓ + 1)2ℓ+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put γ := ε2/(ℓ + 1)2ℓ+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the same argument as above, there is a collection C′′(v0) ⊆ C′(v0) with |C′′(v0)| ≥ γn2/2 such that each vertex is contained in either 0 or at least γn/2 cycles from C′′(v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let W be the set of vertices contained in at least γn/2 cycles from C′′(v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that W ∩ V0 ⊆ N(v0) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, each vertex in W ∩ Vℓ has at least γn/2 neighbors in W ∩ Vℓ, and for each 1 ≤ i ≤ ℓ, each vertex in W ∩ Vi has at least γn/2 neighbors in W ∩ Vi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It follows that W ∩ Vℓ contains at least 1 2|W ∩ Vℓ| · �2k−2ℓ−2 i=0 (γn/2 − i) = poly(γ)n2k−2ℓ paths of length 2k − 2ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now construct a collection of copies of C2k+1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Choose a path yℓ+1, yℓ+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k−ℓ of length 2k − 2ℓ − 1 in W ∩ Vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For each i = ℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , 1, take a neighbor yi ∈ W ∩ Vi−1 of yi+1 and a neighbor y2k−i+1 ∈ W ∩ Vi−1 of y2k−i, such that the vertices y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k are all different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k is a path and y1, y2k ∈ W ∩ V0 ⊆ N(v0), so v0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k is a copy of C2ℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The number of choices for the path yℓ+1, yℓ+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k−ℓ is poly(γ)n2k−2ℓ and the number of choices for each vertex yi, y2k−i+1 ∈ Vi−1 (i = ℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , 1) is at least γn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, the total number of choices for y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k is poly(γ)n2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As there are n choices for v0, we get a total of poly(γ)n2k+1 = polyk(ε)n2k+1 copies of C2k+1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 3 Lower bounds Here we prove the lower bounds in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The lower bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 was proved in [12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For completeness, we include a sketch of the proof: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' δpoly-rem(C2k+1) ≥ 1 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix an arbitrary α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In [2] it was proved that for every ε, there exists a (2k + 1)-partite graph with parts V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , V2k+1 of size αn/(2k + 1) each, with εn2 edge-disjoint copies of C2k+1, but with only εω(1)n2k+1 copies of C2k+1 in total (where the ω(1) term may depend on α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Add sets U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , U2k+1 of size (1 − α)n/(2k + 1) each, and add the complete bipartite graphs (Ui, Vi), 1 ≤ i ≤ 2k + 1, and (Ui, Ui+1), 1 ≤ i ≤ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It is easy to see that this graph has minimum degree (1 − α)n/(2k + 1), and every copy of C2k+1 is contained in V1 ∪ · · · ∪ V2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Letting α → 0, we get that δpoly-rem(C2k+1) ≥ 1 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By combining the fact that δpoly-rem(C3) = 1 3 with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 (with F = C3), we get that δlin-rem(H) ≥ δpoly-rem(H) = 1 3 for every 3-chromatic graph H containing a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This proves the lower bound in the second case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now we prove the lower bounds in the other two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We prove a more general statement for r-chromatic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 3 V2 V3 V4 V5 V1 U2 U3 U4 U5 U1 Figure 1: Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 for C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Heavy edges indicate complete bipartite graphs while dashed edges form the Ruzsa–Szemerédi construction for C5 (see [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph with χ(H) = r ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, 3r−8 3r−5 ≤ δlin-rem(H) ≤ r−2 r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Moreover, δlin-rem(H) = r−2 r−1 if H contains no critical edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Denote h = |V (H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The bound δlin-rem(H) ≤ r−2 r−1 holds for every r-chromatic graph H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' this follows from the Erdős-Simonovits supersaturation theorem, see by [12, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose now that H contains no critical edge, and let us show that δlin-rem(H) ≥ r−2 r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' To this end, we construct, for every small enough ε and infinitely many n, an n-vertex graph G with δ(G) ≥ r−2 r−1n, such that G has at most O(ε2nh) copies of H, but Ω(εn2) edges must be deleted to turn G into an H-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let T (n, r − 1) be the Turán graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' the complete (r − 1)-partite graph with balanced parts V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Add an εn-regular graph inside V1 and let the resulting graph be G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We first claim that G contains O(ε2nh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As H contains no critical edge and χ(H) = r, every copy of H in G contains two edges e and e′ inside V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If e and e′ are disjoint, then there are at most n2(εn)2 = ε2n4 choices for e and e′ and then at most nh−4 choices for the other h− 4 vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Therefore, there are at most ε2nh such H-copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' And if e and e′ intersect, then there are at most n(εn)2 = ε2n3 choices for e and e′ and then at most nh−3 choices for the remaining vertices, again giving at most ε2nh such H-copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So G indeed has O(ε2nh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' On the other hand, we claim that one must delete Ω(εn2) edges to destroy all H-copies in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Observe that G has at least 1 2 |V1|·εn·|V2|·· · ··|Vr−1| = Ωr(εnr) copies of Kr, and every edge participates in at most nr−2 of these copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Thus, deleting cεn2 edges can destroy at most cεnr copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If c is a small enough constant (depending on r), then after deleting any cεn2 edges, there are still Ω(εnr) copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, the remaining graph contains Kr[h], the h-blowup of Kr, and hence H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof that δlin-rem(H) ≥ r−2 r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now prove that δlin-rem(H) ≥ 3r−8 3r−5 for every r-chromatic graph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It suffices to construct, for every small enough ε and infinitely many n, an n-vertex graph G with δ(G) ≥ 3r−8 3r−5n, such that G has at most O(ε2nh) copies of H but at least Ω(εn2) edges must be deleted to turn G into an H-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The vertex set of G consists of r + 1 disjoint sets V0, V1, V2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vr, where |Vi| = n 3r−5 for i = 0, 1, 2, 3 and |Vi| = 3n 3r−5 for i = 4, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=', r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put complete bipartite graphs between V0 and V1, between V0 ∪ V1 and V4 ∪ · · · ∪ Vr, and between Vi to Vj for all 2 ≤ i < j ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put εn-regular bipartite graphs between V1 and V2, and between V1 and V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The resulting graph is G (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It is easy check that δ(G) ≥ 3r−8 3r−5n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, let 0 ≤ i ≤ r and v ∈ Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If 4 ≤ i ≤ r then v is connected to all vertices except for Vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' if i ∈ {2, 3} then v is connected to all vertices except V0 ∪ V1 ∪ Vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' and if i ∈ {0, 1} then v is connected to all vertices except V2 ∪ V3 ∪ Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In any case, the neighborhood of v misses at most 3n 3r−5 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that G has at most O(ε2nh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, observe that if we delete all edges between V1 and V2 then the remaining graph is (r − 1)-colorable with coloring V1 ∪ V2, V0 ∪ V3, V4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, every copy of H must contain an edge e between V1 and V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Similarly, every copy of H must contain an edge e′ between V1 and V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If e, e′ are disjoint then there are at most n2(εn)2 = ε2n4 ways to choose e, e′ and then at most nh−4 ways to choose the remaining vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' And if e and e′ intersect then there are at most n(εn)2 = ε2n3 ways to choose e, e′ and at most nh−3 for the remaining h − 3 vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In both cases, the number of H-copies is at most ε2nh, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now we show that one must delete Ω(εn2) edges to destroy all copies of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Observe that G has |V1| · (εn)2 · |V4| · · · · · |Vr| = Ω(ε2nr) copies of Kr between the sets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that every edge f 4 V1 V2 V3 V0 V1 V2 V3 V4 V0 Figure 2: Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, r = 3 (left) and r = 4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Heavy edges indicate complete bipartite graphs while dashed edges indicate εn-regular bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' participates in at most εnr−2 of these r-cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, by the same argument as above, every copy of Kr containing f must contain an edge e from E(V1, V2) and an edge e′ from E(V1, V3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose without loss of generality that e ̸= f (the case e′ ̸= f is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In the case f ∩ e = ∅, there are at most n · εn = εn2 choices for e and at most nr−4 choices for the remaining vertices of Kr, giving at most εnr−2 copies of Kr containing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' And if f, e intersect, then there are at most εn choices for e and at most nr−3 for the remaining r − 3 vertices, giving again εnr−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We see that deleting cεn2 edges of G can destroy at most cε2nr copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, if c is a small enough constant, then after deleting any cεn2 edges there are still Ω(ε2nr) copies of Kr left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, the remaining graph contains a copy of Kr[h] and hence H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 4 Polynomial removal thresholds: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4 We say that an n-vertex graph G is ε-far from a graph property P (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' being H-free for a given graph H, or being homomorphic to a given graph F) if one must delete at least εn2 edges to make G satisfy P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Trivially, if G has εn2 edge-disjoint copies of H, then it is ε-far from being H-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We need the following result from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For every graph F on f vertices and for every ε > 0, there is q = qF (ε) = poly(f/ε), such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If a graph G is ε-far from being homomorphic to F, then for a sample of q vertices x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq ∈ V (G), taken uniformly with repetitions, it holds that G[{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq}] is not homomorphic to F with probability at least 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 is proved in Section 2 of [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In fact, [21] proves a more general result on property testing of the so-called 0/1-partition properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Such a property is given by an integer f and a function d : [f]2 → {0, 1, ⊥}, and a graph G satisfies the property if it has a partition V (G) = V1 ∪ · · · ∪ Vf such that for every 1 ≤ i, j ≤ f (possibly i = j), it holds that (Vi, Vj) is complete if d(i, j) = 1 and (Vi, Vj) is empty if d(i, j) = 0 (if d(i, j) =⊥ then there are no restrictions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' One can express the property of having a homomorphism into F in this language, simply by setting d(i, j) = 0 for i = j and ij /∈ E(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In [21], the class of these partition properties is denoted GPP0,1, and every such property is shown to be testable by sampling poly(f/ε) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This implies Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that IH is the set of minimal graphs H′ (with respect to inclusion) such that H is homomorphic to H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For convenience, put δ := δhom(IH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Our goal is to show that δpoly-rem(H) ≤ δ+α for every α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So fix α > 0 and let G be a graph with minimum degree δ(G) ≥ (δ + α)n and with εn2 edge-disjoint copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the definition of the homomorphism threshold, there is an IH-free graph F (depending only on IH and α) such that if a graph G0 is IH-free and has minimum degree at least (δ + α 2 ) · |V (G0)|, then G0 is homomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Observe that if a graph G0 is homomorphic to F then G0 is H-free, because F is free of any homomorphic image of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It follows that G is ε-far from being homomorphic to F, because G is ε-far from being H-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now we apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let q = qF (ε) be given by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We assume that q ≫ log(1/α) α2 and n ≫ q2 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Sample q vertices x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq ∈ V (G) with repetition and let X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, G[X] is not homomorphic to F with probability at least 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As n ≫ q2, the vertices x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq are pairwise-distinct with probability at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, for every i ∈ [q], the number of indices j ∈ [q] \\ {i} with xixj ∈ E(G) dominates a binomial 5 distribution B(q − 1, δ(G) n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the Chernoff bound (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' [3, Appendix A]) and as δ(G) ≥ (δ + α)n, the number of such indices is at least (δ + α 2 )q with probability 1 − e−Ω(qα2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Taking the union bound over i ∈ [q], we get that δ(G[X]) ≥ (δ + α 2 )|X| with probability at least 1 − qe−Ω(qα2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='9, as q ≫ log(1/α) α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, with probability at least 1 2 it holds that δ(G[X]) ≥ (δ + α 2 )|X| and G[X] is not homomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If this happens, then G[X] is not IH-free (by the choice of F), hence G[X] contains a copy of some H′ ∈ IH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By averaging, there is H′ ∈ IH such that G[X] contains a copy of H′ with probability at least 1 2|IH|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put k = |V (H′)| and let M be the number of copies of H′ in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The probability that G[X] contains a copy of H′ is at most M( q n)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Using the fact that q = polyH,α( 1 ε), we conclude that M ≥ 1 2|IH| · ( n q )k ≥ polyH,α(ε)nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As H → H′, there exists H′′, a blow-up of H′, such that H′′ have the same number of vertices as H, and that H ⊂ H′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 for H′ with Vi = V (G) for all i, there exist polyH,α(ε)nv(H′′) copies of H′′ in G, and thus polyH,α(ε)nv(H) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 5 Linear removal thresholds: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 Here we prove the upper bounds in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' the lower bounds were proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The first case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, so it remains to prove the other two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We begin with some preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For disjoint sets A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Am, we write � i∈[m] Ai × Ai+1 to denote all pairs of vertices which have one endpoint in Ai and one in Ai+1 for some 1 ≤ i ≤ m, with subscripts always taken modulo m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So a graph G has a homomorphism to the cycle Cm if and only if there is a partition V (G) = A1 ∪ · · · ∪ Am with E(G) ⊆ � i∈[m] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose H is a graph such that χ(H) = 3, H contains a critical edge xy, and odd-girth(H) ≥ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, There is a partition V (H) = A1 ·∪ A2 ·∪ A3 ·∪ B such that A1 = {x}, A2 = {y} and E(H) ⊆ (A3 × B) ∪ (� i∈[3] Ai × Ai+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' if k ≥ 2, there is a partition V (H) = A1 ·∪ A2 ·∪ · · · ·∪ A2k+1 such that A1 = {x}, A2 = {y} and E(H) ⊆ � i∈[2k+1] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In particular, H is homomorphic to C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Write H′ = H − xy, so H′ is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let V (H) = V (H′) = L ·∪ R be a bipartition of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As χ(H) = 3, x and y must both lie in the same side of the bipartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Without loss of generality, assume that x, y ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For the first item, set A1 = {x}, A2 = {y}, A3 = R and B = L\\{x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then every edge of G goes between B and A3 or between two of the sets A1, A2, A3, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose now that k ≥ 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' odd-girth(H) = 2k + 1 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For 1 ≤ i ≤ k, let Xi be the set of vertices at distance (i − 1) from x in H′, and let Yi be the set of vertices at distance (i − 1) from y in H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that X1 = {x} and Y1 = {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, Xi, Yi lie in L if i is odd and in R if i is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Write L′ := L\\ k� i=1 (Xi ∪ Yi), R′ := R\\ k� i=1 (Xi ∪ Yi), We first claim that {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Xk, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Yk, L′, R′} forms a partition of V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The sets X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Xk are clearly pairwise-disjoint, and so are Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, all of these sets are disjoint from L′, R′ by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So we only need to check Xi and Yj are disjoint for every pair 1 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose for contradiction that there exists u ∈ Xi ∩ Yj for some 1 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then i ≡ j (mod 2), because otherwise Xi, Yj are contained in different parts of the bipartition L, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the definition of Xi and Yj, H′ has a path x = x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xi = u and a path y = y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , yj = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, x = x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xi = u = yj, yj−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y1, y, x forms a closed walk of length i+j −1, which is odd as i ≡ j (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, odd-girth(H) ≤ 2k−1, contradicting our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By definition, there are no edges between Xi and Xj for j − i ≥ 2, and similarly for Yi, Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, there are no edges between L′ ∪ R′ and �k−1 i=1 (Xi ∪ Yi) because the vertices in L′ ∪ R′ are at distance more than k to x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Moreover, if k is even then there are no edges between Xk ∪ Yk and R′, and if k is odd then there are no edges between Xk ∪ Yk and L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Next, we show that there are no edges between Xi and Yj for any 1 ≤ i, j ≤ k except (i, j) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, if i = j then e(Xi, Yj) = 0 because Xi, Yj are on the same side 6 x y X2 Y2 L′ R′ x y X2 Y2 X3 Y3 R′ L′ Figure 3: Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, k = 2 (left) and k = 3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Edges indicate bipartite graphs where edges can be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' of the bipartition L, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So suppose that i ̸= j, say i < j, and assume by contradiction that there is an edge uv with u ∈ Xi, v ∈ Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then v is at distance at most i + 1 ≤ k from x, implying that Yj intersects X1 ∪ · · · ∪ Xi+1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Finally, we define the partition A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , A2k+1 that satisfies the assertion of the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If k is even then take A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , A2k+1 to be X1, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Yk−1, Yk∪R′, L′, Xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , X2, and if k is odd then take A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , A2k+1 to be X1, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Yk−1, Yk ∪ L′, R′, Xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' See Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the above, in both cases it holds that E(H) ⊆ � i∈[2k+1] Ai × Ai+1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For vertex u ∈ V (G), denote by NG(u) the neighborhood of u and let degG(u) = |NG(u)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For vertices u, v ∈ V (G), denote by NG(u, v) the common neighborhood of u, v and let degG(u, v) = |NG(u, v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph on h vertices such that χ(H) = 3 and H contains a critical edge xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be a graph on n vertices with δ(G) ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let ab ∈ E(G) such that degG(a, b) ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, there are at least poly(α)nh−2 copies of H in G mapping xy ∈ E(H) to ab ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the first item of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, there is a partition V (H) = A1 ·∪ A2 ·∪ A3 ·∪ B such that A1 = {x}, A2 = {y} and E(H) ⊆ (A3 × B) ∪ � i∈[3] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let s = |A3| and t = |B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each u ∈ NG(a, b) has at least αn − 2 ≥ αn 2 neighbors not equal to a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, there are at least 1 2 · |NG(a, b)| · αn 2 ≥ α2n2 4 edges uv with u ∈ NG(a, b) and v /∈ {a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 with H = K2, V1 = NG(a, b) and V2 = V (G)\\{a, b}, we see that there are poly(α)ns+t pairs of disjoint sets (S, T ) such that |S| = s, |T | = t, S ⊆ NG(a, b), a, b /∈ T , and S, T form a complete bipartite graph in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Given any such pair, it is safe to map x to a, y to b, A3 to S and B to T to obtain an H-copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, G contains at least poly(α)ns+t = poly(α)nh−2 copies of H mapping xy to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph on h vertices such that χ(H) = 3, H contains a critical edge xy, and odd-girth(H) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be a graph on n vertices, let ab ∈ E(G), and suppose that there exists A ⊂ NG(a) and B ⊂ NG(b) such that |A| , |B| ≥ αn and |NG(a′, b′)| ≥ αn for all distinct a′ ∈ A and b′ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then there are at least poly(α)nh−2 copies of H in G mapping xy ∈ E(H) to ab ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 (using odd-girth(H) ≥ 5), there exists a partition V (H) = A1 ·∪ · · · ·∪ A5 such that A1 = {x}, A2 = {y}, and E(H) ⊆ � i∈[5] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put si = |Ai| for i ∈ [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' There are at least (|A||B| − |A|)/2 ≥ α2n2/3 pairs {a′, b′} of distinct vertices with a′ ∈ A, b′ ∈ B (the factor of 2 is due to the fact that each pair in A ∩ B is counted twice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each such pair a′, b′ has at least αn − 2 ≥ αn/2 common neighbors c′ /∈ {a, b}, by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Therefore, there are at least α2n2 3 αn 2 = α3n3 6 triples (a′, b′, c′) such that a′ ∈ A, b′ ∈ B, and c′ ̸= a, b is a common neighbor of a′, b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 with H = K2,1 and V1 = A, V2 = B, V3 = V (G)\\{a, b}, there are at least poly(α)ns3+s4+s5 corresponding copies of K2,1[s3, s5, s4], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=', triples of disjoint sets (R, S, T ) such that R ⊆ A, S ⊆ B, a, b /∈ T , |R| = s5, |S| = s3, |T | = s4, and (R, T ) and (S, T ) form complete bipartite graphs in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Given any such 7 triple, we can safely map A1 = {x} to a, A2 = {y} to b, A5 to R, A3 to S, and A4 to T to obtain a copy of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Thus, there are at least poly(α)ns3+s4+s5 = poly(α)nh−2 copies of H mapping xy to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In the following theorem we prove the upper bound in the second case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph such that χ(H) = 3, H has a critical edge xy, and H contains a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, δlin-rem(H) ≤ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Write h = v(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix an arbitrary α > 0, and let G be an n-vertex graph with minimum degree δ(G) ≥ ( 1 3 + α)n and with a collection C = {H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Hm} of m := εn2 edge-disjoint copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , m, there exist u, v, w ∈ V (Hi) forming a triangle (because H contains a triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As degG(u) + degG(v) + degG(w) ≥ 3δ(G) ≥ (1 + 3α)n, two of u, v, w have at least αn common neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We denote these two vertices by ai and bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, G has at least poly(α)nh−2 copies of H which map xy to aibi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The edges a1b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , ambm are distinct because H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Hm are edge-disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, summing over all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , m, we see that G contains at least εn2 · poly(α)nh−2 = poly(α)εnh copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This proves that δlin-rem(H) ≤ 1 3 + α, and taking α → 0 gives δlin-rem(H) ≤ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In what follows, we need the following very well-known observation, originating in the work of Andrásfai, Erdős and Sós, see [4, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If δ(G) > 2 2k+1n and odd-girth(G) ≥ 2k + 1 for k ≥ 2, then G is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose by contradiction that G is not bipartite and take a shortest odd cycle C in G, so |C| ≥ 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As � x∈C deg(x) ≥ (2k+1)δ(G) > 2n, there exists a vertex v /∈ C with at least 3 neighbors on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then there are two neighbors x, y ∈ C of v such that the distance of x, y along C is not equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then by taking the odd path between x, y along C and adding the edges vx, vy, we get a shorter odd cycle, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We will also use the following result of Letzter and Snyder, see [17, Corollary 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='6 ([17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be a {C3, C5}-free graph on n vertices with δ(G) > n 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then G is homomorphic to C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We can now finally prove the upper bound in the last case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph such that χ(H) = 3, H contains critical edge xy, and odd-girth(H) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then δlin-rem(H) ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Denote h = |V (H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Write odd-girth(G) = 2k + 1 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the second item of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, there is a partition V (H) = A1 ·∪ A2 ·∪ · · · ·∪ A2k+1 such that |A1| = |A2| = 1, and E(H) ⊆ � i∈[2k+1] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Denote si = |Ai| for each i ∈ [2k + 1], so H is a subgraph of the blow-up C2k+1[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , s2k+1] of C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let c1 = c1(C2k+1, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , s2k+1) > 0 and c2 = c2(k) > 0 be the constants given by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' According to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, δpoly-rem(C2k+1) = 1 2k+1 < 1 4, and hence there exists a constant c3 = c3(k) > 0 such that if G is a graph on n vertices with δ(G) ≥ n 4 and at least εn2 edge-disjoint C2k+1-copies, then G contains at least c3ε 1 c3 n2k+1 copies of C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Set c := c1 · min(c2, c3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let α > 0 and ε be small enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' it suffices to assume that ε < � α2 200k(k+2) �1/c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be a graph on n vertices with δ(G) ≥ ( 1 4 + α)n which contains at least εn2 edge-disjoint copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Our goal is to show that G contains ΩH,α(εnh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose first that G contains at least εcn2 edge-disjoint copies of C2ℓ+1 for some 1 ≤ ℓ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If ℓ < k, then G contains Ωk(εc/c2n2k+1) = Ωk(εc1n2k+1) copies of C2k+1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 and the choice of c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' And if ℓ = k, then G contains Ωk(εc/c3n2k+1) = Ωk(εc1n2k+1) copies of C2k+1 by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 and the choice of c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In either case, G contains Ωk(εc1n2k+1) copies of C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' But then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 (with V1 = · · · = V2k+1 = V (G)), G contains at least ΩH(εc1/c1nh) = ΩH(εnh) copies of C2k+1[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , s2k+1], and hence ΩH,α(εnh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This concludes the proof of this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' From now on, assume that G contains at most εcn2 edge-disjoint C2ℓ+1-copies for every ℓ ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let Cℓ be a maximal collection of edge-disjoint C2ℓ+1-copies in G, so |Cℓ| ≤ εcn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let Ec be the set of edges which 8 are contained in one of the cycles in C1 ∪ · · · ∪ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let S be the set of vertices which are incident with at least αn 10 edges from Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then |Ec| ≤ k � ℓ=1 (2ℓ + 1)εcn2 = k(k + 2)εcn2 and |S| ≤ 2 |Ec| αn/10 ≤ 20k(k + 2)εc α n < αn 10 , (1) where the last inequality holds by our assumed bound on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G′ be the subgraph of G obtained by deleting the edges in Ec and the vertices in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that G′ ⊆ G − Ec is {C3, C5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , C2k+1}-free because for every 1 ≤ ℓ ≤ k, we removed all edges from a maximal collection of edge-disjoint C2ℓ+1-copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' |V (G′)| > (1 − α 10)n and δ(G′) > ( 1 4 + 4α 5 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The first inequality follows from (1) as |V (G′)| = n − |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each v ∈ V (G)\\S has at most αn 10 incident edges from Ec, and at most |S| < αn 10 neighbors in S, thereby degG′(v) > degG(v) − αn 5 ≥ ( 1 4 + 4α 5 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, δ(G′) > ( 1 4 + 4α 5 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' G′ is homomorphic to C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Moreover, G′ is bipartite unless k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that G′ is {C3, C5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , C2k+1}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As k ≥ 2, G′ is {C3, C5}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, δ(G′) > n 4 ≥ |V (G′)| 4 by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So G′ is homomorphic to C7 by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If k ≥ 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' odd-girth(H) ≥ 7, then odd-girth(G′) ≥ 2k + 3 ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As δ(G′) > n 4 , G′ is bipartite by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The rest of the proof is divided into two cases based whether or not G′ is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' These cases are handled by Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose that G′ is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then G has ΩH,α(εnh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let (L′, R′) be a bipartition of G′, so V (G) = L′ ·∪ R′ ·∪ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let L1 ⊆ S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' R1 ⊆ S) be the set of vertices of S having at most αn 5 neighbors in L′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' R′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G′′ be the bipartite subgraph of G induced by the bipartition (L′′, R′′) := (L′ ·∪ L1, R′ ·∪ R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let S′′ = V (G)\\(L′′ ·∪ R′′), so V (G) = L′′ ·∪ R′′ ·∪ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that δ(G′′) ≥ ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' First, as G′ is a subgraph of G′′, we have degG′′(v) > ( 1 4 + 4α 5 )n for each v ∈ V (G′) ⊆ V (G′′) by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now we consider vertices in V (G′′) \\ V (G′) = L1 ∪ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each v ∈ L1 has at most |S| ≤ αn 10 neighbors in S and at most αn 5 neighbors in L′, by the definition of L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, v has at least degG(v) − 3α 10 n ≥ ( 1 4 + α 2 )n neighbors in R′ ⊆ V (G′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the symmetric argument for vertices v ∈ R1, we get that δ(G′′) ≥ ( 1 4 + α 2 )n, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For an edge uv ∈ E(G)\\E(G′′), we say uv is of type I if u, v ∈ L′′ or u, v ∈ R′′, and we say that uv is of type II if u ∈ S′′ or v ∈ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Every edge in E(G)\\E(G′′) is of type I or II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Since χ(H) = 3 and G′′ is bipartite, each copy of H in G must contain an edge of type I or an edge of type II (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As G has εn2 edge-disjoint H-copies, G contains at least εn2 2 edges of type I or at least εn2 2 edges of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now consider these two cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that xy ∈ E(H) denotes a critical edge of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Case 1: G contains εn2 2 edges of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any edge ab ∈ E(G) of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Without loss of generality, assume a, b ∈ L′′ (the case a, b ∈ R′′ is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that G has poly(α)nh−2 copies of H mapping xy ∈ E(H) to ab ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If degG(a, b) ≥ αn 2 then this holds by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Otherwise, degG(a, b) < αn 2 , and thus |R′′| ≥ |NG′′(a) ∪ NG′′(b)| ≥ degG′′(a) + degG′′(b) − degG(a, b) > 2δ(G′′) − αn 2 > n 2 , using that δ(G′′) ≥ ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Thus, |L′′| < n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This implies that for all a′ ∈ NG′′(a), b′ ∈ NG′′(b), degG′′(a′, b′) ≥ 2δ(G′′) − |L′′| ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 (with A = NG′′(a) and B = NG′′(b)), there are poly(α)nh−2 copies of H mapping xy to ab, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Summing over all edges ab of type I, we get εn2 2 · poly(α)nh−2 = poly(α)εnh copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof in Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 9 L′′ R′′ a b L′′ R′′ a b a′ b′ L′′ R′′ S′′ a b a′ b′ Figure 4: Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10: Case 1 with degG(a, b) ≥ αn 2 (left), Case 1 with degG(a, b) < αn 2 (middle) and Case 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The red part is the common neighborhood of a and b (or a′ and b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Case 2: G contains εn2 2 edges of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that the number of edges of type II is trivially at most |S′′| n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Thus, |S′′| ≥ εn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix some a ∈ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the definition of L′′, R′′ and S′′, v has at least αn 5 neighbors in L′ ⊆ L′′ and at least αn 5 neighbors in R′ ⊆ R′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Without loss of generality, assume |L′′| ≤ |R′′|, thereby |L′′| ≤ n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now fix any b ∈ L′′ adjacent to a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' there are at least αn 5 choices for b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We have |NG(a) ∩ R′′| ≥ αn 5 and |NG′′(b)| ≥ δ(G′′) > n 4 , and for all a′ ∈ NG(a) ∩ R′′, b′ ∈ NG′′(b) ⊆ R′′ it holds that degG′′(a′, b′) ≥ 2δ(G′′) − |L′′| ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Therefore, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3, G has poly(α)nh−2 copies of H mapping xy to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Enumerating over all a ∈ S′′ and b ∈ NG(a) ∩ L′′, we again get ΩH,α(εnh) copies of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose G′ is non-bipartite but homomorphic to C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then G has ΩH,α(εnh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='9 we must have k = 2 , so odd-girth(H) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The proof is similar to that of Proposi- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10, but instead of a bipartition of G′, we use a partition corresponding to a homomorphism into C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let V (G)\\S = V (G′) = V ′ 1 ·∪ V ′ 2 ·∪ · · · ·∪ V ′ 7 be a partition of V (G′) such that E(G′) ⊆ � i∈[7] V ′ i × V ′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Here and later, all subscripts are modulo 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We have V ′ i ̸= ∅ for all i ∈ [7], because otherwise G′ would be bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For i ∈ [7], let Si be the set of vertices in S having at most 2αn 5 neighbors in V (G′)\\ (V ′ i−1 ∪V ′ i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In case v lies in multiple Si’s, we put v arbitrarily in one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Set V ′′ i := V ′ i ∪ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G′′ be the 7-partite subgraph of G with parts V ′′ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , V ′′ 7 and with all edges of G between V ′′ i and V ′′ i+1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By definition, G′ is a subgraph of G′′, and G′′ is homomorphic to C7 via the homomorphism V ′′ i �→ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put S′′ := V (G)\\V (G′′) = S \\ �7 i=1 Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now collect the following useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The following holds: (i) δ(G′′) ≥ ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (ii) For every i ∈ [7] and for every u, v ∈ V ′′ i or u ∈ V ′′ i , v ∈ V ′′ i+2, it holds that degG′′(u, v) ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (iii) For every i ∈ [7], every v ∈ V ′′ i has at least αn neighbors in V ′′ i−1 and at least αn neighbors in V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (iv) For every a ∈ S′′, there are i, j with j − i ≡ 1, 3 (mod 7) and |NG(a) ∩ V ′′ i | , ��NG(a) ∩ V ′′ j �� > 2αn 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' fds (i) Let i ∈ [7] and v ∈ V ′′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If v ∈ V (G′), then degG′′(v) ≥ degG′(v) ≥ δ(G′) > ( 1 4 + α 2 )n, using Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Otherwise, v ∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By definition, v has at most 2αn 5 neighbours in V (G′)\\(V ′ i−1 ∪V ′ i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, v has at most |S| ≤ αn 10 neighbours in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It follows that v has at least degG(v)− 2αn 5 − αn 10 ≥ ( 1 4 + α 2 )n neighbors in V ′′ i−1 ∪ V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, degG′′(v) > ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (ii) First, observe that |V ′′ i | + ��V ′′ i+2 �� ≥ �1 4 + α 2 � n (2) 10 for all i ∈ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, V ′′ i+1 is non-empty, and fixing any v ∈ V ′′ i+1, we have |V ′′ i | + ��V ′′ i+2 �� ≥ degG′′(v) ≥ δ(G′′) ≥ ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By applying (2) to the pairs (i + 2, i + 4) and (i − 2, i), we get ��V ′′ i−1 �� + ��V ′′ i+1 �� + ��V ′′ i+3 �� ≤ n − ( ��V ′′ i+2 �� + ��V ′′ i+4 ��) − ( ��V ′′ i−2 �� + |V ′′ i |) ≤ n − 2 �1 4 + α 2 � n < n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (3) Now let i ∈ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For u, v ∈ V ′′ i we have NG′′(u) ∪ NG′′(v) ⊆ V ′′ i−1 ∪ V ′′ i+1, and for u ∈ V ′′ i , v ∈ V ′′ i+2 we have NG′′(u) ∪ NG′′(v) ⊆ V ′′ i−1 ∪ V ′′ i+1 ∪ V ′′ i+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In both cases, |NG′′(u) ∪ NG′′(v)| < n 2 by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As degG′′(u) + degG′′(v) ≥ 2δ(G′′) ≥ ( 1 2 + α)n, we have degG′′(u, v) > αn, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (iii) We first argue that |V ′′ i | ≤ ( 1 4 − 3α 2 )n for each i ∈ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, by applying (2) to the pairs (i − 1, i + 1), (i + 2, i + 4), (i + 3, i + 5), we get |V ′′ i | ≤ n − ( ��V ′′ i−1 �� + ��V ′′ i+1 ��) − ( ��V ′′ i+2 �� + ��V ′′ i+4 ��) − ( ��V ′′ i+3 �� + ��V ′′ i+5 ��) ≤ n − 3 �1 4 + α 2 � n = �1 4 − 3α 2 � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now, for every v ∈ V ′′ i , we have NG′′(v) ⊆ V ′′ i−1 ∪ V ′′ i+1 and ��V ′′ i−1 �� , ��V ′′ i+1 �� < ( 1 4 − 3α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, v has at least degG′′(v) − ( 1 4 − 3α 2 )n ≥ αn neighbors in each of V ′′ i−1, V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (iv) Let I be the set of i with |NG(a) ∩ V ′′ i | ≥ 2αn 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If I is empty, then a has less than 5 · 2αn 25 = 2αn 5 neighbors in every V (G′)\\(V ′ i−1 ∪V ′ i+1) and therefore can not be in S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose for contradiction that there exist no i, j ∈ I with j − i ≡ 1, 3 (mod 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that there is j ∈ [7] such that I ⊆ {j, j + 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix an arbitrary i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, i ± 1, i ± 3 /∈ I by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, at most one of i + 2, i − 2 is in I, because (i − 2) − (i + 2) ≡ 3 (mod 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So I ⊆ {i, i + 2} or I ⊆ {i − 2, i}, proving our claim that I ⊆ {j, j + 2} for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the definition of I, a has at most 5 · 2αn 25 = 2αn 5 neighbors in V (G′)\\(V ′ j ∪ V ′ j+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, a ∈ Sj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This contradicts the fact that a ∈ S′′, as S′′ ∩ Si+1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We continue with the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that the edges in E(G) \\ E(G′′) are precisely the edges of G not belonging to � i∈[7] V ′′ i × V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For an edge ab ∈ E(G)\\E(G′′), we say ab is of type I if a, b ∈ V (G′′), and of type II if a ∈ S′′ or b ∈ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Clearly, every edge in E(G)\\E(G′′) is either of type I or of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Since odd-girth(H) = 5 and C5 is not homomorphic to C7, every H-copy in G must contain some edge of type I or of type II (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As G has εn2 edge-disjoint H-copies, G must have at least εn2 2 edges of type I or at least εn2 2 edges of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We consider these two cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 5 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that xy ∈ E(H) denotes a critical edge of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Case 1: G contains εn2 2 edges of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any edge ab of type I, where a ∈ V ′′ i and b ∈ V ′′ j for i, j ∈ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now show that G has poly(α)nh−2 copies of H mapping xy ∈ E(H) to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As ab /∈ E(G′′), we have i−j ≡ 0, ±2, ±3 (mod 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' When j−i ≡ 0, ±2 (mod 7), we have degG(a, b) ≥ degG′′(a, b) > αn by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, G has poly(α)nh−2 copies of H mapping xy to ab, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now suppose that j−i ≡ ±3 (mod 7), say j ≡ i+3 (mod 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Denote A := NG(a)∩V ′′ i−1 and B := NG(b)∩V ′′ j+1 = NG(b)∩V ′′ i−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We have that |A| , |B| ≥ αn by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (iii), and |NG(a′, b′)| > αn for all a′ ∈ A, b′ ∈ B by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3, G has poly(α)nh−2 copies of H mapping xy to ab, proving our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Summing over all edges ab of type I, we get εn2 2 · poly(α)nh−2 = ΩH,α(εnh) copies of H in G, finishing this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Case 2: G contains εn2 2 edges of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Notice that the number edges incident to S′′ is at most |S′′| n, meaning that |S′′| ≥ εn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any a ∈ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (iv), there exist i, j ∈ [7] with j − i ≡ 1, 3 (mod 7) and |NG(a) ∩ V ′′ i | , ��NG(a) ∩ V ′′ j �� > 2αn 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any b ∈ NG(a) ∩ V ′′ i (there are at least 2αn 25 choices for b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Take A = NG(a)∩V ′′ j and B = NG(b)∩V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We have that |A| ≥ 2αn 25 , and |B| ≥ αn by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Further, as j − (i + 1) ≡ 0, 2 (mod 7), Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (ii) implies that |NG(a′, b′)| > αn for all a′ ∈ A, b′ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3, G has poly(α)nh−2 copies of H mapping xy to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Summing over all choices of a ∈ S′′ and b ∈ V ′′ i , we acquire |S′′| · 2αn 25 · poly(α)nh−2 = ΩH,α(εnh) copies of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof of Case 2, and hence the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11 imply the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 11 V ′′ 1 V ′′ 2 V ′′ 3 V ′′ 4 V ′′ 5 V ′′ 6 V ′′ 7 a b V ′′ 1 V ′′ 2 V ′′ 3 V ′′ 4 V ′′ 5 V ′′ 6 V ′′ 7 a b a′ b′ V ′′ 1 V ′′ 2 V ′′ 3 V ′′ 4 V ′′ 5 V ′′ 6 V ′′ 7 S′′ a b a′ b′ Figure 5: Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11: Case 1 for j = i + 2 (left), Case 1 for j = i + 3 (middle) and Case 2 for j = i + 3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The red part is the common neighborhood of a and b (or a′ and b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 6 Concluding remarks and open questions It would be interesting to determine the possible values of δpoly-rem(H) for 3-chromatic graphs H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So far we know that 1 2k+1 is a value for each k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Is there a graph H with 1 5 < δpoly-rem(H) < 1 3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, is it true that δpoly-rem(H) > 1 5 if H is not homomorphic to C5?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Another question is whether the inequality in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4 is always tight, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' is it always true that δpoly-rem(H) = δhom(IH)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Finally, we wonder whether the parameters δpoly-rem(H) and δlin-rem(H) are monotone, in the sense that they do not increase when passing to a subgraph of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We are not aware of a way of proving this without finding δpoly-rem(H), δlin-rem(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Allen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Böttcher, S.' 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sha256:6503929c15108ce7ea45b4c8accf863438e2f04a9512a8567c8c763233d742b0 +size 4063277 diff --git a/5dA0T4oBgHgl3EQfNv_S/content/tmp_files/2301.02152v1.pdf.txt b/5dA0T4oBgHgl3EQfNv_S/content/tmp_files/2301.02152v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..53e852ffdec379a4ce783036ed3e2606b1727cf0 --- /dev/null +++ b/5dA0T4oBgHgl3EQfNv_S/content/tmp_files/2301.02152v1.pdf.txt @@ -0,0 +1,2837 @@ +L-HYDRA: MULTI-HEAD PHYSICS-INFORMED NEURAL +NETWORKS +ZONGREN ZOU∗ AND GEORGE EM KARNIADAKIS† +Abstract. We introduce multi-head neural networks (MH-NNs) to physics-informed machine +learning, which is a type of neural networks (NNs) with all nonlinear hidden layers as the body and +multiple linear output layers as multi-head. Hence, we construct multi-head physics-informed neural +networks (MH-PINNs) as a potent tool for multi-task learning (MTL), generative modeling, and +few-shot learning for diverse problems in scientific machine learning (SciML). MH-PINNs connect +multiple functions/tasks via a shared body as the basis functions as well as a shared distribution +for the head. The former is accomplished by solving multiple tasks with MH-PINNs with each head +independently corresponding to each task, while the latter by employing normalizing flows (NFs) for +density estimate and generative modeling. To this end, our method is a two-stage method, and both +stages can be tackled with standard deep learning tools of NNs, enabling easy implementation in +practice. MH-PINNs can be used for various purposes, such as approximating stochastic processes, +solving multiple tasks synergistically, providing informative prior knowledge for downstream few-shot +learning tasks such as meta-learning and transfer learning, learning representative basis functions, +and uncertainty quantification. We demonstrate the effectiveness of MH-PINNs in five benchmarks, +investigating also the possibility of synergistic learning in regression analysis. We name the open- +source code “Lernaean Hydra” (L-HYDRA), since this mythical creature possessed many heads for +performing important multiple tasks, as in the proposed method. +Key words. +PINNs, meta-learning, multi-tasking, transfer learning, generative models, nor- +malizing flows, stochastic problems +MSC codes. 34F05, 62M45, 65L99, 65M99, 65N99 +1. Introduction. Learning across tasks has drawn great attention recently in +deep learning and is an emerging theme in scientific machine learning (SciML), due +to the fact that several classes of scientific problems are similar and/or related in- +trinsically by their common physics. +Intuitively, if tasks are similar, e.g., in the +context of approximating stochastic processes [44], learning solution operators of ordi- +nary/partial differential equations (ODEs/PDEs) [28], and solving parametric PDEs +[42, 19, 4], it may be beneficial to relate them in the modeling, algorithm design, +and/or solving procedure. In this regard, machine learning solvers, developed rapidly +in the past few years, are considerably more flexible and of higher potential compared +to traditional numerical solvers. Significant progress has been witnessed in the general +area, including meta-learning for solving ODEs/PDEs [30, 27, 34, 6], transfer learning +for physics-informed neural networks (PINNs) [3, 7], transfer learning for domain shift +in solving PDEs [14], multi-task learning for PINNs [40], and generative methods for +solving stochastic differential equations (SDEs) [44, 46, 15]. More recently, operator +learning [28, 24] in which direct operator mapping is learned and subsequently used +for other tasks in one-shot format has attracted a lot of attention. +Multi-head neural networks (MH-NNs) fit perfectly different scenarios of learning +across tasks. They were originally proposed as members of hard-parameter sharing +neural networks (NNs) for deep multi-task learning (MTL) [5], in which multiple +tasks, denoted as Tk, k = 1, ..., M, where M is the number of total tasks, are solved +simultaneously. The general goals of using MH-NNs in MTL are diverse: achieving +∗Division of Applied Mathematics, +Brown University, +Providence, +RI 02912, +USA (zon- +gren zou@brown.edu). +†Corresponding author. +Division of Applied Mathematics, Brown University, Providence, RI +02912, USA (george karniadakis@brown.edu). +1 +arXiv:2301.02152v1 [cs.LG] 5 Jan 2023 + +2 +Z. ZOU AND G. E. KARNIADAKIS +better performance for all tasks, learning good and useful representations for down- +stream tasks, and/or boosting the learning of main tasks with the help of auxiliary +tasks. Moreover, although originally designed for solving multiple tasks, MH-NNs in +recent years have also been extensively used for meta-learning. For example, in [41], +the connection between MTL and meta-learning was analyzed, and meta-learning al- +gorithms for MH-NN were discussed; in [25], it was shown that MH-NNs, trained in +MTL fashion also perform task-specific adaptation in meta-learning; [37] argued that +the effectiveness of model-agnostic meta-learning [10], a well-known meta-learning al- +gorithm, may be due to successfully learned good representations rather than learned +adaptation, and MH-NNs were used to study the detailed contributions of NNs in +fast task adaptations. Overall, it is commonly acknowledged in the literature that +when used to solve previous tasks, MH-NNs are capable of distilling useful shared +information and storing it in their bodies and heads. +In this paper, we develop MH-NNs for physics-informed machine learning [17], +propose multi-head physics-informed neural networks (MH-PINNs), and further in- +vestigate their applicability and capabilities to MTL, generative modeling, and meta- +learning. A MH-PINN, as shown in Fig. 1, is built upon a conventional MH-NN and +consists of two main parts, the body and multiple heads, and each head connects to +a specific ODE/PDE task. Many architecture splitting strategies for MH-NNs are +adopted in different applications scenarios; e.g., for some computer vision problems, +a NN is split such that the body consists of convolutional layers and is followed by +fully-connected layers as heads. In this paper, however, we choose the simplest one, +i.e., the body consists of all nonlinear layers and the head is the last linear layer, +for the following two reasons: (1) the dimensionality of the head is reduced, which +enables fast density estimation (see next section); and (2) the body spontaneously +provides a set of basis functions. +Fig. 1. Schematic view of the structure of multi-head physics-informed neural networks (MH- +PINNs) with M different heads, which are built upon conventional multi-head neural networks. +The shared layers are often referred to as body and the task-specific layer as head. +Generally, +uk, k = 1, ..., M represent M solutions to M different ODEs/PDEs, formulated in Eq. (2.1), which +may differ in source terms fk, boundary/initial condition terms bk, or differential operator Fk. +The novelty and major contributions of this work are as follows: +1. We propose a new physics-informed generative method using MH-PINNs for +learning stochastic processes from data and physics. +2. We propose a new method for physics-informed few-shot regression problems +with uncertainty quantification using MH-PINNs. +3. We study and demonstrate the effectiveness of MTL and synergistic learning +with MH-NNs in regression problems. +The paper is organized as follows. In Sec. 2, we present the problem formulation, + +Fiui(α)/ = fi(α), Biui(α)/ = bi(α) +head +task, +1 +F2[u2(α)] = f2(α), B2[u2(α)] = b2(α) +head, +u2 +Body +- +FM[uM(α)] = fM(α), BM[uM(c)] = bM(c) +head +uM +task, +ML-HYDRA +3 +details of MH-PINNs, and the general methodology, including how to use MH-PINNs +for MTL, generative modeling, downstream few-shot physics-informed learning with +uncertainty quantification (UQ). In Sec. 3, we discuss existing research closely related +to our work and compare them conceptually. In Sec. 4, we test MH-PINNs with five +benchmarks, each of which corresponds to one or more learning purposes, e.g., MTL +and generative modeling. +In Sec. 5, we investigate MTL and synergistic learning +with the function approximation example. We conclude and summarize in Sec. 6. +The details of our experiments, such as NN architectures and training strategies, can +be found in Appendix A and B, as well as in the L-HYDRA open-source codes on +GitHub, which will be released once the paper is accepted. +2. Methodology. We assume that we have a family of tasks, {Tk}M +k=1, each of +which is associated with data Dk, k = 1, ..., M. The primary focus of this paper is on +scientific computing and ODEs/PDEs, and therefore we further assume {Tk}M +k=1 are +physics-informed regression problems [17]. +Consider a PDE of the following form: +Fk[uk(x)] = fk(x), x ∈ Ωk, +(2.1a) +Bk[uk(x)] = bk(x), x ∈ ∂Ωk, +(2.1b) +where k denotes the index of the task and k = 1, ..., M, x is the general spatial- +temporal coordinate of Dx dimensions, Ωk are bounded domains, fk and uk are the +Du-dimensional source terms and solutions to the PDE, respectively, Fk are general +differential operators, Bk are general boundary/initial condition operators, and bk are +boundary/initial condition terms. For simplicity, throughout this paper, the domain +and the boundary/initial operator, denoted as Ω and B, are assumed to be the same +for all tasks, and the solutions uk to be task-specific. The task Tk is described as +approximating uk, and/or fk, and/or Fk, and/or bk, from data Dk and Eq. (2.1). +Traditional numerical solvers often tackle {Tk}M +k=1 independently, without lever- +aging or transferring knowledge across tasks. The PINN method [38] was designed to +solve ODEs/PDEs independently using NNs, which, however, yields M uncorrelated +results. In this paper instead we treat {Tk}M +k=1 as a whole and connect them with MH- +PINNs, the architecture of which, shown in Fig. 1, enforces basis-functions-sharing +predictions on the solutions uk. In addition to the informative representation/body, +we further relate {Tk}M +k=1 by assuming that their corresponding heads in MH-PINNs, +denoted as {Hk}M +k=1, are samples of a random variable with unknown probability +density function (PDF), denoted as H and p(H), respectively. The shared body and +a generative model of H immediately form a generative model of the solution u, and +generators of the source term f and the boundary/initial term b as well by substitut- +ing u into Eq. (2.1) and automatic differentiation [1], from which a generative method +for approximating stochastic processes is seamlessly developed. +Generators of u, f and b, as discussed in [30], are able to provide an informative +prior distribution in physics-informed Bayesian inference [43, 26] as well as in UQ +for SciML [47, 36], where the informative prior compensates for the insufficiency of +observational data to address the physics-informed learning problems with even a few +noisy measurements. In this paper, we generalize such problem to deterministic cases +as well, where the data is noiseless and methods and results are deterministic, and +refer to it as few-shot physics-informed learning. The general idea is to apply prior +knowledge learned from connecting {Tk}M +k=1 with MH-PINNs to new tasks, denoted +as ˜T , associated with insufficient data ˜D, for accurate and trustworthy predictions. + +4 +Z. ZOU AND G. E. KARNIADAKIS +The schematic view of the learning framework is illustrated in Fig. 2, and the details +are explained next. +Fig. 2. Schematic view of the learning framework and the proposed method. Three general +types of learning are addressed: physics-informed learning, generative modeling, and few-shot learn- +ing. The physics-informed learning is performed with MH-PINNs; the generative modeling is done +afterwards by density estimate over the head via normalizing flows (NFs); in the end the few-shot +physics-informed learning is accomplished with prior knowledge obtained from previous two via either +fine-tuning with the learned regularization or Bayesian inference with the learned prior distribution. +The body represents the set of basis functions learned from solving {Tk}M +k=1 with MH-PINNs, and +the density of the head, estimated from its samples using NFs, acts as the regularization, the prior +distribution, or the generator together with the body, depending on the usage of MH-PINNs in ap- +plications. +2.1. Multi-head physics-informed neural networks (MH-PINNs). Hard +parameter sharing is the most commonly used approach when MTL with NNs are +considered, and MH-NNs, as its simplest instance, are frequently adopted [5, 39]. +A MH-PINN, as described earlier, is composed of a body and multiple heads. We +denote by Φ the body and by Hk the head for Tk. Notice that here Φ : RDx → RL is +a function parameterized by a neural network with parameter θ, and Hk ∈ RL+1 is a +vector, where L is the number of neurons on the last layer of the body. Let us define +Hk = [h0 +k, h1 +k, ..., hL +k ]T , Φ(x) = [φ1(x), ..., φL(x)]T , where φ : RDx → R, and then the +surrogate for the solution in Tk can be rewritten as ˆuk(x) = h0 +k+�L +l=1 hl +kφl(x), ∀x ∈ Ω. +The approximated source terms and boundary/initial terms are derived from Eq. (2.1) +accordingly. In the MTL framework , given data {Dk}M +k=1 and physics Eq. (2.1), the +loss function L is formulated as follows: +(2.2) +L({Dk}M +k=1; θ, {Hk}M +k=1) = 1 +M +M +� +k=1 +Lk(Dk; θ, Hk), +where Lk denotes the common loss function in PINNs. Conventionally, the data for +Tk is expressed as Dk = {Df +k, Db +k, Du +k}, where Df +k = {xi +k, f i +k} +N f +k +i=1, Db +k = {xi +k, bi +k}N b +k +i=1 and + +samples of head +Learned distribution of head +1.2 +1.2 +1.0 +1.0 - +0.8 +0.8 +learned by normalizing +0.6 +0.6 +flows +0.4 +0.4 - +0.2 +0.2 +0.0 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +1.0 +-0.5 +0.0 +0.5 +1.0 +generative modeling { Hkl +prior knowledge +new tasks T +Fine-tuning +Body +Bayesian inference +1.00-0.750.500.250.000.250.500.751.00L-HYDRA +5 +Du +k = {xi +k, ui +k}N u +k +i=1, and Lk as follows: +(2.3) +Lk(Dk; θ, Hk) = wf +k +N f +k +N f +k +� +i=1 +||Fk(ˆuk(xi +k)) − f i +k||2 + wb +k +N b +k +N b +k +� +i=1 +||B(ˆuk(xi +k)) − bi +k||2 ++ wu +k +N u +k +N u +k +� +i=1 +||ˆuk(xi +k) − ui +k||2 + R(θ, Hk), +where || · || represents a properly chosen norm, R(·) is a regularization method over +the parameters of NNs, N f +k , N b +k, N u +k are the numbers of data points for fk, bk, uk, and +wf +k, wb +k, wu +k are weights to balance different terms in the loss function. +2.2. Generative modeling and normalizing flows (NFs). As mentioned +earlier, MH-PINNs connect {Tk}M +k=1 by making two assumptions: (1) the solutions +uk, k = 1, ..., M share the same set of basis functions, Φ; and (2) the corresponding +coefficients are samples of the same random variable, H. In [7], Φ was used as a carrier +of prior knowledge from {Tk}M +k=1 in downstream physics-informed learning tasks. In +this work, we extend it by utilizing the information from the head as well by estimating +the PDF and a generator of H from its samples, {Hk}M +k=1, using normalizing flows +(NFs). The interested readers are directed to [32, 22] for reviews of NFs as well as +[9, 33, 20] for developments of some popular NFs. +We choose NFs over other commonly used generative models, e.g., generative +adversarial networks (GANs) [13], variational auto-encoders (VAEs) [21], or diffusion +models [16], because the NF serves as both a density estimator and a generator. The +former is able to provide proper regularization in the downstream few-shot physics- +informed learning tasks, while the latter leads to a physics-informed generative method +for approximating stochastic processes. It is worth noting that in previous works on +physics-informed generative methods [44, 46, 15], NNs are trained by measurements +over uk, and/or fk, and/or bk. Our model, on the other hand, learns through samples +of the head, which is obtained from MTL in the first step. This learning strategy +brings two substantial advantages: (1) flexibility in dealing with unstructured data, +e.g., inconsistent measurements across tasks; (2) simplicity and controlability of the +training by decoupling the physics-informed learning and the generative modeling. +2.3. Prior knowledge utilized in the downstream tasks. Here, we describe +details on how to utilize the prior knowledge stored in MH-PINNs, for downstream +few-shot physics-informed learning task, ˜T , which is defined the same as all other +tasks in the upstream training, but with much fewer measurements. Training of MH- +PINNs and NFs yield a body, Φ, samples of heads, {Hk}M +k=1, and an estimated PDF +of the head, ˆp(H) ≈ p(H). In solving ˜T with ˜D, we fix the body Φ and find the head +˜H that best explains the data ˜D and the physics in Eq. (2.1). Noiseless and noisy data +are considered in this paper: for noiseless data, regular NN training is performed on +the head for new tasks to provide deterministic predictions, where the learned PDF +of the head, ˆp(H), acts as a regularization term in the loss function; for noisy data, +Bayesian inference is performed on the head as well, in which ˆp(H) denotes the prior +distribution. Details are presented in the following. +2.3.1. Regularization in optimization. Limited data in few-shot learning +often leads to over-fitting and/or poor inter-/extrapolation performance. In this re- +gard, regularizing the head according to its PDF is able to prevent over-fitting and + +6 +Z. ZOU AND G. E. KARNIADAKIS +provide additional prior knowledge for better inter-/extrapolation performance. The +optimization problem is cast as +(2.4) +˜H∗ = arg min +˜ +H +L∗( ˜D; ˜H), where L∗( ˜D; ˜H) = L( ˜D; ˜H) − α log p( ˜H) +≈ L( ˜D; ˜H) − α log ˆp( ˜H), +where L is the regular loss function in physics-informed learning for data ˜D and param- +eter ˜H, and α ≥ 0 is the coefficient to adjust the regularization effect. Problem (2.4) +in this work is solved with gradient descent. +2.3.2. Prior distribution in Bayesian inference. As opposed to point esti- +mate obtained by solving the optimization problem (2.4), the posterior distribution +of the head in ˜T is obtained using Bayesian inference. Similar as in [30, 47], the +posterior distribution of ˜H is established as follows: +(2.5) +p( ˜H| ˜D) ∝ p( ˜D| ˜H)p( ˜H) ≈ p( ˜D| ˜H)ˆp( ˜H), +where p( ˜H| ˜D) is the posterior distribution, p( ˜D| ˜H) is the likelihood distribution, +which is often assumed to be independent Gaussian over all measurements in ˜D, +and ˆp is the estimated PDF of the head via NFs. Intractability of distribution (2.5) +requires approximation methods, among which Markov chain Monte Carlo methods, +such as Hamiltonian Monte Carlo (HMC) [31], generally provide the most accurate +estimation. +Moreover, the relatively low dimension of ˜H also enables the use of +Laplace’s approximation (LA) [18], which is employed in this paper as an alternative +to HMC. +3. Related works. Deep NNs in recent years have been extensively investigated +for solutions of ODEs/PDEs, SDEs as well as operator learning. Although not explic- +itly introduced as MH-PINNs, MH-NNs were first used to solve ODEs/PDEs by [7], +in which MH-PINNs were pre-trained on multiple similar tasks, and then the heads +were discarded while the body was kept and transferred to solving new tasks, by either +least square estimate for linear ODEs/PDEs, or fine-tuning with gradient descent for +nonlinear ones. In [7], a one-shot transfer learning algorithm for linear problems was +proposed but other potential uses of MH-NNs, e.g., MTL and generative modeling, +were not discussed, as opposed to the work presented herein. Furthermore, [7] focused +only on fast and deterministic predictions with high accuracy using sufficient clean +data, while in this paper, we study the applicability of MH-NNs to few-shot physics- +informed learning as well, where data is insufficient and/or noisy, and address such +cases with UQ. We note that MH-NN was also used as a multi-output NN in [45], +which, however, focused on solving single tasks and obtaining uncertainties. +Generative modeling in the context of scientific computing has also been studied +recently, and a few attempts for adopting deep generative NNs to SciML problems +have been made in [44, 46, 15], most of which have focused on approximating stochas- +tic processes and on solving SDEs. We propose a new physics-informed generative +method, as an alternative to the current ones, using MH-PINNs, and test it in approxi- +mating stochastic processes. In this regard, our method is functionally the same as the +current ones, but technically different. All previous methods address physics-informed +generative modeling using end-to-end learning strategies by coupling two dissimilar +types of learning, physics-informed learning and generative modeling, which may be +problematic for implementation and usage in practice when either type of learning +becomes more complicated. Our method, on the other hand, addresses the problem in + +L-HYDRA +7 +an entirely new angle by decoupling those two: physics-informed learning is performed +first and is followed by learning generators. To this end, our method is a two-step +method, and with the help of well-developed algorithms from both fields, our method +has advantages both in flexibility and simplicity in implementation. +4. Results. In this section, we test our method using five benchmarks. The first +one is a pedagogical function regression, in which we aim to demonstrate the basic +applicability and capabilities of our method, showing the importance of incorporating +the distribution of the head in the downstream tasks and in obtaining results with +or without uncertainty. The second example is a nonlinear ODE system, in which +we test our method in approximating stochastic processes through a differential op- +erator, compare different NFs, and eventually compare our method with another +well-known physics-informed generative model, physics-informed GANs (PI-GANs) +[44] in generative modeling. The third is a 1-D nonlinear reaction-diffusion equation, +the fourth is a 2-D nonlinear Allen-Cahn equation, and the fifth is the 2-D stochastic +Helmholtz equation with 20 dimensions. In all examples unless stated otherwise, data +for {Dk}M +k=1 are noise-free and task-wisely sufficient, while ˜D in downstream tasks is +insufficient, which makes the downstream tasks of the few-shot type. In addition, ex- +cept for the first example, results from Bayesian inference are obtained by employing +HMC, and the predicted mean denoted as µ and predicted standard deviation denoted +as σ are computed from the posterior samples of functions or unknown parameters. +The predicted uncertainty is defined as 2σ in this paper. +4.1. Function approximation. We start with a function regression problem +using only data and no physics, which is a degenerate instance of Eq. (2.1) with Fk +being fixed as an identity operator, no B and bk, and uk = fk being task-specific. In +this case, Dk and ˜D are given as {(xi +k, f i +k)}Nk +i=1 and {(xi, f i)}N +i=1, respectively, and Tk +and ˜T are defined as approximating functions fk and ˜f from Dk and ˜D, respectively. +The stochastic function f in this example is defined as follows: +(4.1) +f(x) = A cos(ωx) + 2βx, x ∈ [−1, 1], +A ∼ U[1, 3), ω ∼ U[2π, 4π), P(β = ±1) = 0.5, +where U stands for uniform distribution and P(Ξ) is defined as the probability of +the event Ξ. Our goal is to approximate f from data with MH-NNs and NFs, and +solving the downstream few-shot regression tasks ˜T as well, in which two functions, +2 cos(2πx)+2x and 2 cos(4πx)−2x, are regressed from 4 and 5 measurements equidis- +tantly distributed on [−0.9, −0.1], respectively. +For the training of MH-NNs and NFs, 1, 000 f subject to Eq. (4.1) are sampled, +each of which forms a regression task with 40 measurements sampled equidistantly on +[−1, 1] as data. Samples of f for training are displayed in Fig. 3(a). Both noiseless +and noisy data cases are considered in the few-shot regression tasks. As described in +Sec. 2.3, the former is solved by fine-tuning the head using gradient descent, while +the latter is solved by estimating the posterior distribution (Eq. (2.5)) using HMC +and LA. The noise ε is assumed to be independent additive Gaussian noise with scale +0.2, i.e., ε ∼ N(0, 0.22). In the downstream few-shot regression tasks we compare our +method with two other approaches, the transfer learning (TL) method from [7], which +only transfers the body, Φ, and the regular NN method, in which no prior knowledge +is employed and all parameters of NN are trained. +Results for approximating f and solving the downstream tasks are presented in +Fig. 3. As shown in Fig. 3(a), our method approximates the stochastic function f + +8 +Z. ZOU AND G. E. KARNIADAKIS +(a) +(b) +(c) +Fig. 3. Results for approximating the stochastic function defined in Eq. (4.1) and solving the +downstream few-shot regression tasks. (a) Left: 1, 000 samples generated from the exact distribu- +tion; middle: 1, 000 samples generated from the learned generator; right: statistics computed from +samples, in which we refer to the interval of mean ± 2 standard deviations as bound. (b)/(c) Results +for the downstream tasks. Left: results for noiseless cases using our method, the transfer learning +(TL) method in [7], and regular NN method; middle: results for noisy case using our method with +HMC for posterior estimate; right: results for the same noisy case using our method with LA for +posterior estimate. +well, demonstrating the capability of MH-NNs in generative modeling. In solving the +downstream tasks with noiseless data, L2 regularization is imposed in the TL method +and regular NN method, to prevent over-fitting when only 4 or 5 measurements are +available. As we can see from Figs. 3(b) and (c), our approach yields accurate predic- +tions and performs significantly better than the other two in both tasks, particularly +in the region where there are no measurements. By comparing our approach with +the NN method, we can see that prior knowledge of f is learned from {Tk}M +k=1 and +transferred successfully to new downstream tasks. By comparing our approach with +the TL method, we can see that the prior knowledge is stored in both the body and +(the distribution of) the head. For the noisy cases, it is shown that, for both tasks and +both posterior estimating methods, the predictions are accurate and trustworthy: the +predicted means agree with the references and the errors are bounded by the predicted +uncertainties. It is worth noting that the predicted uncertainties do not develop in the +interval [0, 1] and show periodic patterns, even if there are no measurements. That is +because an informative prior, which is learned by MH-NNs and NFs, is imposed on +the head in Bayesian inference. +The target functions in downstream tasks considered previously are chosen to be +in-distribution. They are regressed well with insufficient data, mainly because they + +4 +3 +2 +0 +-1 +-2 +4 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.84 +3 +2 +0 +-1 +-2 +-3 +4 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.86 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.86 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8Predicted mean +Predicted bound +Reference mean +Reference bound +2 +.2 +6 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.85 +Measurements +Reference +Ours +3 +TL +2 +NN +0 +1 +-2 +-3 +4 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.85 +2 std +4 +Measurements +Reference +3 +Mean +2 +0 +-1 +-2 +-3 +4 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.85 +2 std +4 +Measurements +Reference +3 +Mean +2 +0 +-1 +-2 +-3 +4 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.85 +4 +0 +-1 +-2 +-3 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8L-HYDRA +9 +belong to the space of functions, on which the generator is trained. However, when +functions in the downstream tasks are out-of-distribution (OOD), our approach fails +to produce good predictions, even if the data is sufficient, as shown in Fig. 4. Here, +the target function is chosen to be 2 cos(4.5π) + x with both ω and β being OOD. +Fluctuations are predicted but do not match the reference. In Fig. 4, we can further +see that when data is sufficient, a NN trained from scratch significantly outperforms +our approach, showing that, for OOD functions, the more we rely on the learned +regularization, which is indicated by the value of α in Eq. (2.4), the more erroneous +the prediction is. +Fig. 4. Results for regression on an out-of-distribution function. Left: few-shot regression with +clean data using our approach; middle: few-shot regression with noisy data using our approach with +HMC for posterior estimate; right: regression with sufficient clean data using regular NN method +and our approach with different regularization terms, α in Eq. (2.4). +4.2. Nonlinear ODE system. In this example, we consider the following ODE +system [28], which describes the motion of a pendulum with an external force: +(4.2) +du1 +dt = u2, +du2 +dt = −λ sin(u1) + f(t), +with initial condition u1(0) = u2(0) = 0. In Eq. (4.2), f is the external force and +λ is a constant. Here, to demonstrate and study the capability of our method in +generative modeling, we first consider the case where f is a Gaussian process and +λ = 1 is known, which is referred to as the forward problem. Different from previous +studies [44, 46, 15], in which a stochastic process is approximated directly by the +output of NNs, in this example we place the differential operator right after NNs and +approximate the stochastic process f as the source term in Eq. (2.1). We also test +our method on the inverse problem, where the values of λ in Eq. (4.2) are unknown in +{Tk}M +k=1 and ˜T . The forward problem corresponds to Eq. (2.1) with Fk, bk being the +same for all tasks and uk, fk being task-specific, while the inverse problem corresponds +to Eq. (2.1) with bk being the same and uk, fk, and the differential operator Fk being +different as a consequence of task-specific λ. +4.2.1. Forward problem. We first assume λ = 1 in Eq. (4.2) is known, and +the data on f is available, i.e., Dk = {(xi +k, f i +k)}Nk +i=1, k = 1, ..., M. As described before, +we employ MH-PINNs to solve {Tk}M +k=1 all at once and then employ NFs to learn the +distribution of the head. Consequently, we obtain generators of f and u. In this case, +f is assumed to be a Gaussian process with squared kernel function: +(4.3) +f(t) ∼ GP(0, K), t ∈ [0, 1], K(x, x′) = exp(−|x − x′|2 +2l2 +), +where the correlation length l is set to 0.1, 0.25, 0.5. As discussed in Sec. 2.2, many +types of NFs have been developed in the past decade for generative modeling and + +5 +Measurements +Reference +Ours +3 +0 +-2 +-3 +-0.8 +-0.6 +-0.4 +-1 +-0.2 +0 +0.2 +0.4 +0.6 +0.85 +2 std +4 +Measurements +Reference +3 +Mean +2 +-2 +-3 +4 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.85 +Measurements +4 +Reference +NN +3 += 10-6 +2 += 10-4 +10-2 +0 +-2 +-3 +4 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +110 +Z. ZOU AND G. E. KARNIADAKIS +density estimate. MH-PINNs, when used as generators, are compatible with all NFs. +In this regard, we compare three popular NFs, RealNVP [9], MAF [33] and IAF [20], +and eventually compare MH-PINNs (with NFs) against PI-GAN [44] in approximating +the Gaussian process defined in Eq. (4.3) with different correlation lengths. +For the training of MH-PINNs and NFs as well as PI-GANs, 2, 000 f are sampled +with respect to Eq. (4.3), each of which forms a physics-informed regression task with +65 measurements of f equidistantly sampled on [0, 1] as data. Notice that the ODE +system in Eq. (4.2) can be rewritten in a simpler format as follows: +(4.4) +utt = −λ sin(u) + f(t), t ∈ [0, 1], +with initial conditions u(0) = ut(0) = 0. Hence, we choose to use Eq. (4.4) to build +the loss function for physics-informed learning. +Results for comparisons are shown in Fig. 5 and Table 1. +From the spectral +analysis of the approximated Gaussian processes shown in Fig. 5, we can see that +MH-PINNs with MAF and IAF are comparable with PI-GANs while MH-PINNs +with RealNVP fall marginally behind. As shown in Table 1, the computational costs +of MH-PINNs with MAF and RealNVP are significantly lower than PI-GANs, while +MH-PINNs with IAF is more expensive than PI-GANs. We note that in this example +we also record the computational cost for sampling using different generators. As +shown in Table 1, PI-GANs are significantly faster in generating samples. That is +because, generally, GANs require relatively shallow NNs as opposite to NFs, for which +a deep architecture is needed to keep up the expressivity. Among three NFs, IAF is +the fastest in sampling while MAF is the lowest, as opposed to training, which is +consistent with the properties of those two NFs: MAF is slow for the forward pass, +which is used to generate samples, and fast for the inverse pass, which is used to +compute the density, while IAF is the opposite. Despite the fact that MAF is slow in +sampling, considering its fast training and good performance, we equip MH-PINNs +with MAF as the density estimator and the generator for all other examples in this +paper. +Fig. 5. +Approximating Gaussian processes as the source term in Eq. (4.2) using different +models: +spectra of the correlation structure for the learned generators, for different correlation +lengths, l. The covariance matrix is constructed using 10, 000 generated samples, and eigen-values +are averaged over 10 generators trained independently. +4.2.2. Inverse problem. Next, we assume λ in Eq. (4.2) is unknown, and some +measurements of u are available, in addition to f, i.e., Dk = {{xi +k, f i +k} +N f +k +i=1, {xi +k, ui +k}N u +k +i=1}. +MH-PINNs are first employed to infer uk as well as λk from data Dk and physics, and +NFs are employed afterwards to learn from samples of Hk and λk. To this end, the +generative model is for the joint distribution of u, f and λ. Here, we assume f follows +a truncated Karhuen-Loeve (KL)-expansion, with 5 leading terms, of the Gaussian +process with squared kernel function and correlation length 0.1, and for each task Tk, + +1 = 0.1 +0.3 +e-Reference +PI-GAN +0.25 +MAF +IAF +0.2 +RealNVP +eigenvalues +0.15 +0.1 +0.05 +0 +0 +5 +10 +15 +components1 = 0.25 +0.7 +0.6 +0.5 +eigenvalues +0.4 +0.3 +0.2 +0.1 +0 +0 +5 +10 +15 +components1 = 0.5 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +5 +10 +15 +componentsL-HYDRA +11 +MAF +IAF +RealNVP +PI-GAN +Phase 1 +134s +134s +134s +N/A +Phase 2 +252s +3939s +245s +N/A +Total +386s +4073s +379s +3243s +Sampling +1.98 × 10−1s +1.48 × 10−2s +1.50 × 10−2s +2.29 × 10−3s +Table 1 +Computational time for different models to approximate Gaussian process with correlation +length l = 0.1. +The MH-PINN method is a two-step method and hence its computation is de- +composed into two parts: training MH-PINNs referred to as phase 1 and training NFs referred to +as phase 2. Sampling time is defined to be the average time needed to generate 10, 000 samples of u. +λk = 1 +2 exp( +� +[0,1] f 2 +k(t)dt). As for the downstream task ˜T , the target is to infer u and +λ from insufficient data of u and f. +For the training of MH-PINNs and NFs, 2, 000 samples of f are generated and +displayed in Fig. 6(a). For each task, we assume 33 measurements of fk and 9 mea- +surements of uk, equidistantly distributed on [0, 1], are available, and initial conditions +are hard-encoded in NN modeling. For the downstream task, we assume 1 random +measurement of u and 8 random measurements of f are available with hard-encoded +initial conditions as well. +For the case with noisy measurements, we assume the +noises εf and εu to be additive Gaussian, with 0.05 noise scale for measurements of +f and 0.005 noise scale for measurements of u, respectively, i.e. εf ∼ N(0, 0.052) +and εu ∼ N(0, 0.0052). The reference solution as well as the clean data of uk are +generated by solving Eq. (4.2) for each task Tk, with corresponding fk and λk using +Matlab ode45. +Results are shown in Fig. 6 and Table 2, from which we can see our method is able +to approximate the stochastic process as a source term well and produce accurate and +trustworthy predictions, for u, f and also λ in the downstream task with limited data, +in both noiseless and noisy cases. As shown, the PINN method yields unacceptable +estimate over both u and λ due to lack of data, while our approach is of much higher +accuracy by integrating prior knowledge from {Tk}M +k=1 with MH-PINNs. +PINN +MH-PINN +λ +0.8440 +2.5428 +Error (%) +63.99 +1.21 +Table 2 +Estimate of λ and L2 relative error of u for the downstream inverse problem on Eq. (4.2) with +clean data, using our approach and the regular PINN method. The reference value for λ is 2.3609. +4.3. 1-D nonlinear reaction-diffusion equation. We now test our method +on a 1-D nonlinear time-dependent reaction-diffusion equation, which is commonly +referred to as Fisher’s equation [2]: +ut = Duxx + ku(1 − u), t ∈ [0, 1], x ∈ [−1, 1], +(4.5) +u(t, −1) = u(t, 1) = 0, t ∈ [0, 1], +(4.6) +u(0, x) = u0(x), x ∈ [−1, 1], +(4.7) +where D = 0.1, k = 0.1 and u0(x) is the initial condition function. In this example, +we assume that the initial condition function is a stochastic process with the following + +12 +Z. ZOU AND G. E. KARNIADAKIS +(a) +(b) +(c) +Fig. 6. Results for the inverse problem of the ODE system (4.2), with initial conditions hard- +encoded in NN modeling. (a) Left: 1, 000 samples of f, generated from the exact distribution; middle: +1, 000 samples of f, generated from the learned generator; right: statistics computed from samples. +The bound is defined as the same as in the caption of Fig. 3. (b) Predicted f and u using PINNs +and our approach, for the downstream inverse problem with noiseless data. (c) Predicted f, u and +λ with uncertainties using our approach, for the downstream inverse problem with noisy data. The +predicted mean and standard deviation of λ is 2.4663 and 0.1501, while the reference value is 2.3609. +distribution: +(4.8) +u0(x) = (x2 − 1) +5 +5 +� +j=1 +ξj(cos2(jx) − 1), x ∈ [−1, 1], +where ξj, j = 1, ..., 5 are independent and identically distributed (i.i.d.) random vari- +ables subject to uniform distribution on [0, 1), i.e., ξj ∼ U[0, 1). +Unlike previous +examples, the stochasticity comes from the initial condition rather than the source +term. This example corresponds to Eq. (2.1) with Fk and fk being the same for all +tasks, and uk and bk being task-specific. In addition to measurements of u0, points +on which the PDE residuals are computed are also required in both {Tk}M +k=1 and ˜T . +Hence, the data is Dk = {{(ti +k, xi +k), 0} +N f +k +i=1, {(0, xi +k), bi +k}N b +k +i=1}. +For the training, 2, 000 samples of u0(x) are generated, displayed in Fig. 7(a), +and each sample forms a physics-informed regression task with 41 measurements of +u0 equidistantly sampled on [−1, 1] as data for initial condition. Besides, for all tasks, +a uniform mesh 21 × 41 on temporal-spatial domain [0, 1] × [−1, 1] is used to com- +pute the PDE residual loss. For the downstream tasks ˜T , 5 random measurements +of u0 are available and the same uniform mesh is applied. The boundary conditions +are hard-encoded in NN modeling in both {Tk}M +k=1 and ˜T . For the noisy case, the + +4 +3 +2 +-3 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.94 +3 +2 +-3 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.94 +Predicted mean +Predicted bound +3 +Reference mean +Reference bound +2 +-3 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +t2 +Measurements +1.5 +Reference + Ours +1 +..- PINN +0.5 +0 +-0.5 +-1 +-1.5 +-2 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +t0.15 +0.1 +U +0.05 +0 +-0.05 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +t2 +2 std +1.5 +Measurements +Reference +Mean +0.5 +0 +-0.5 +-1 +-1.5 +-2 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +t0.15 +0.1 +0.05 +0 +-0.05 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +t2.5 +Prior (learned from physics) +Posterior +2 +1.5 +0.5 +2 +3 +4 +5 +6L-HYDRA +13 +noise ε is assumed to be independent additive Gaussian noise with 0.02 noise scale +for both measurements of u0 and the PDE residual, i.e., ε ∼ N(0, 0.022). Results are +presented in Fig. 7 and Table 3. We can see that our method estimates a good gen- +erator of the stochastic processes from data and physics, which provides informative +prior knowledge in the downstream few-shot physics-informed regression tasks. The +prediction is accurate in both noiseless and noisy cases, and the errors in the noisy +case are bounded by the predicted uncertainty. The L2 error of u, shown in Table 3, +indicates that our approach outperforms the PINN method by a significant amount, +hence demonstrating the effectiveness of bringing prior knowledge into solving similar +tasks. +(a) +(b) +(c) +Fig. 7. +Generator learning and few-shot physics-informed learning on 1-D time-dependent +reaction-diffusion equation (4.5), with boundary conditions hard-encoded in NN modeling. (a) Left: +1, 000 training samples of u0; middle: 1, 000 samples of u(0, ·) from the learned generator; right: +statistics computed from samples. The bound is defined as the same as in the caption of Fig. 3. (b) +Predicted u at t = 0, 0.5, 1 using our approach and the PINN method with noiseless measurements. +(c) Predicted mean and uncertainty of u at t = 0, 0.5, 1 using our approach with HMC for posterior +estimate, with noisy measurements. +PINN +MH-PINN +Error (%) +78.77 +0.22 +Table 3 +L2 relative error of u for the downstream few-shot physics-informed learning task on Eq. (4.5) +with clean data of u0 using our approach and the PINN method. + +0.5 +0.4 +0.3 +0=4 +0.2 +u +0.1 +0 +-0.1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.5 +0.4 +0.3 +0=4 +0.2 +u +0.1 +0 +-0.1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.5 +Predicted mean +Predicted bound +Reference mean +0.4 +Reference bound +0.3 +0=1 +0.2 +0.1 +-0.1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.5 +Measurements +Reference +0.4 +Ours +..-PINN +0.3 +0=1 +0.2 +0.1 +0 +-0.1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.5 +0.4 +0.3 +t=0.5 +0.2 +0.1 +0 +-0.1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.5 +0.4 +0.3 +t=1 +0.2 +u +0.1 +0 +-0.1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.5 +2 std +Measurements +0.4 +Reference +Mean +0.3 +0=1 +0.2 +f +0.1 +0 +-0.1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.5 +0.4 +0.3 +t=0.5 +0.2 +0.1 +0 +-0.1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.5 +0.4 +0.3 +t=1 +0.2 +u +0.1 +0 +-0.1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.814 +Z. ZOU AND G. E. KARNIADAKIS +4.4. 2-D nonlinear Allen-Cahn equation. We now move to a 2-D steady +nonlinear Allen-Cahn equation with Dirichlet boundary conditions [43]: +λ∆u + u(u2 − 1) = f, x, y ∈ [0, 1], +(4.9) +u(x, 0) = u(x, 1) = u(0, y) = u(1, y) = 0, +(4.10) +where λ = 0.1 is a constant and f is the source term. Here, we impose a distribution +to f, which is derived from Eq. (4.9) and the following distribution of the solution u: +(4.11) +u(x, y) = 1 +5 +5 +� +j=1 +ξj +sin(jπx) sin(jπy) +j2π2 +, x, y ∈ [0, 1], +with i.i.d. random variables ξj, j = 1, ..., 5, subject to uniform distribution, i.e. ξj ∼ +U[0, 1). In this example, we wish to use our method to learn generators of both u and f +from data of f and physics in Eq. (4.9), and use it to solve the downstream task ˜T with +insufficient data ˜D. This example corresponds to Eq. (2.1) with Fk, bk being the same +among tasks and fk, uk being task-specific, and the data is Dk = {(xi +k, yi +k), f i +k} +N f +k +i=1. +To train the MH-PINNs and NFs, we sample 2, 000 f from its distribution, each +of which is resolved with a 51 × 51 uniform mesh on 2-D spatial domain [0, 1] × [0, 1]. +As for the downstream task, 100 random measurements of f on the uniform mesh are +assumed to be available. The noise is assumed to be independent additive Gaussian +noise with 0.05 noise scale. In both Tk and ˜T , the boundary conditions are hard- +encoded in NN modeling. Results as well as the locations of the measurements are +presented in Fig. 8 and Table 4. +Similar to all previous examples, our approach +delivers accurate and trustworthy predictions, showing that prior knowledge is learned +and transferred well in both deterministic and Bayesian inferences. +(a) +(b) +Fig. 8. Results for few-shot physics-informed learning on the 2-D nonlinear Allen-Cahn equa- +tion Eq. (4.9) with noisy measurements of f. Predicted mean µ and standard deviation σ are com- +puted over 1, 000 posterior samples from HMC. The absolute error is defined as the absolute value +of difference between the reference and µ. Black crosses represent the locations of the measurements +on f. + +reference of f +0.9 +0.2 +0.8 +0.7 +0.1 +0.6 +9 +0.5 +0 +0.4 +-0.1 +0.3 +0.2 +-0.2 +0.1 +0 +-0.3 +0 +0.2 +0.4 +0.6 +0.8predicted mean of f +1 +x +X +X +X +0.9 +X +X +0.2 +X +0.8 +X +XX +X ++ +X +X +0.7 +XX +X +0.1 +X +× +X +X +X +0.6 +X +X +X +++ +X +x +X +9 +0.5 +X +0 +X +X +0.4 +X +X +-0.1 +X +X +X +0.3 +× XX +XX +0.2× +X +XX +-0.2 +X +X +0.1 +X +X +X +X +X +X +X +0 +-0.3 +0 +0.2 +0.4 +0.6 +0.8 +1absolute error of j +1 +0.05 +交 +X +XX +X +XX +0.9 +X +0.045 +X +X +X +0.8 +0.04 +X +X +X +X +X +0.7 +XX +0.035 +X +X +X +X +0.6 +X +X +0.03 +X +X +++ +X +X +9 +0.5 +X +0.025 +X +X +X +X +0.4 +XX +X +X +X +X +0.02 +X +X +X +X +0.3 +X +0.015 +XX +XX +XX × +X +XX +X +0.2× +0.01 +X +XX +X +X +0.1 +X +X +0.005 +X +X +X +X +X ++ +0 +0 +0 +0.2 +0.4 +0.6 +0.8 +1predicted uncertainty of f +0.05 +0.9 +0.045 +0.8 +0.04 +0.7 +0.035 +0.6 +0.03 +9 +0.5 +0.025 +0.4 +0.02 +0.3 +0.015 +0.2 +0.01 +0.1 +0.005 +0 +0 +0 +0.2 +0.4 +0.6 +0.8reference of u +0.01 +0.9 +0 +0.8 +0.7 +-0.01 +0.6 +9 +0.5 +-0.02 +0.4 +-0.03 +0.3 +0.2 +-0.04 +0.1 +0 +-0.05 +0 +0.2 +0.4 +0.6 +0.8 +1predicted mean of u +0.01 +0.9 +0 +0.8 +0.7 +-0.01 +0.6 +9 +0.5 +-0.02 +0.4 +-0.03 +0.3 +0.2 +-0.04 +0.1 +0 +-0.05 +0 +0.2 +0.4 +0.6 +0.8 +1absolute error of u +0.01 +0.9 +0.009 +0.8 +0.008 +0.7 +0.007 +0.6 +0.006 +9 +0.5 +0.005 +0.4 +0.004 +0.3 +0.003 +0.2 +0.002 +0.1 +0.001 +0 +0 +0 +0.2 +0.4 +0.6 +0.8 +1predicted uncertainty of u +0.01 +0.9 +0.009 +0.8 +0.008 +0.7 +0.007 +0.6 +0.006 +9 +0.5 +0.005 +0.4 +0.004 +0.3 +0.003 +0.2 +0.002 +0.1 +0.001 +0 +0 +0 +0.2 +0.4 +0.6 +0.8 +1L-HYDRA +15 +PINN +MH-PINN +Error (%) +12.82 +0.30 +Table 4 +L2 relative error of u for the downstream few-shot physics-informed learning task on Eq. (4.9) +with clean data of f, using our approach and the PINN method. +4.5. 2-D Stochastic Helmholtz equation. The last example we test in this +paper is the 2-D Helmholtz equation with stochastic source term and Dirichlet bound- +ary conditions [35]: +(λ2 − ∇2)u = f, x, y ∈ [0, 2π], +(4.12) +u(x, 0) = u(x, 2π) = u(0, y) = u(2π, y) = 0, +(4.13) +where λ2 is the Helmholtz constant and f is defined as follows: +(4.14) +f(x, y) = 2 +d{ +d/4 +� +i=1 +ξi sin(ix) + ξi+d cos(ix) + ξi+2d sin(iy) + ξi+3d cos(iy)}, +where ξj, j = 1, ..., d are i.i.d. random variables subject to uniform distribution U[0, 1) +and d represents the dimension of the randomness. For demonstration purposes, we +consider the case where d = 20 in this paper, unlike the one in [35] with d = 100. +The first case we study is the forward problem with λ2 = 1 known. This setup +corresponds to Eq. (2.1) with Fk, bk being shared among tasks and uk, fk being task- +specific. Next, we study the inverse problem with unknown λ, where data on u and f +are available, which corresponds to Eq. (2.1) with only bk being the same and uk, fk +and operator Fk being task-specific. The downstream tasks are defined as the same +as {Tk}M +k=1 in both cases, but with fewer measurements. +For both the forward and inverse problems, 10, 000 f are sampled from its distri- +bution, and hence 10, 000 tasks are solved with MH-PINNs with boundary conditions +hard-encoded in NN modeling. We display the samples of a slice of f in Fig. 9(a). +For the forward problem, Dk only contains measurements of the source term fk, i.e. +Dk = {{(xi +k, yi +k), f i +k} +N f +k +i=1}, while for the inverse problem Dk also contains measure- +ments of the sought solution uk: Dk = {{(xi +k, yi +k), f i +k} +N f +k +i=1, {(xi +k, yi +k), ui +k}N u +k +i=1}. For the +training in the forward problem, each sample of f is resolved by a 50 × 50 uniform +mesh on 2-D spatial domain (0, 2π)×(0, 2π) with boundary excluded. For the inverse +problem, the same 10, 000 samples of f are used, but this time they are resolved with +a 21 × 21 uniform mesh. In addition, for each task Tk, measurements of uk on a 6 × 6 +uniform mesh are available. The reference solution and measurements of u are gen- +erated by solving Eq. (4.12) with λ2 +k = +� +[0,2π]2 f 2 +k(x, y)dxdy using the finite difference +method with five-point stencil. For the downstream tasks, 100 random measurements +of f are available for the forward problem, and 50 random measurements of f and +10 random measurements of u are available for the inverse problem. The noise is +assumed to be independent additive Gaussian noise with 0.05 noise scale. +Results are displayed in Tables 5 and 6, and Figs. 9 and 10. +As shown, the +learned generator is able to produce samples of f with high quality as well as providing +informative prior knowledge for the downstream tasks, in both the forward and inverse +problems. As for the noisy case with Bayesian inference and UQ, the predicted means +agree with the references and the absolute errors are mostly bounded by the predicted +uncertainties. The effectiveness of our approach for few-shot physics-informed learning + +16 +Z. ZOU AND G. E. KARNIADAKIS +and the applicability to both deterministic optimization and Bayesian inference have +been consistently demonstrated in the past five examples. +(a) +(b) +(c) +Fig. 9. Generator learning and few-shot physics-informed learning on the stochastic Helmholtz +equation (4.12). (a) Left: 1, 000 training samples of a slice of f at y = π; middle: 1, 000 samples of +a slice of f at y = π from the learned generator; right: statistics computed from samples. (b)/(c) +Results for the downstream forward problem with 100 random noisy measurements on f, using our +approach with HMC. From left to right are reference, predicted mean, absolute error, and predicted +uncertainty of f/u. Black crosses represent the locations of the measurements of f. +PINN +MH-PINN +Error (%) +21.14 +1.12 +Table 5 +L2 relative error of u for the downstream forward problem on Eq. (4.12) with clean data of f, +using our approach and the PINN method. +PINN +MH-PINN +λ +1.9328 +1.0170 +Error (%) +59.92 +2.58 +Table 6 +Estimate of λ and L2 relative error of u for the downstream inverse problem on Eq. (4.12) with +clean data. The reference value of λ is 1.0042. +5. Multi-task learning with multi-head neural networks. So far we have +mostly focused on using MH-NNs together with NFs to estimate stochastic generators + +0.4 +0.2 +=/ +9. +0 +U +-0.2 +-0.4 +-0.6 +0 +2 +3 +4 +5 +60.4 +0.2 += +9. +0 +-0.2 +-0.4 +-0.6 +0 +2 +3 +4 +5 +6Predicted mean +Predicted bound +Reference mean +0.4 +Reference bound +0.2 +-0.2 +-0.4 +-0.6 +0 +2 +3 +4 +5 +6reference of f +6 +5 +0.5 +4 +9 +3 +0 +2 +1 +-0.5 +2 +3 +4 +5 +6 +1predicted mean of f +6 +X +X +X +X +X +X +XX +5 +X +X +X +X +X +X +0.5 +4 +X +X +X +X +9 +X +3 +X +X +X +XX +0 +2 × +X +X +1 +X +X +X +-0.5 +2 +3 +4 +5 +6absolute error of t +0.1 +6 +0.09 +XX +5 +0.08 +0.07 +0.06 +9 +0.05 +3 +0.04 +2× +0.03 ++ +0.02 +0.01 +X +0 +2 +3 +4 +5 +6predicted uncertainty of f +0.1 +6 +0.09 +0.08 +0.07 +4 +0.06 +9 +0.05 +3 +0.04 +2 +0.03 +0.02 +1 +0.01 +0 +2 +3 +4 +5 +6reference of u +6 +0.04 +5 +0.02 +0 +4 +-0.02 +9 +3 +-0.04 +2 +-0.06 +1 +-0.08 +-0.1 +1 +2 +3 +4 +5 +6predicted mean of u +6 +0.04 +5 +0.02 +0 +4 +-0.02 +9 +3 +-0.04 +2 +-0.06 +1 +-0.08 +-0.1 +2 +3 +4 +5 +6 +1absolute error of u +0.02 +6 +0.018 +5 +0.016 +0.014 +4 +0.012 +9 +0.01 +3 +0.008 +2 +0.006 +0.004 +1 +0.002 +0 +1 +2 +3 +4 +5 +6predicted uncertainty of u +0.02 +6 +0.018 +5 +0.016 +0.014 +4 +0.012 +9 +0.01 +3 +0.008 +2 +0.006 +0.004 +1 +0.002 +0 +1 +2 +3 +4 +5 +6L-HYDRA +17 +(a) +(b) +Fig. 10. +Results for the downstream inverse problem on the stochastic Helmholtz equa- +tion (4.12), with 50 random noisy measurements of f and 10 random noisy measurements of u. +λ is estimated as 1.0785 ± 0.0307 in the format of predicted mean ± predicted standard deviation, +while the reference value is 1.0042. (a)/(b) From left to right are reference, predicted mean, absolute +error, and predicted uncertainty of f/u. Black crosses represent locations of the measurements of f +or u. +and learn informative prior knowledge from {Tk}M +k=1. +This was achieved by first +training MH-NNs in a MTL fashion and then training NFs to estimate the PDF of the +head. Intuitively, the capability of MH-NNs when trained in MTL in capturing shared +information is the key to the success in generative modeling and few-shot learning. +For physics-informed MTL with MH-PINNs, ODEs/PDEs are solved simultaneously, +and assuming the solutions to share the same set of basis functions gives us samples +of the set of coefficients, which enables the generative modeling, followed by few-shot +learning, which is the whole point of the method proposed in this paper. However, +the cost and/or the benefit of imposing the same set of basis functions to all solutions +have not been explicitly discussed yet. On one hand, the shared body relates the +training of tasks, which may be helpful if tasks are similar in certain ways. On the +other hand, forcing all solutions to share the same basis functions may also be harmful +when they behave differently. In particular, for tasks with sufficient data and physics, +forcing them to share the same body with all other tasks may act as a negative +regularization, and single-task learning (STL) may outperform MTL in terms of the +prediction accuracy in those specific tasks. In this section, we investigate the effect +of MTL using MH-NNs and provide preliminary results and analysis by revisiting the +simple function approximation example in Sec. 4.1, which, hopefully, could provide +useful information and insight for future more rigorous research. +5.1. Basis function learning and synergistic learning. As discussed before, +the quality and behavior of basis functions learned in MTL are crucial to genera- +tive modeling and learning the relation and the representative information of tasks +{Tk}M +k=1. We consistently noticed from numerical examples that the initialization of +the head in MH-NNs has great impact on the average accuracy of MTL, the learning of +the basis functions, and the distribution of the head. Here, we test three initialization +strategies, random normal method with 0.05 standard deviation referred to as RN + +reference of f +6 +5 +0.5 +4 +9 +3 +0 +2 +1 +-0.5 +1 +2 +3 +4 +5 +6predicted mean of f +x +6 +X +X +X +X +X + X +X +X +5 +X +X +X +X +X +X +X +0.5 +4 +X +X +X +X +X +X +X +9 +3 +X +X +X +X +X +X +0 +2 +X +X +X +X +X +X +X +X +1 +X +X +X +-0.5 +1 +2 +3 +4 +5 +6absolute error of i +0.1 +6 +0.09 +5 +0.08 +0.07 +4 +X +0.06 +9 +0.05 +3 +X +X + X +0.04 +2 +0.03 +0.02 +0.01 +0 +2 +3 +4 +5 +6predicted uncertainty of f +0.1 +6 +0.09 +5 +0.08 +0.07 +4 +0.06 +9 +0.05 +3 +0.04 +2 +0.03 +0.02 +0.01 +0 +2 +3 +4 +5 +6reference of u +6 +0.04 +5 +0.02 +0 +4 +-0.02 +9 +3 +-0.04 +2 +-0.06 +1 +-0.08 +-0.1 +1 +2 +3 +4 +5 +6predicted mean of u +6 +0.04 +X +5 +0.02 +X +X +0 +4 +-0.02 +X +X +9 +3 +-0.04 +2 +X +X +X +-0.06 +1 +-0.08 +X +-0.1 +1 +2 +3 +4 +5 +6absolute error of u +0.02 +6 +0.018 +5 +0.016 +0.014 +4 +0.012 +9 +x +X +0.01 +3 +0.008 +2 +X +0.006 +0.004 +1 +X +0.002 +0 +1 +2 +3 +4 +5 +6predicted uncertainty of u +0.02 +6 +0.018 +5 +0.016 +0.014 +4 +0.012 +9 +0.01 +3 +0.008 +2 +0.006 +0.004 +1 +0.002 +0 +1 +2 +3 +4 +5 +618 +Z. ZOU AND G. E. KARNIADAKIS +(0.05), Glorot uniform method [12] referred to as GU, and random normal method +with 1 standard deviation referred to as RN (1). In the downstream few-shot learning +tasks, we fine-tune the head without the learned PDF, which is in fact the TL method +from [7], by which the information from the distribution of the head is excluded and +the prediction accuracy is fully determined by the level of prior knowledge contained +in the basis functions. +As shown in Fig. 11, RN (0.05) yields the least informative basis functions, whose +behavior is dominated by the hyperbolic tangent activation function of NNs. This is +further demonstrated in the downstream few-shot learning tasks using the TL method. +It also provides the worst prediction accuracy on average in MTL, as presented in +Table 8. GN and RN (1) perform similarly. Plots of some basis functions seemingly +indicate that RN (1) yields better basis functions, whose behaviors are more similar +to the family of functions displayed in Fig. 11(b), which, however, does not necessarily +imply richer prior knowledge in the downstream tasks, as shown in Fig. 11(c). +It is shown empirically that compared to other two initialization strategies, MH- +NNs with RN (0.05) does not deliver accurate MTL nor synergistic learning in basis +functions. However, we noticed that, in generative modeling, it performs significantly +better in terms of accuracy and convergence speed. As shown in Fig. 11(d), samples +from the learned generator are of higher quality. We consistently found that initializ- +ing heads with relatively small values often led to easy and fast training of NFs and +accurate learning of the generative models. We conjecture that this happens because +MH-NNs in MTL tend to contain the representative and informative information in +the heads when heads are initialized with small values, while contain it in the basis +functions when heads are initialized with relatively large values. +RN (0.05) +GU +RN (1) +Error (%) +0.8373 ± 0.2341 +0.1907 ± 0.0690 +0.3131 ± 0.0937 +Table 7 +L2 relative errors, from MTL, for 1, 000 tasks, using different initialization methods. +The +errors are displayed in the format of mean ± standard deviation, computed over all tasks. +5.2. Multi-Task Learning (MTL) versus Single-Task Learning (STL). +As discussed earlier, MTL with MH-NNs does not necessarily result in synergistic +learning nor higher accuracy for all tasks on average. Here, we use again the function +approximation example in Sec. 4.1, to investigate the effectiveness of MTL with MH- +NNs, as compared to STL. The first case we consider here is assuming that the data is +sufficient. For that, we randomly choose 100 out of the 1, 000 training samples, each +one of which is approximated by a NN trained independently, and compare the results +with MH-NNs in terms of prediction accuracy. Note that in this case, a MH-NN is +trained on 1, 000 functions as before and tested on the chosen 100 functions, while a +single-head NN with the same architecture is trained on 100 functions directly. Results +are shown in Table 8, from which it is verified empirically that MTL is outperformed +by STL under certain circumstances, e.g., when the random normal initialization +methods are used. +The second case we consider is assuming that the data is sufficient for some tasks +while insufficient for other tasks. For that, we split equally the 1, 000 tasks into two +subsets of tasks. For the first 500 tasks, we assume we only have 10 measurements +randomly sampled on [−1, 1], while for the other 500 tasks, we assume we have full 40 +measurements equidistantly distributed on [−1, 1]. MTL with MH-NNs is performed +on those 1, 000 regression tasks all at once, and the tasks are treated as equal. The + +L-HYDRA +19 +(a) +(b) +(c) +(d) +Fig. 11. The effect of different initialization methods of the head, in basis functions learning, +few-shot learning, and generator learning. (a) Samples of 20 basis functions from MH-NNs, trained +for approximating 1, 000 f generated from Eq. (4.1), using, from left to right, RN (0.05), GU and +RN (1) initialization methods. (b) 1, 000 training samples of f. (c) Results for two downstream +few-shot regression tasks, using TL method without regularization informed by the learned PDF, as +opposite to the proposed approach. (d) Results for generator learning, using, from left to right, RN +(0.05), GU and RN (1) initialization methods. +RN (0.05) +GU +RN (1) +STL +Error (%) +0.7575 ± 0.2477 +0.1362 ± 0.0259 +0.3664 ± 0.1031 +0.2102 ± 0.0794 +Table 8 +L2 relative errors of f, from MTL with MH-NNs and STL with NNs, on 100 tasks. Different +initialization methods are used for the heads in MH-NNs. The errors are displayed in the format of +mean ± standard deviation, computed over all 100 tasks. +results are presented in Table 9 and Fig. 12. We can see that, compared to STL, MTL +improves the prediction accuracy on tasks with insufficient data, providing empirical +evidence of synergistic learning. +Also, interestingly, RN (1) initialization method, +which yields the worst generative models, performs the best among all three, which +agrees with our previous conjecture on the basis functions learning with MH-NNs, +that heads initialized with large values tend to force representative and informative +information to be encoded in the basis functions. +6. Discussion. We have developed multi-head neural networks (MH-NNs) for +physics-informed machine learning, and proposed multi-head physics-informed neural +networks (MH-PINNs) as a new method, implemented in the L-HYDRA code. The +primary focus of this work is on MH-NNs and MH-PINNs for various learning prob- + +6 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.8 +0.6 +0.4 +0.2 +0 +-0.2 +-0.4 +-0.6 +-0.8 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.80.8 +0.6 +0.4 +0.2 +0 +-0.2 +-0.4 +-0.6 +-0.8 +1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.81 +0.8 +0.6 +0.4 +0.2 +0 +-0.2 +-0.4 +-0.6 +-0.8 +1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8Measurements +Reference +RN (0.05) +3 +GU +2 +RN +-2 +-3 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.84 +3 +2 +0 +-1 +-2 +-3 +4 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.86 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.86 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.820 +Z. ZOU AND G. E. KARNIADAKIS +RN (0.05) +GU +RN (1) +Error (%) +63.60 ± 24.08 +40.49 ± 20.49 +16.91 ± 11.08 +Table 9 +L2 relative errors of f, from MTL with MH-NNs, on 500 tasks equipped with insufficient data. +The errors are displayed in the format of mean ± standard deviation, computed over all 500 tasks. +Fig. 12. Results for 3 tasks with insufficient data from MTL with MH-NNs, using different +initialization methods over the head, and from STL with NNs with the same architecture. We note +that tasks with sufficient data and tasks with insufficient data are treated equally in MTL. +lems in scientific machine learning, including multi-task learning (MTL), stochastic +processes approximation, and few-shot regression learning. We first formulated the +problem in Eq. (2.1), introduced the architecture design of MH-PINNs, and proposed +a method to transform MH-NNs and MH-PINNs to generative models with the help +of normalizing flows (NFs) for density estimation and generative modeling. We then +studied the applicability and capabilities of MH-PINNs in solving ordinary/paritial +differential equations (ODEs/PDEs) as well as approximating stochastic processes. +We completed the paper with preliminary and empirical explorations of MH-NNs +in synergistic learning, and examined the potential benefits and cost of MTL with +MH-NNs. +This paper can be used in various ways: it proposes a NN approach for MTL in +solving ODEs/PDEs; it provides a new approach to approximate stochastic processes; +it presents a method to address few-shot physics-informed learning problems, which +are often encountered in the context of meta-learning and transfer learning; it contains +a systematic study of applying MH-NNs to scientific computing problems; it presents +the first empirical evidence of synergistic learning. +However, there are a few major problems on MH-NNs we did not address, one +of which is the expressivity of MH-NNs, or more generally hard-parameter sharing +NNs in approximating complicated stochastic processes. Intuitively, if two functions +behave very differently, forcing them to share the same basis functions would affect +adversely the approximation accuracy. The second problem is the balancing issue of +different terms in the loss function in MTL. It is shown in the literature [29] that +PINNs, trained in single-task learning, are already deeply influenced by the weights +in front of different terms in the loss function, e.g., data loss, boundary condition loss, +PDE residual loss. This issue may be more complex in training MH-PINNs, because +in MTL the loss function is commonly defined as weighted summation of task-specific +loss. The last major problem is MH-PINNs for synergistic learning. In this paper, we +only studied one example in function approximation and presented empirical evidence. +More work for the understanding of synergistic learning with MH-PINNs along both +the theoretical and computational directions should be pursued in the future. +Acknowledgments. We would like to thank Professor Xuhui Meng of Huazhong +University of Science and Technology for helpful discussions. This work was supported +by: the Vannevar Bush Faculty Fellowship award (GEK) from ONR (N00014-22- + +Measurements +Reference +RN (0.05) +3 +GU +2 +RN(1) +NN +3 +-0.8 +-0.6 +-1 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.85 +4 +3 +2 +0 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.85 +4 +3 +0 +-1 +-2 +-3 +4 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8L-HYDRA +21 +1-2795); the U.S. Department of Energy, Advanced Scientific Computing Research +program, under the Scalable, Efficient and Accelerated Causal Reasoning Operators, +Graphs and Spikes for Earth and Embedded Systems (SEA-CROGS) project, DE- +SC0023191; and by the MURI/AFOSR FA9550-20-1-0358 project. +REFERENCES +[1] M. Abadi, A. Agarwal, P. Barham, and et. al., TensorFlow: Large-scale machine learning +on heterogeneous systems, 2015, https://www.tensorflow.org/. +Software available from +tensorflow.org. +[2] M. J. 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Foster, Multi-output physics-informed neural networks for forward and +inverse PDE problems with uncertainties, Computer Methods in Applied Mechanics and +Engineering, (2022), p. 115041. +[46] W. Zhong and H. Meidani, PI-VAE: Physics-Informed Variational Auto-Encoder for stochas- +tic differential equations, arXiv preprint arXiv:2203.11363, (2022). +[47] Z. Zou, X. Meng, A. F. Psaros, and G. E. Karniadakis, NeuralUQ: A comprehensive library +for uncertainty quantification in neural differential equations and operators, arXiv preprint +arXiv:2208.11866, (2022). +Appendix A. Details of NN architectures and training hyperparam- +eters. +For all examples in Secs. 4 and 5, MH-PINNs are implemented as fully- +connected NNs (FNNs) with 3 nonlinear hidden layers, each of which is equipped +with 50 neurons and hyperbolic tangent activation function. The number of heads is +the same as the number of tasks in the corresponding examples: 1, 000 in Sec. 4.1, +2, 000 in Secs. 4.2, 4.3 and 4.4, and 10, 000 in Sec. 4.5. Weights in the body of MH- +PINNs are initialized with Glorot uniform initialization [12] and biases are initialized +with zero, while heads are initialized by sampling from random normal distribution +with 0.05 standard deviation, for fast training of NFs and better performance of the +learned generators. +Except for the forward problem in Sec. 4.2, NFs in this paper are chosen to be +MAF [33] with 10 bijectors, i.e. the invertible map in NFs, each of which is a MADE +[11], a NN with masked dense layers, with two nonlinear hidden layers equipped with +100 neurons and ReLU activation function. The RealNVP [9] and IAF [20] used in +the forward problem in Sec. 4.2 also have 10 bijectors, each of which is a NN with +two nonlinear hidden layers equipped with 100 neurons and ReLU activation function. +The implementation mostly follows the instructions of TensorFlow Probability library +[8] for NFs. +PI-GANs [44] implemented in Sec. 4.2 have the following architecture: the dis- +criminator is a FNN with 3 nonlinear hidden layers, each of which is equipped with +128 neurons and Leaky ReLU activation function; the generator that takes as input +t is a FNN with 3 nonlinear hidden layers, each of which is equipped with 50 neu- +rons and hyperbolic tangent activation function; the other generator takes as input a +Gaussian random variable in 50 dimensions with zero mean and identity covariance +matrix, and is implemented as a FNN with 3 nonlinear hidden layers, each of which +has 128 neurons and hyperbolic tangent activation function. The input dimensions of +those 3 FNNs are 65, 1 and 50, and the output dimensions are 1, 50, 50, respectively. +For the training of MH-PINNs, full-batch training is deployed with Adam opti- +mizer for 50, 000 iterations. For the training of NFs, except for the forward problem +in Sec. 4.2, mini-batch training is deployed with batch size being 100 and Adam op- +timizer for 1, 000 epochs. +NFs in the forward problem in Sec. 4.2 are trained for +500 epochs instead, and L2 regularization is imposed to the parameters of RealNVP +for better performance. For all NFs, to achieve stable training, a hyperbolic tangent +function is imposed on the logarithm of the scale, computed from each bijector, such +that the logarithm of the scale lies in (−1, 1). For the training of PI-GANs, min- +batch training is deployed with batch size being 100 and Adam optimizer for 100, 000 +iterations. Besides, the same as in [44, 30], physics-informed Wasserstein GANs (PI- +WGANs) with gradient penalty are employed, in which the coefficient for gradient +penalty is set to be 0.1. Iteratively, 5 updates of the discriminator are performed and +followed by 1 update of the generators. Except in training PI-GANs, the learning +rate of Adam optimizer is set to be 10−3 and other hyperparameters of Adam are set +as default. In training PI-GANs, the learning rate is set to be 10−4, β1 = 0.5 and +β2 = 0.9 in Adam optimizer for both discriminator and generators. + +24 +Z. ZOU AND G. E. KARNIADAKIS +Training of MH-PINNs, NFs, and PI-GANs was all performed on a single NVIDIA +TITAN Xp GPU. The L-HYDRA code for TensorFlow implementation along with +some representative examples will be released on GitHub once the paper is accepted. +Appendix B. Details for performing Bayesian inference. +Hamiltonian +Monte Carlo (HMC) [31] is employed in all Bayesian inference examples for uncer- +tainty quantification (UQ) while Laplace approximation [18] is only employed in the +first example. In this paper, HMC with adaptive step size [23] is used, in which the +initial step size is set to be either 0.1 or 0.01, tuned for better acceptance rate. The +number of burn-in samples and the number of posterior samples are set to be 1, 000. +The number of steps for the leapfrog scheme is set to be either 30 or 50, also tuned for +better acceptance rate. NeuralUQ library [47] for UQ in scientific machine learning is +used as a tool for physics-informed Bayesian inference in the downstream tasks. The +ideal acceptance rate in HMC, as discussed in [30, 47], is around 60%. In this paper, +we found chains with acceptance rate from 50% to 80% acceptable. + diff --git a/5dA0T4oBgHgl3EQfNv_S/content/tmp_files/load_file.txt b/5dA0T4oBgHgl3EQfNv_S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bcb57967683568752016a9acf8dda5fd891d4806 --- /dev/null +++ b/5dA0T4oBgHgl3EQfNv_S/content/tmp_files/load_file.txt @@ -0,0 +1,1939 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf,len=1938 +page_content='L-HYDRA: MULTI-HEAD PHYSICS-INFORMED NEURAL NETWORKS ZONGREN ZOU∗ AND GEORGE EM KARNIADAKIS† Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We introduce multi-head neural networks (MH-NNs) to physics-informed machine learning, which is a type of neural networks (NNs) with all nonlinear hidden layers as the body and multiple linear output layers as multi-head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Hence, we construct multi-head physics-informed neural networks (MH-PINNs) as a potent tool for multi-task learning (MTL), generative modeling, and few-shot learning for diverse problems in scientific machine learning (SciML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' MH-PINNs connect multiple functions/tasks via a shared body as the basis functions as well as a shared distribution for the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The former is accomplished by solving multiple tasks with MH-PINNs with each head independently corresponding to each task, while the latter by employing normalizing flows (NFs) for density estimate and generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' To this end, our method is a two-stage method, and both stages can be tackled with standard deep learning tools of NNs, enabling easy implementation in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' MH-PINNs can be used for various purposes, such as approximating stochastic processes, solving multiple tasks synergistically, providing informative prior knowledge for downstream few-shot learning tasks such as meta-learning and transfer learning, learning representative basis functions, and uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We demonstrate the effectiveness of MH-PINNs in five benchmarks, investigating also the possibility of synergistic learning in regression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We name the open- source code “Lernaean Hydra” (L-HYDRA), since this mythical creature possessed many heads for performing important multiple tasks, as in the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' PINNs, meta-learning, multi-tasking, transfer learning, generative models, nor- malizing flows, stochastic problems MSC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 34F05, 62M45, 65L99, 65M99, 65N99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Learning across tasks has drawn great attention recently in deep learning and is an emerging theme in scientific machine learning (SciML), due to the fact that several classes of scientific problems are similar and/or related in- trinsically by their common physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Intuitively, if tasks are similar, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', in the context of approximating stochastic processes [44], learning solution operators of ordi- nary/partial differential equations (ODEs/PDEs) [28], and solving parametric PDEs [42, 19, 4], it may be beneficial to relate them in the modeling, algorithm design, and/or solving procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this regard, machine learning solvers, developed rapidly in the past few years, are considerably more flexible and of higher potential compared to traditional numerical solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Significant progress has been witnessed in the general area, including meta-learning for solving ODEs/PDEs [30, 27, 34, 6], transfer learning for physics-informed neural networks (PINNs) [3, 7], transfer learning for domain shift in solving PDEs [14], multi-task learning for PINNs [40], and generative methods for solving stochastic differential equations (SDEs) [44, 46, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' More recently, operator learning [28, 24] in which direct operator mapping is learned and subsequently used for other tasks in one-shot format has attracted a lot of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Multi-head neural networks (MH-NNs) fit perfectly different scenarios of learning across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' They were originally proposed as members of hard-parameter sharing neural networks (NNs) for deep multi-task learning (MTL) [5], in which multiple tasks, denoted as Tk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', M, where M is the number of total tasks, are solved simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The general goals of using MH-NNs in MTL are diverse: achieving ∗Division of Applied Mathematics, Brown University, Providence, RI 02912, USA (zon- gren zou@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Division of Applied Mathematics, Brown University, Providence, RI 02912, USA (george karniadakis@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='02152v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='LG] 5 Jan 2023 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS better performance for all tasks, learning good and useful representations for down- stream tasks, and/or boosting the learning of main tasks with the help of auxiliary tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Moreover, although originally designed for solving multiple tasks, MH-NNs in recent years have also been extensively used for meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For example, in [41], the connection between MTL and meta-learning was analyzed, and meta-learning al- gorithms for MH-NN were discussed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' in [25], it was shown that MH-NNs, trained in MTL fashion also perform task-specific adaptation in meta-learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' [37] argued that the effectiveness of model-agnostic meta-learning [10], a well-known meta-learning al- gorithm, may be due to successfully learned good representations rather than learned adaptation, and MH-NNs were used to study the detailed contributions of NNs in fast task adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Overall, it is commonly acknowledged in the literature that when used to solve previous tasks, MH-NNs are capable of distilling useful shared information and storing it in their bodies and heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this paper, we develop MH-NNs for physics-informed machine learning [17], propose multi-head physics-informed neural networks (MH-PINNs), and further in- vestigate their applicability and capabilities to MTL, generative modeling, and meta- learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' A MH-PINN, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 1, is built upon a conventional MH-NN and consists of two main parts, the body and multiple heads, and each head connects to a specific ODE/PDE task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Many architecture splitting strategies for MH-NNs are adopted in different applications scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', for some computer vision problems, a NN is split such that the body consists of convolutional layers and is followed by fully-connected layers as heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this paper, however, we choose the simplest one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', the body consists of all nonlinear layers and the head is the last linear layer, for the following two reasons: (1) the dimensionality of the head is reduced, which enables fast density estimation (see next section);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' and (2) the body spontaneously provides a set of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Schematic view of the structure of multi-head physics-informed neural networks (MH- PINNs) with M different heads, which are built upon conventional multi-head neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The shared layers are often referred to as body and the task-specific layer as head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Generally, uk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', M represent M solutions to M different ODEs/PDEs, formulated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1), which may differ in source terms fk, boundary/initial condition terms bk, or differential operator Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The novelty and major contributions of this work are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We propose a new physics-informed generative method using MH-PINNs for learning stochastic processes from data and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We propose a new method for physics-informed few-shot regression problems with uncertainty quantification using MH-PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We study and demonstrate the effectiveness of MTL and synergistic learning with MH-NNs in regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2, we present the problem formulation, Fiui(α)/ = fi(α), Biui(α)/ = bi(α) head task, 1 F2[u2(α)] = f2(α), B2[u2(α)] = b2(α) head, u2 Body FM[uM(α)] = fM(α), BM[uM(c)] = bM(c) head uM task, ML-HYDRA 3 details of MH-PINNs, and the general methodology, including how to use MH-PINNs for MTL, generative modeling, downstream few-shot physics-informed learning with uncertainty quantification (UQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3, we discuss existing research closely related to our work and compare them conceptually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4, we test MH-PINNs with five benchmarks, each of which corresponds to one or more learning purposes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', MTL and generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 5, we investigate MTL and synergistic learning with the function approximation example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We conclude and summarize in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The details of our experiments, such as NN architectures and training strategies, can be found in Appendix A and B, as well as in the L-HYDRA open-source codes on GitHub, which will be released once the paper is accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We assume that we have a family of tasks, {Tk}M k=1, each of which is associated with data Dk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The primary focus of this paper is on scientific computing and ODEs/PDEs, and therefore we further assume {Tk}M k=1 are physics-informed regression problems [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Consider a PDE of the following form: Fk[uk(x)] = fk(x), x ∈ Ωk, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1a) Bk[uk(x)] = bk(x), x ∈ ∂Ωk, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1b) where k denotes the index of the task and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', M, x is the general spatial- temporal coordinate of Dx dimensions, Ωk are bounded domains, fk and uk are the Du-dimensional source terms and solutions to the PDE, respectively, Fk are general differential operators, Bk are general boundary/initial condition operators, and bk are boundary/initial condition terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For simplicity, throughout this paper, the domain and the boundary/initial operator, denoted as Ω and B, are assumed to be the same for all tasks, and the solutions uk to be task-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The task Tk is described as approximating uk, and/or fk, and/or Fk, and/or bk, from data Dk and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Traditional numerical solvers often tackle {Tk}M k=1 independently, without lever- aging or transferring knowledge across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The PINN method [38] was designed to solve ODEs/PDEs independently using NNs, which, however, yields M uncorrelated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this paper instead we treat {Tk}M k=1 as a whole and connect them with MH- PINNs, the architecture of which, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 1, enforces basis-functions-sharing predictions on the solutions uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In addition to the informative representation/body, we further relate {Tk}M k=1 by assuming that their corresponding heads in MH-PINNs, denoted as {Hk}M k=1, are samples of a random variable with unknown probability density function (PDF), denoted as H and p(H), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The shared body and a generative model of H immediately form a generative model of the solution u, and generators of the source term f and the boundary/initial term b as well by substitut- ing u into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) and automatic differentiation [1], from which a generative method for approximating stochastic processes is seamlessly developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Generators of u, f and b, as discussed in [30], are able to provide an informative prior distribution in physics-informed Bayesian inference [43, 26] as well as in UQ for SciML [47, 36], where the informative prior compensates for the insufficiency of observational data to address the physics-informed learning problems with even a few noisy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this paper, we generalize such problem to deterministic cases as well, where the data is noiseless and methods and results are deterministic, and refer to it as few-shot physics-informed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The general idea is to apply prior knowledge learned from connecting {Tk}M k=1 with MH-PINNs to new tasks, denoted as ˜T , associated with insufficient data ˜D, for accurate and trustworthy predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS The schematic view of the learning framework is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2, and the details are explained next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Schematic view of the learning framework and the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Three general types of learning are addressed: physics-informed learning, generative modeling, and few-shot learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The physics-informed learning is performed with MH-PINNs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' the generative modeling is done afterwards by density estimate over the head via normalizing flows (NFs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' in the end the few-shot physics-informed learning is accomplished with prior knowledge obtained from previous two via either fine-tuning with the learned regularization or Bayesian inference with the learned prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The body represents the set of basis functions learned from solving {Tk}M k=1 with MH-PINNs, and the density of the head, estimated from its samples using NFs, acts as the regularization, the prior distribution, or the generator together with the body, depending on the usage of MH-PINNs in ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Multi-head physics-informed neural networks (MH-PINNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Hard parameter sharing is the most commonly used approach when MTL with NNs are considered, and MH-NNs, as its simplest instance, are frequently adopted [5, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' A MH-PINN, as described earlier, is composed of a body and multiple heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We denote by Φ the body and by Hk the head for Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Notice that here Φ : RDx → RL is a function parameterized by a neural network with parameter θ, and Hk ∈ RL+1 is a vector, where L is the number of neurons on the last layer of the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Let us define Hk = [h0 k, h1 k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', hL k ]T , Φ(x) = [φ1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', φL(x)]T , where φ : RDx → R, and then the surrogate for the solution in Tk can be rewritten as ˆuk(x) = h0 k+�L l=1 hl kφl(x), ∀x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The approximated source terms and boundary/initial terms are derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In the MTL framework , given data {Dk}M k=1 and physics Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1), the loss function L is formulated as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2) L({Dk}M k=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' θ, {Hk}M k=1) = 1 M M � k=1 Lk(Dk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' θ, Hk), where Lk denotes the common loss function in PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Conventionally, the data for Tk is expressed as Dk = {Df k, Db k, Du k}, where Df k = {xi k, f i k} N f k i=1, Db k = {xi k, bi k}N b k i=1 and samples of head Learned distribution of head 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 learned by normalizing 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0 generative modeling { Hkl prior knowledge new tasks T Fine-tuning Body Bayesian inference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='00-0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='00L-HYDRA 5 Du k = {xi k, ui k}N u k i=1, and Lk as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3) Lk(Dk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' θ, Hk) = wf k N f k N f k � i=1 ||Fk(ˆuk(xi k)) − f i k||2 + wb k N b k N b k � i=1 ||B(ˆuk(xi k)) − bi k||2 + wu k N u k N u k � i=1 ||ˆuk(xi k) − ui k||2 + R(θ, Hk), where || · || represents a properly chosen norm, R(·) is a regularization method over the parameters of NNs, N f k , N b k, N u k are the numbers of data points for fk, bk, uk, and wf k, wb k, wu k are weights to balance different terms in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Generative modeling and normalizing flows (NFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As mentioned earlier, MH-PINNs connect {Tk}M k=1 by making two assumptions: (1) the solutions uk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', M share the same set of basis functions, Φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' and (2) the corresponding coefficients are samples of the same random variable, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In [7], Φ was used as a carrier of prior knowledge from {Tk}M k=1 in downstream physics-informed learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this work, we extend it by utilizing the information from the head as well by estimating the PDF and a generator of H from its samples, {Hk}M k=1, using normalizing flows (NFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The interested readers are directed to [32, 22] for reviews of NFs as well as [9, 33, 20] for developments of some popular NFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We choose NFs over other commonly used generative models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', generative adversarial networks (GANs) [13], variational auto-encoders (VAEs) [21], or diffusion models [16], because the NF serves as both a density estimator and a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The former is able to provide proper regularization in the downstream few-shot physics- informed learning tasks, while the latter leads to a physics-informed generative method for approximating stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' It is worth noting that in previous works on physics-informed generative methods [44, 46, 15], NNs are trained by measurements over uk, and/or fk, and/or bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Our model, on the other hand, learns through samples of the head, which is obtained from MTL in the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' This learning strategy brings two substantial advantages: (1) flexibility in dealing with unstructured data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', inconsistent measurements across tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2) simplicity and controlability of the training by decoupling the physics-informed learning and the generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Prior knowledge utilized in the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Here, we describe details on how to utilize the prior knowledge stored in MH-PINNs, for downstream few-shot physics-informed learning task, ˜T , which is defined the same as all other tasks in the upstream training, but with much fewer measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Training of MH- PINNs and NFs yield a body, Φ, samples of heads, {Hk}M k=1, and an estimated PDF of the head, ˆp(H) ≈ p(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In solving ˜T with ˜D, we fix the body Φ and find the head ˜H that best explains the data ˜D and the physics in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Noiseless and noisy data are considered in this paper: for noiseless data, regular NN training is performed on the head for new tasks to provide deterministic predictions, where the learned PDF of the head, ˆp(H), acts as a regularization term in the loss function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' for noisy data, Bayesian inference is performed on the head as well, in which ˆp(H) denotes the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Details are presented in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Regularization in optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Limited data in few-shot learning often leads to over-fitting and/or poor inter-/extrapolation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this re- gard, regularizing the head according to its PDF is able to prevent over-fitting and 6 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS provide additional prior knowledge for better inter-/extrapolation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The optimization problem is cast as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4) ˜H∗ = arg min ˜ H L∗( ˜D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ˜H), where L∗( ˜D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ˜H) = L( ˜D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ˜H) − α log p( ˜H) ≈ L( ˜D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ˜H) − α log ˆp( ˜H), where L is the regular loss function in physics-informed learning for data ˜D and param- eter ˜H, and α ≥ 0 is the coefficient to adjust the regularization effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4) in this work is solved with gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Prior distribution in Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As opposed to point esti- mate obtained by solving the optimization problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4), the posterior distribution of the head in ˜T is obtained using Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Similar as in [30, 47], the posterior distribution of ˜H is established as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5) p( ˜H| ˜D) ∝ p( ˜D| ˜H)p( ˜H) ≈ p( ˜D| ˜H)ˆp( ˜H), where p( ˜H| ˜D) is the posterior distribution, p( ˜D| ˜H) is the likelihood distribution, which is often assumed to be independent Gaussian over all measurements in ˜D, and ˆp is the estimated PDF of the head via NFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Intractability of distribution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5) requires approximation methods, among which Markov chain Monte Carlo methods, such as Hamiltonian Monte Carlo (HMC) [31], generally provide the most accurate estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Moreover, the relatively low dimension of ˜H also enables the use of Laplace’s approximation (LA) [18], which is employed in this paper as an alternative to HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Deep NNs in recent years have been extensively investigated for solutions of ODEs/PDEs, SDEs as well as operator learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Although not explic- itly introduced as MH-PINNs, MH-NNs were first used to solve ODEs/PDEs by [7], in which MH-PINNs were pre-trained on multiple similar tasks, and then the heads were discarded while the body was kept and transferred to solving new tasks, by either least square estimate for linear ODEs/PDEs, or fine-tuning with gradient descent for nonlinear ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In [7], a one-shot transfer learning algorithm for linear problems was proposed but other potential uses of MH-NNs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', MTL and generative modeling, were not discussed, as opposed to the work presented herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Furthermore, [7] focused only on fast and deterministic predictions with high accuracy using sufficient clean data, while in this paper, we study the applicability of MH-NNs to few-shot physics- informed learning as well, where data is insufficient and/or noisy, and address such cases with UQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We note that MH-NN was also used as a multi-output NN in [45], which, however, focused on solving single tasks and obtaining uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Generative modeling in the context of scientific computing has also been studied recently, and a few attempts for adopting deep generative NNs to SciML problems have been made in [44, 46, 15], most of which have focused on approximating stochas- tic processes and on solving SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We propose a new physics-informed generative method, as an alternative to the current ones, using MH-PINNs, and test it in approxi- mating stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this regard, our method is functionally the same as the current ones, but technically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' All previous methods address physics-informed generative modeling using end-to-end learning strategies by coupling two dissimilar types of learning, physics-informed learning and generative modeling, which may be problematic for implementation and usage in practice when either type of learning becomes more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Our method, on the other hand, addresses the problem in L-HYDRA 7 an entirely new angle by decoupling those two: physics-informed learning is performed first and is followed by learning generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' To this end, our method is a two-step method, and with the help of well-developed algorithms from both fields, our method has advantages both in flexibility and simplicity in implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this section, we test our method using five benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The first one is a pedagogical function regression, in which we aim to demonstrate the basic applicability and capabilities of our method, showing the importance of incorporating the distribution of the head in the downstream tasks and in obtaining results with or without uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The second example is a nonlinear ODE system, in which we test our method in approximating stochastic processes through a differential op- erator, compare different NFs, and eventually compare our method with another well-known physics-informed generative model, physics-informed GANs (PI-GANs) [44] in generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The third is a 1-D nonlinear reaction-diffusion equation, the fourth is a 2-D nonlinear Allen-Cahn equation, and the fifth is the 2-D stochastic Helmholtz equation with 20 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In all examples unless stated otherwise, data for {Dk}M k=1 are noise-free and task-wisely sufficient, while ˜D in downstream tasks is insufficient, which makes the downstream tasks of the few-shot type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In addition, ex- cept for the first example, results from Bayesian inference are obtained by employing HMC, and the predicted mean denoted as µ and predicted standard deviation denoted as σ are computed from the posterior samples of functions or unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The predicted uncertainty is defined as 2σ in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We start with a function regression problem using only data and no physics, which is a degenerate instance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) with Fk being fixed as an identity operator, no B and bk, and uk = fk being task-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this case, Dk and ˜D are given as {(xi k, f i k)}Nk i=1 and {(xi, f i)}N i=1, respectively, and Tk and ˜T are defined as approximating functions fk and ˜f from Dk and ˜D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The stochastic function f in this example is defined as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) f(x) = A cos(ωx) + 2βx, x ∈ [−1, 1], A ∼ U[1, 3), ω ∼ U[2π, 4π), P(β = ±1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5, where U stands for uniform distribution and P(Ξ) is defined as the probability of the event Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Our goal is to approximate f from data with MH-NNs and NFs, and solving the downstream few-shot regression tasks ˜T as well, in which two functions, 2 cos(2πx)+2x and 2 cos(4πx)−2x, are regressed from 4 and 5 measurements equidis- tantly distributed on [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the training of MH-NNs and NFs, 1, 000 f subject to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) are sampled, each of which forms a regression task with 40 measurements sampled equidistantly on [−1, 1] as data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Samples of f for training are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Both noiseless and noisy data cases are considered in the few-shot regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3, the former is solved by fine-tuning the head using gradient descent, while the latter is solved by estimating the posterior distribution (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5)) using HMC and LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The noise ε is assumed to be independent additive Gaussian noise with scale 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', ε ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In the downstream few-shot regression tasks we compare our method with two other approaches, the transfer learning (TL) method from [7], which only transfers the body, Φ, and the regular NN method, in which no prior knowledge is employed and all parameters of NN are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results for approximating f and solving the downstream tasks are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3(a), our method approximates the stochastic function f 8 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results for approximating the stochastic function defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) and solving the downstream few-shot regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (a) Left: 1, 000 samples generated from the exact distribu- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' middle: 1, 000 samples generated from the learned generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' right: statistics computed from samples, in which we refer to the interval of mean ± 2 standard deviations as bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (b)/(c) Results for the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Left: results for noiseless cases using our method, the transfer learning (TL) method in [7], and regular NN method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' middle: results for noisy case using our method with HMC for posterior estimate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' right: results for the same noisy case using our method with LA for posterior estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' well, demonstrating the capability of MH-NNs in generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In solving the downstream tasks with noiseless data, L2 regularization is imposed in the TL method and regular NN method, to prevent over-fitting when only 4 or 5 measurements are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As we can see from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3(b) and (c), our approach yields accurate predic- tions and performs significantly better than the other two in both tasks, particularly in the region where there are no measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' By comparing our approach with the NN method, we can see that prior knowledge of f is learned from {Tk}M k=1 and transferred successfully to new downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' By comparing our approach with the TL method, we can see that the prior knowledge is stored in both the body and (the distribution of) the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the noisy cases, it is shown that, for both tasks and both posterior estimating methods, the predictions are accurate and trustworthy: the predicted means agree with the references and the errors are bounded by the predicted uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' It is worth noting that the predicted uncertainties do not develop in the interval [0, 1] and show periodic patterns, even if there are no measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' That is because an informative prior, which is learned by MH-NNs and NFs, is imposed on the head in Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The target functions in downstream tasks considered previously are chosen to be in-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' They are regressed well 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8Predicted mean Predicted bound Reference mean Reference bound 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 6 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='85 Measurements Reference Ours 3 TL 2 NN 0 1 2 3 4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='85 4 0 1 2 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8L-HYDRA 9 belong to the space of functions, on which the generator is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' However, when functions in the downstream tasks are out-of-distribution (OOD), our approach fails to produce good predictions, even if the data is sufficient, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Here, the target function is chosen to be 2 cos(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5π) + x with both ω and β being OOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Fluctuations are predicted but do not match the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4, we can further see that when data is sufficient, a NN trained from scratch significantly outperforms our approach, showing that, for OOD functions, the more we rely on the learned regularization, which is indicated by the value of α in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4), the more erroneous the prediction is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results for regression on an out-of-distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Left: few-shot regression with clean data using our approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' middle: few-shot regression with noisy data using our approach with HMC for posterior estimate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' right: regression with sufficient clean data using regular NN method and our approach with different regularization terms, α in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Nonlinear ODE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this example, we consider the following ODE system [28], which describes the motion of a pendulum with an external force: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2) du1 dt = u2, du2 dt = −λ sin(u1) + f(t), with initial condition u1(0) = u2(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2), f is the external force and λ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Here, to demonstrate and study the capability of our method in generative modeling, we first consider the case where f is a Gaussian process and λ = 1 is known, which is referred to as the forward problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Different from previous studies [44, 46, 15], in which a stochastic process is approximated directly by the output of NNs, in this example we place the differential operator right after NNs and approximate the stochastic process f as the source term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We also test our method on the inverse problem, where the values of λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2) are unknown in {Tk}M k=1 and ˜T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The forward problem corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) with Fk, bk being the same for all tasks and uk, fk being task-specific, while the inverse problem corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) with bk being the same and uk, fk, and the differential operator Fk being different as a consequence of task-specific λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Forward problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We first assume λ = 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2) is known, and the data on f is available, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', Dk = {(xi k, f i k)}Nk i=1, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As described before, we employ MH-PINNs to solve {Tk}M k=1 all at once and then employ NFs to learn the distribution of the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Consequently, we obtain generators of f and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this case, f is assumed to be a Gaussian process with squared kernel function: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3) f(t) ∼ GP(0, K), t ∈ [0, 1], K(x, x′) = exp(−|x − x′|2 2l2 ), where the correlation length l is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2, many types of NFs have been developed in the past decade for generative modeling and 5 Measurements Reference Ours 3 0 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='85 2 std 4 Measurements Reference 3 Mean 2 2 3 4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='85 Measurements 4 Reference NN 3 = 10-6 2 = 10-4 10-2 0 2 3 4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 110 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS density estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' MH-PINNs, when used as generators, are compatible with all NFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this regard, we compare three popular NFs, RealNVP [9], MAF [33] and IAF [20], and eventually compare MH-PINNs (with NFs) against PI-GAN [44] in approximating the Gaussian process defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3) with different correlation lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the training of MH-PINNs and NFs as well as PI-GANs, 2, 000 f are sampled with respect to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3), each of which forms a physics-informed regression task with 65 measurements of f equidistantly sampled on [0, 1] as data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Notice that the ODE system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2) can be rewritten in a simpler format as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4) utt = −λ sin(u) + f(t), t ∈ [0, 1], with initial conditions u(0) = ut(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Hence, we choose to use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4) to build the loss function for physics-informed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results for comparisons are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 5 and Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' From the spectral analysis of the approximated Gaussian processes shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 5, we can see that MH-PINNs with MAF and IAF are comparable with PI-GANs while MH-PINNs with RealNVP fall marginally behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As shown in Table 1, the computational costs of MH-PINNs with MAF and RealNVP are significantly lower than PI-GANs, while MH-PINNs with IAF is more expensive than PI-GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We note that in this example we also record the computational cost for sampling using different generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As shown in Table 1, PI-GANs are significantly faster in generating samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' That is because, generally, GANs require relatively shallow NNs as opposite to NFs, for which a deep architecture is needed to keep up the expressivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Among three NFs, IAF is the fastest in sampling while MAF is the lowest, as opposed to training, which is consistent with the properties of those two NFs: MAF is slow for the forward pass, which is used to generate samples, and fast for the inverse pass, which is used to compute the density, while IAF is the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Despite the fact that MAF is slow in sampling, considering its fast training and good performance, we equip MH-PINNs with MAF as the density estimator and the generator for all other examples in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Approximating Gaussian processes as the source term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2) using different models: spectra of the correlation structure for the learned generators, for different correlation lengths, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The covariance matrix is constructed using 10, 000 generated samples, and eigen-values are averaged over 10 generators trained independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Next, we assume λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2) is unknown, and some measurements of u are available, in addition to f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', Dk = {{xi k, f i k} N f k i=1, {xi k, ui k}N u k i=1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' MH-PINNs are first employed to infer uk as well as λk from data Dk and physics, and NFs are employed afterwards to learn from samples of Hk and λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' To this end, the generative model is for the joint distribution of u, f and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Here, we assume f follows a truncated Karhuen-Loeve (KL)-expansion, with 5 leading terms, of the Gaussian process with squared kernel function and correlation length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1, and for each task Tk, 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3 e-Reference PI-GAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='25 MAF IAF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 RealNVP eigenvalues 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 0 0 5 10 15 components1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 eigenvalues 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0 0 5 10 15 components1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0 0 5 10 15 componentsL-HYDRA 11 MAF IAF RealNVP PI-GAN Phase 1 134s 134s 134s N/A Phase 2 252s 3939s 245s N/A Total 386s 4073s 379s 3243s Sampling 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='98 × 10−1s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='48 × 10−2s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='50 × 10−2s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='29 × 10−3s Table 1 Computational time for different models to approximate Gaussian process with correlation length l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The MH-PINN method is a two-step method and hence its computation is de- composed into two parts: training MH-PINNs referred to as phase 1 and training NFs referred to as phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Sampling time is defined to be the average time needed to generate 10, 000 samples of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' λk = 1 2 exp( � [0,1] f 2 k(t)dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As for the downstream task ˜T , the target is to infer u and λ from insufficient data of u and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the training of MH-PINNs and NFs, 2, 000 samples of f are generated and displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For each task, we assume 33 measurements of fk and 9 mea- surements of uk, equidistantly distributed on [0, 1], are available, and initial conditions are hard-encoded in NN modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the downstream task, we assume 1 random measurement of u and 8 random measurements of f are available with hard-encoded initial conditions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the case with noisy measurements, we assume the noises εf and εu to be additive Gaussian, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 noise scale for measurements of f and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='005 noise scale for measurements of u, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' εf ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='052) and εu ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0052).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The reference solution as well as the clean data of uk are generated by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2) for each task Tk, with corresponding fk and λk using Matlab ode45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 6 and Table 2, from which we can see our method is able to approximate the stochastic process as a source term well and produce accurate and trustworthy predictions, for u, f and also λ in the downstream task with limited data, in both noiseless and noisy cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As shown, the PINN method yields unacceptable estimate over both u and λ due to lack of data, while our approach is of much higher accuracy by integrating prior knowledge from {Tk}M k=1 with MH-PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' PINN MH-PINN λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8440 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5428 Error (%) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='21 Table 2 Estimate of λ and L2 relative error of u for the downstream inverse problem on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2) with clean data, using our approach and the regular PINN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The reference value for λ is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 1-D nonlinear reaction-diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We now test our method on a 1-D nonlinear time-dependent reaction-diffusion equation, which is commonly referred to as Fisher’s equation [2]: ut = Duxx + ku(1 − u), t ∈ [0, 1], x ∈ [−1, 1], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5) u(t, −1) = u(t, 1) = 0, t ∈ [0, 1], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6) u(0, x) = u0(x), x ∈ [−1, 1], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='7) where D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1, k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 and u0(x) is the initial condition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this example, we assume that the initial condition function is a stochastic process with the following 12 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results for the inverse problem of the ODE system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2), with initial conditions hard- encoded in NN modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (a) Left: 1, 000 samples of f, generated from the exact distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' middle: 1, 000 samples of f, generated from the learned generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' right: statistics computed from samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The bound is defined as the same as in the caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (b) Predicted f and u using PINNs and our approach, for the downstream inverse problem with noiseless data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (c) Predicted f, u and λ with uncertainties using our approach, for the downstream inverse problem with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The predicted mean and standard deviation of λ is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4663 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1501, while the reference value is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' distribution: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8) u0(x) = (x2 − 1) 5 5 � j=1 ξj(cos2(jx) − 1), x ∈ [−1, 1], where ξj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', 5 are independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=') random vari- ables subject to uniform distribution on [0, 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', ξj ∼ U[0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Unlike previous examples, the stochasticity comes from the initial condition rather than the source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' This example corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) with Fk and fk being the same for all tasks, and uk and bk being task-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In addition to measurements of u0, points on which the PDE residuals are computed are also required in both {Tk}M k=1 and ˜T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Hence, the data is Dk = {{(ti k, xi k), 0} N f k i=1, {(0, xi k), bi k}N b k i=1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the training, 2, 000 samples of u0(x) are generated, displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 7(a), and each sample forms a physics-informed regression task with 41 measurements of u0 equidistantly sampled on [−1, 1] as data for initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Besides, for all tasks, a uniform mesh 21 × 41 on temporal-spatial domain [0, 1] × [−1, 1] is used to com- pute the PDE residual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the downstream tasks ˜T , 5 random measurements of u0 are available and the same uniform mesh is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The boundary conditions are hard-encoded in NN modeling in both {Tk}M k=1 and ˜T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the noisy case, the 4 3 2 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='94 3 2 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='94 Predicted mean Predicted bound 3 Reference mean Reference bound 2 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9 t2 Measurements 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 Reference Ours 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='.- PINN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9 t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9 1 t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 Prior (learned from physics) Posterior 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 2 3 4 5 6L-HYDRA 13 noise ε is assumed to be independent additive Gaussian noise with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='02 noise scale for both measurements of u0 and the PDE residual, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', ε ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 7 and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We can see that our method estimates a good gen- erator of the stochastic processes from data and physics, which provides informative prior knowledge in the downstream few-shot physics-informed regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The prediction is accurate in both noiseless and noisy cases, and the errors in the noisy case are bounded by the predicted uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The L2 error of u, shown in Table 3, indicates that our approach outperforms the PINN method by a significant amount, hence demonstrating the effectiveness of bringing prior knowledge into solving similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Generator learning and few-shot physics-informed learning on 1-D time-dependent reaction-diffusion equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5), with boundary conditions hard-encoded in NN modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (a) Left: 1, 000 training samples of u0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' middle: 1, 000 samples of u(0, ·) from the learned generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' right: statistics computed from samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The bound is defined as the same as in the caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (b) Predicted u at t = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5, 1 using our approach and the PINN method with noiseless measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (c) Predicted mean and uncertainty of u at t = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5, 1 using our approach with HMC for posterior estimate, with noisy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' PINN MH-PINN Error (%) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='22 Table 3 L2 relative error of u for the downstream few-shot physics-informed learning task on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5) with clean data of u0 using our approach and the PINN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 Predicted mean Predicted bound Reference mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 Reference bound 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3 0=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='814 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2-D nonlinear Allen-Cahn equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We now move to a 2-D steady nonlinear Allen-Cahn equation with Dirichlet boundary conditions [43]: λ∆u + u(u2 − 1) = f, x, y ∈ [0, 1], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9) u(x, 0) = u(x, 1) = u(0, y) = u(1, y) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='10) where λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 is a constant and f is the source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Here, we impose a distribution to f, which is derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9) and the following distribution of the solution u: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='11) u(x, y) = 1 5 5 � j=1 ξj sin(jπx) sin(jπy) j2π2 , x, y ∈ [0, 1], with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' random variables ξj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', 5, subject to uniform distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ξj ∼ U[0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this example, we wish to use our method to learn generators of both u and f from data of f and physics in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9), and use it to solve the downstream task ˜T with insufficient data ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' This example corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) with Fk, bk being the same among tasks and fk, uk being task-specific, and the data is Dk = {(xi k, yi k), f i k} N f k i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' To train the MH-PINNs and NFs, we sample 2, 000 f from its distribution, each of which is resolved with a 51 × 51 uniform mesh on 2-D spatial domain [0, 1] × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As for the downstream task, 100 random measurements of f on the uniform mesh are assumed to be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The noise is assumed to be independent additive Gaussian noise with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 noise scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In both Tk and ˜T , the boundary conditions are hard- encoded in NN modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results as well as the locations of the measurements are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 8 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Similar to all previous examples, our approach delivers accurate and trustworthy predictions, showing that prior knowledge is learned and transferred well in both deterministic and Bayesian inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results for few-shot physics-informed learning on the 2-D nonlinear Allen-Cahn equa- tion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9) with noisy measurements of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Predicted mean µ and standard deviation σ are com- puted over 1, 000 posterior samples from HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The absolute error is defined as the absolute value of difference between the reference and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Black crosses represent the locations of the measurements on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' reference of f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 9 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 1L-HYDRA 15 PINN MH-PINN Error (%) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='30 Table 4 L2 relative error of u for the downstream few-shot physics-informed learning task on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9) with clean data of f, using our approach and the PINN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 2-D Stochastic Helmholtz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The last example we test in this paper is the 2-D Helmholtz equation with stochastic source term and Dirichlet bound- ary conditions [35]: (λ2 − ∇2)u = f, x, y ∈ [0, 2π], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='12) u(x, 0) = u(x, 2π) = u(0, y) = u(2π, y) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='13) where λ2 is the Helmholtz constant and f is defined as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='14) f(x, y) = 2 d{ d/4 � i=1 ξi sin(ix) + ξi+d cos(ix) + ξi+2d sin(iy) + ξi+3d cos(iy)}, where ξj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', d are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' random variables subject to uniform distribution U[0, 1) and d represents the dimension of the randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For demonstration purposes, we consider the case where d = 20 in this paper, unlike the one in [35] with d = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The first case we study is the forward problem with λ2 = 1 known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' This setup corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) with Fk, bk being shared among tasks and uk, fk being task- specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Next, we study the inverse problem with unknown λ, where data on u and f are available, which corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1) with only bk being the same and uk, fk and operator Fk being task-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The downstream tasks are defined as the same as {Tk}M k=1 in both cases, but with fewer measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For both the forward and inverse problems, 10, 000 f are sampled from its distri- bution, and hence 10, 000 tasks are solved with MH-PINNs with boundary conditions hard-encoded in NN modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We display the samples of a slice of f in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the forward problem, Dk only contains measurements of the source term fk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Dk = {{(xi k, yi k), f i k} N f k i=1}, while for the inverse problem Dk also contains measure- ments of the sought solution uk: Dk = {{(xi k, yi k), f i k} N f k i=1, {(xi k, yi k), ui k}N u k i=1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the training in the forward problem, each sample of f is resolved by a 50 × 50 uniform mesh on 2-D spatial domain (0, 2π)×(0, 2π) with boundary excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the inverse problem, the same 10, 000 samples of f are used, but this time they are resolved with a 21 × 21 uniform mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In addition, for each task Tk, measurements of uk on a 6 × 6 uniform mesh are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The reference solution and measurements of u are gen- erated by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='12) with λ2 k = � [0,2π]2 f 2 k(x, y)dxdy using the finite difference method with five-point stencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the downstream tasks, 100 random measurements of f are available for the forward problem, and 50 random measurements of f and 10 random measurements of u are available for the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The noise is assumed to be independent additive Gaussian noise with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 noise scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results are displayed in Tables 5 and 6, and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As shown, the learned generator is able to produce samples of f with high quality as well as providing informative prior knowledge for the downstream tasks, in both the forward and inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As for the noisy case with Bayesian inference and UQ, the predicted means agree with the references and the absolute errors are mostly bounded by the predicted uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The effectiveness of our approach for few-shot physics-informed learning 16 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS and the applicability to both deterministic optimization and Bayesian inference have been consistently demonstrated in the past five examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Generator learning and few-shot physics-informed learning on the stochastic Helmholtz equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (a) Left: 1, 000 training samples of a slice of f at y = π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' middle: 1, 000 samples of a slice of f at y = π from the learned generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' right: statistics computed from samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (b)/(c) Results for the downstream forward problem with 100 random noisy measurements on f, using our approach with HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' From left to right are reference, predicted mean, absolute error, and predicted uncertainty of f/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Black crosses represent the locations of the measurements of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' PINN MH-PINN Error (%) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='12 Table 5 L2 relative error of u for the downstream forward problem on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='12) with clean data of f, using our approach and the PINN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' PINN MH-PINN λ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9328 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0170 Error (%) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='58 Table 6 Estimate of λ and L2 relative error of u for the downstream inverse problem on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='12) with clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The reference value of λ is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Multi-task learning with multi-head neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' So far we have mostly focused on using MH-NNs together with NFs to estimate stochastic generators 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 =/ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 0 U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0 2 3 4 5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0 2 3 4 5 6Predicted mean Predicted bound Reference mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 Reference bound 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0 2 3 4 5 6reference of f 6 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 4 9 3 0 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 2 3 4 5 6 1predicted mean of f 6 X X X X X X XX 5 X X X X X X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 4 X X X X 9 X 3 X X X XX 0 2 × X X 1 X X X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 2 3 4 5 6absolute error of t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='09 XX 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='06 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='04 2× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='03 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='01 X 0 2 3 4 5 6predicted uncertainty of f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='07 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='06 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='04 2 0.' 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measurements of f and 10 random noisy measurements of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' λ is estimated as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0785 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0307 in the format of predicted mean ± predicted standard deviation, while the reference value is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (a)/(b) From left to right are reference, predicted mean, absolute error, and predicted uncertainty of f/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Black crosses represent locations of the measurements of f or u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' and learn informative prior knowledge from {Tk}M k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' This was achieved by first training MH-NNs in a MTL fashion and then training NFs to estimate the PDF of the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Intuitively, the capability of MH-NNs when trained in MTL in capturing shared information is the key to the success in generative modeling and few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For physics-informed MTL with MH-PINNs, ODEs/PDEs are solved simultaneously, and assuming the solutions to share the same set of basis functions gives us samples of the set of coefficients, which enables the generative modeling, followed by few-shot learning, which is the whole point of the method proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' However, the cost and/or the benefit of imposing the same set of basis functions to all solutions have not been explicitly discussed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' On one hand, the shared body relates the training of tasks, which may be helpful if tasks are similar in certain ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' On the other hand, forcing all solutions to share the same basis functions may also be harmful when they behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In particular, for tasks with sufficient data and physics, forcing them to share the same body with all other tasks may act as a negative regularization, and single-task learning (STL) may outperform MTL in terms of the prediction accuracy in those specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this section, we investigate the effect of MTL using MH-NNs and provide preliminary results and analysis by revisiting the simple function approximation example in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1, which, hopefully, could provide useful information and insight for future more rigorous research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Basis function learning and synergistic learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As discussed before, the quality and behavior of basis functions learned in MTL are crucial to genera- tive modeling and learning the relation and the representative information of tasks {Tk}M k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We consistently noticed from numerical examples that the initialization of the head in MH-NNs has great impact on the average accuracy of MTL, the learning of the basis functions, and the distribution of the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Here, we test three initialization strategies, random normal method with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 standard deviation referred to as RN reference of f 6 5 0.' metadata={'source': 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+page_content='012 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='01 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='008 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='004 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='002 0 1 2 3 4 5 618 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05), Glorot uniform method [12] referred to as GU, and random normal method with 1 standard deviation referred to as RN (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In the downstream few-shot learning tasks, we fine-tune the head without the learned PDF, which is in fact the TL method from [7], by which the information from the distribution of the head is excluded and the prediction accuracy is fully determined by the level of prior knowledge contained in the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 11, RN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05) yields the least informative basis functions, whose behavior is dominated by the hyperbolic tangent activation function of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' This is further demonstrated in the downstream few-shot learning tasks using the TL method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' It also provides the worst prediction accuracy on average in MTL, as presented in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' GN and RN (1) perform similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Plots of some basis functions seemingly indicate that RN (1) yields better basis functions, whose behaviors are more similar to the family of functions displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 11(b), which, however, does not necessarily imply richer prior knowledge in the downstream tasks, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 11(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' It is shown empirically that compared to other two initialization strategies, MH- NNs with RN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05) does not deliver accurate MTL nor synergistic learning in basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' However, we noticed that, in generative modeling, it performs significantly better in terms of accuracy and convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 11(d), samples from the learned generator are of higher quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We consistently found that initializ- ing heads with relatively small values often led to easy and fast training of NFs and accurate learning of the generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We conjecture that this happens because MH-NNs in MTL tend to contain the representative and informative information in the heads when heads are initialized with small values, while contain it in the basis functions when heads are initialized with relatively large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' RN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05) GU RN (1) Error (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8373 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1907 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0690 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3131 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0937 Table 7 L2 relative errors, from MTL, for 1, 000 tasks, using different initialization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The errors are displayed in the format of mean ± standard deviation, computed over all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Multi-Task Learning (MTL) versus Single-Task Learning (STL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' As discussed earlier, MTL with MH-NNs does not necessarily result in synergistic learning nor higher accuracy for all tasks on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Here, we use again the function approximation example in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1, to investigate the effectiveness of MTL with MH- NNs, as compared to STL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The first case we consider here is assuming that the data is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For that, we randomly choose 100 out of the 1, 000 training samples, each one of which is approximated by a NN trained independently, and compare the results with MH-NNs in terms of prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Note that in this case, a MH-NN is trained on 1, 000 functions as before and tested on the chosen 100 functions, while a single-head NN with the same architecture is trained on 100 functions directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results are shown in Table 8, from which it is verified empirically that MTL is outperformed by STL under certain circumstances, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', when the random normal initialization methods are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The second case we consider is assuming that the data is sufficient for some tasks while insufficient for other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For that, we split equally the 1, 000 tasks into two subsets of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the first 500 tasks, we assume we only have 10 measurements randomly sampled on [−1, 1], while for the other 500 tasks, we assume we have full 40 measurements equidistantly distributed on [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' MTL with MH-NNs is performed on those 1, 000 regression tasks all at once, and the tasks are treated as equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The L-HYDRA 19 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The effect of different initialization methods of the head, in basis functions learning, few-shot learning, and generator learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (a) Samples of 20 basis functions from MH-NNs, trained for approximating 1, 000 f generated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1), using, from left to right, RN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05), GU and RN (1) initialization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (b) 1, 000 training samples of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (c) Results for two downstream few-shot regression tasks, using TL method without regularization informed by the learned PDF, as opposite to the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (d) Results for generator learning, using, from left to right, RN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05), GU and RN (1) initialization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' RN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05) GU RN (1) STL Error (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='7575 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1362 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3664 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2102 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='0794 Table 8 L2 relative errors of f, from MTL with MH-NNs and STL with NNs, on 100 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Different initialization methods are used for the heads in MH-NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The errors are displayed in the format of mean ± standard deviation, computed over all 100 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' results are presented in Table 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We can see that, compared to STL, MTL improves the prediction accuracy on tasks with insufficient data, providing empirical evidence of synergistic learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Also, interestingly, RN (1) initialization method, which yields the worst generative models, performs the best among all three, which agrees with our previous conjecture on the basis functions learning with MH-NNs, that heads initialized with large values tend to force representative and informative information to be encoded in the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We have developed multi-head neural networks (MH-NNs) for physics-informed machine learning, and proposed multi-head physics-informed neural networks (MH-PINNs) as a new method, implemented in the L-HYDRA code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The primary focus of this work is on MH-NNs and MH-PINNs for various learning prob- 6 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='820 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS RN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05) GU RN (1) Error (%) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='60 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='08 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='49 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='49 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='91 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='08 Table 9 L2 relative errors of f, from MTL with MH-NNs, on 500 tasks equipped with insufficient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The errors are displayed in the format of mean ± standard deviation, computed over all 500 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Results for 3 tasks with insufficient data from MTL with MH-NNs, using different initialization methods over the head, and from STL with NNs with the same architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We note that tasks with sufficient data and tasks with insufficient data are treated equally in MTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' lems in scientific machine learning, including multi-task learning (MTL), stochastic processes approximation, and few-shot regression learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We first formulated the problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1), introduced the architecture design of MH-PINNs, and proposed a method to transform MH-NNs and MH-PINNs to generative models with the help of normalizing flows (NFs) for density estimation and generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We then studied the applicability and capabilities of MH-PINNs in solving ordinary/paritial differential equations (ODEs/PDEs) as well as approximating stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We completed the paper with preliminary and empirical explorations of MH-NNs in synergistic learning, and examined the potential benefits and cost of MTL with MH-NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' This paper can be used in various ways: it proposes a NN approach for MTL in solving ODEs/PDEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' it provides a new approach to approximate stochastic processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' it presents a method to address few-shot physics-informed learning problems, which are often encountered in the context of meta-learning and transfer learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' it contains a systematic study of applying MH-NNs to scientific computing problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' it presents the first empirical evidence of synergistic learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' However, there are a few major problems on MH-NNs we did not address, one of which is the expressivity of MH-NNs, or more generally hard-parameter sharing NNs in approximating complicated stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Intuitively, if two functions behave very differently, forcing them to share the same basis functions would affect adversely the approximation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The second problem is the balancing issue of different terms in the loss function in MTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' It is shown in the literature [29] that PINNs, trained in single-task learning, are already deeply influenced by the weights in front of different terms in the loss function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=', data loss, boundary condition loss, PDE residual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' This issue may be more complex in training MH-PINNs, because in MTL the loss function is commonly defined as weighted summation of task-specific loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The last major problem is MH-PINNs for synergistic learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this paper, we only studied one example in function approximation and presented empirical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' More work for the understanding of synergistic learning with MH-PINNs along both the theoretical and computational directions should be pursued in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' We would like to thank Professor Xuhui Meng of Huazhong University of Science and Technology for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' This work was supported by: the Vannevar Bush Faculty Fellowship award (GEK) from ONR (N00014-22- Measurements Reference RN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05) 3 GU 2 RN(1) NN 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='8 0.' metadata={'source': 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comprehensive library for uncertainty quantification in neural differential equations and operators, arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='11866, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Details of NN architectures and training hyperparam- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For all examples in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4 and 5, MH-PINNs are implemented as fully- connected NNs (FNNs) with 3 nonlinear hidden layers, each of which is equipped with 50 neurons and hyperbolic tangent activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The number of heads is the same as the number of tasks in the corresponding examples: 1, 000 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1, 2, 000 in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='4, and 10, 000 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Weights in the body of MH- PINNs are initialized with Glorot uniform initialization [12] and biases are initialized with zero, while heads are initialized by sampling from random normal distribution with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='05 standard deviation, for fast training of NFs and better performance of the learned generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Except for the forward problem in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2, NFs in this paper are chosen to be MAF [33] with 10 bijectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' the invertible map in NFs, each of which is a MADE [11], a NN with masked dense layers, with two nonlinear hidden layers equipped with 100 neurons and ReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The RealNVP [9] and IAF [20] used in the forward problem in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 also have 10 bijectors, each of which is a NN with two nonlinear hidden layers equipped with 100 neurons and ReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The implementation mostly follows the instructions of TensorFlow Probability library [8] for NFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' PI-GANs [44] implemented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 have the following architecture: the dis- criminator is a FNN with 3 nonlinear hidden layers, each of which is equipped with 128 neurons and Leaky ReLU activation function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' the generator that takes as input t is a FNN with 3 nonlinear hidden layers, each of which is equipped with 50 neu- rons and hyperbolic tangent activation function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' the other generator takes as input a Gaussian random variable in 50 dimensions with zero mean and identity covariance matrix, and is implemented as a FNN with 3 nonlinear hidden layers, each of which has 128 neurons and hyperbolic tangent activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The input dimensions of those 3 FNNs are 65, 1 and 50, and the output dimensions are 1, 50, 50, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the training of MH-PINNs, full-batch training is deployed with Adam opti- mizer for 50, 000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the training of NFs, except for the forward problem in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2, mini-batch training is deployed with batch size being 100 and Adam op- timizer for 1, 000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' NFs in the forward problem in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='2 are trained for 500 epochs instead, and L2 regularization is imposed to the parameters of RealNVP for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For all NFs, to achieve stable training, a hyperbolic tangent function is imposed on the logarithm of the scale, computed from each bijector, such that the logarithm of the scale lies in (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' For the training of PI-GANs, min- batch training is deployed with batch size being 100 and Adam optimizer for 100, 000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Besides, the same as in [44, 30], physics-informed Wasserstein GANs (PI- WGANs) with gradient penalty are employed, in which the coefficient for gradient penalty is set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Iteratively, 5 updates of the discriminator are performed and followed by 1 update of the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Except in training PI-GANs, the learning rate of Adam optimizer is set to be 10−3 and other hyperparameters of Adam are set as default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In training PI-GANs, the learning rate is set to be 10−4, β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='5 and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='9 in Adam optimizer for both discriminator and generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' 24 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' ZOU AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' KARNIADAKIS Training of MH-PINNs, NFs, and PI-GANs was all performed on a single NVIDIA TITAN Xp GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The L-HYDRA code for TensorFlow implementation along with some representative examples will be released on GitHub once the paper is accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Details for performing Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' Hamiltonian Monte Carlo (HMC) [31] is employed in all Bayesian inference examples for uncer- tainty quantification (UQ) while Laplace approximation [18] is only employed in the first example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this paper, HMC with adaptive step size [23] is used, in which the initial step size is set to be either 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='1 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content='01, tuned for better acceptance rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The number of burn-in samples and the number of posterior samples are set to be 1, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The number of steps for the leapfrog scheme is set to be either 30 or 50, also tuned for better acceptance rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' NeuralUQ library [47] for UQ in scientific machine learning is used as a tool for physics-informed Bayesian inference in the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' The ideal acceptance rate in HMC, as discussed in [30, 47], is around 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} +page_content=' In this paper, we found chains with acceptance rate from 50% to 80% acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dA0T4oBgHgl3EQfNv_S/content/2301.02152v1.pdf'} diff --git a/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf b/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..35e9f641478a6b81671267ba5535648927da1135 --- /dev/null +++ b/69E3T4oBgHgl3EQfpwru/content/2301.04646v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Supply and Demand with Dynamic Pricing: +Problem Formalisation and Conceptual Analysis +Thibaut Théatea,∗, Antonio Suterab, Damien Ernsta,c +aDepartment of Electrical Engineering and Computer Science, University of Liège, Liège, Belgium +bHaulogy, Intelligent Systems Solutions, Braine-le-Comte, Belgium +cInformation Processing and Communications Laboratory, Institut Polytechnique de Paris, Paris, France +Abstract +The energy transition is expected to significantly increase the share of renewable energy sources whose production is +intermittent in the electricity mix. Apart from key benefits, this development has the major drawback of generating a +mismatch between power supply and demand. The innovative dynamic pricing approach may significantly contribute to +mitigating that critical problem by taking advantage of the flexibility offered by the demand side. At its core, this idea +consists in providing the consumer with a price signal which is evolving over time, in order to influence its consumption. +This novel approach involves a challenging decision-making problem that can be summarised as follows: how to determine +a price signal maximising the synchronisation between power supply and demand under the constraints of maintaining +the producer/retailer’s profitability and benefiting the final consumer at the same time? As a contribution, this research +work presents a detailed formalisation of this particular decision-making problem. Moreover, the paper discusses the +diverse algorithmic components required to efficiently design a dynamic pricing policy: different forecasting models +together with an accurate statistical modelling of the demand response to dynamic prices. +Keywords: +Matching of supply and demand, dynamic pricing, demand response, power producer/retailer. +1. Introduction +Climate change is undeniably a major challenge facing +humanity in the 21st century [1]. +An ambitious trans- +formation is required in all sectors to significantly lower +their respective carbon footprints. Electricity generation +is no exception, with the burning of fossil fuels, mainly coal +and gas, being by far the dominant power source in the +world today [2]. This sector has to undergo an important +transformation of the global electricity mix by promoting +power sources with a significantly lower carbon footprint. +Belonging to that category are nuclear power, hydroelec- +tricity, biomass or geothermal energy which are relatively +controllable, but also the energy directly extracted from +wind and sun which is conversely intermittent in nature. +Since wind turbines and photovoltaic panels are expected +to play a key role in the energy transition, solutions are +required to address their variable production. Interesting +technical avenues are the interconnection of power grids [3] +and the development of storage capacities such as battery, +pumped hydroelectricity or hydrogen [4]. Another promis- +ing and innovative solution is to influence the behaviour +of consumers through the use of dynamic pricing (DP), so +that power supply and demand are better synchronised. +∗Corresponding author. +Email addresses: thibaut.theate@uliege.be (Thibaut +Théate), dernst@uliege.be (Damien Ernst) +The dynamic pricing approach consists in continuously +adapting the electricity price that the final consumer has +to pay in order to influence its consumption behaviour. +Basically, when demand exceeds supply, the power price +would be increased in order to take down consumption. +Conversely, a reduced price would be provided when there +is excessive production compared to consumption. From a +graphical perspective, the objective is not only to shift the +daily consumption curve but also to change its shape in +order to better overlap with the intermittent production +curve of renewable energy sources. This is illustrated in +Figure 1 for a representative situation. +The innovative dynamic pricing approach relies on two +important assumptions. Firstly, the final consumer has to +be equipped with a smart metering device to measure its +production in real-time and with communication means +for the price signal. Secondly, the final consumer has to +be able to provide a certain amount of flexibility regarding +its power consumption. Moreover, it has to be sufficiently +receptive to the incentives offered to reduce its electricity +bill in exchange for a behaviour change. If these require- +ments are met, the major strength of the dynamic pricing +approach is its potential benefits for both the consumer +and the producer/retailer. Moreover, these benefits would +not only be in terms of economy, but also potentially in +terms of ecology and autonomy. In fact, dynamic prices +reward the flexibility of the demand side. +arXiv:2301.11587v1 [q-fin.TR] 27 Jan 2023 + +12:00 +06:00 +18:00 +12:00 +06:00 +18:00 +Supply +Demand +Time +Time +Power +Power +Figure 1: Illustration of the dynamic pricing approach’s potential +to shift and change the shape of a typical daily consumption curve +(blue) so that there is a better synchronisation with the daily inter- +mittent production curve of renewable energy sources (red). +The contributions of this research work are twofold. +Firstly, the complex decision-making problem faced by +a producer/retailer willing to develop a dynamic pricing +strategy is presented and rigorously formalised. Secondly, +the diverse algorithmic components required to efficiently +design a dynamic pricing policy are thoroughly discussed. +To the authors’ knowledge, demand response via dynamic +pricing has received considerable attention from the re- +search community, but from the perspective of the demand +side alone. Therefore, the present research may be consid- +ered as a pioneer work studying dynamic pricing from the +perspective of the supply side for taking advantage of the +flexibility of the power consumers. +This research paper is structured as follows. First of +all, the scientific literature about both dynamic pricing +and demand response is concisely reviewed in Section 2. +Then, Section 3 presents a detailed formalisation of the +decision-making problem behind the novel dynamic pric- +ing approach. Afterwards, Section 4 analyses the algorith- +mic components necessary for the development of dynamic +pricing policies. Subsequently, a fair performance assess- +ment methodology is introduced in Section 5 to quanti- +tatively evaluate the performance of a dynamic pricing +policy. To end this paper, Section 6 discusses interesting +research avenues for future work and draws conclusions. +2. Literature review +Over the last decade, the management of the demand +side in the scope of the energy transition has received in- +creasing attention from the research community. In fact, +there exist multiple generic approaches when it comes to +demand response. Without getting into too many details, +the scientific literature includes some surveys summarising +and discussing the different techniques available together +with their associated challenges and benefits [5, 6, 7, 8, 9]. +In this research work, the focus is exclusively set on the +demand response induced by dynamic power prices. +As previously mentioned, the scientific literature about +demand response via dynamic pricing is primarily focused +on the perspective of the demand side. Multiple techniques +have already been proposed to help the consumer provide +flexibility and take advantage of behavioural changes to +lower its electricity bill. For instance, [10] presents a power +scheduling method based on a genetic algorithm to opti- +mise residential demand response via an energy manage- +ment system, so that the electricity cost is reduced. In [11], +a technique based on dynamic programming is introduced +for determining the optimal schedule of residential con- +trollable appliances in the context of time-varying power +pricing. One can also mention [12] that proposes an energy +sharing model with price-based demand response for mi- +crogrids of peer-to-peer prosumers. The approach is based +on a distributed iterative algorithm and has been shown +to lower the prosumers’ costs and improve the sharing of +photovoltaic energy. More recently, (deep) reinforcement +learning techniques have been proven to be particularly +relevant for controlling the residential demand response in +the context of dynamic power prices [13, 14]. +On the contrary, the question of inducing a residential +demand response based on a dynamic pricing approach +from the perspective of the supply side has not received +a lot of attention from the research community yet. Still, +there are a few works in the scientific literature about the +mathematical modelling of the demand response caused by +dynamic power prices, which is a key element in achieving +that objective. To begin with, [15] presents a simulation +model highlighting the evolution of electricity consump- +tion profiles when shifting from a fixed tariff to dynamic +power prices. The same objective is pursued by [16] which +introduces a fully data-driven approach relying on the data +collected by smart meters and exogenous variables. The +resulting simulation model is based on consumption pro- +files clustering and conditional variational autoencoders. +Alternatively, [17] presents a functional model of residen- +tial power consumption elasticity under dynamic pricing to +assess the impact of different electricity price levels, based +on a Bayesian probabilistic approach. In addition to these +mathematical models, one can also mention some real-life +experiments conducted to assess the responsiveness of res- +idential electricity demand to dynamic pricing [18, 19]. +2 + +3. Problem formalisation +This section presents a mathematical formalisation of +the challenging sequential decision-making problem related +to the dynamic pricing approach for inducing a residential +demand response. +To begin with, the contextualisation +considered for studying this particular problem is briefly +described, followed by an overview of the decision-making +process. Then, a discretisation of the continuous timeline +is introduced. Subsequently, the formal definition of a dy- +namic pricing policy is presented. Lastly, the input and +output spaces of a dynamic pricing policy are described, +together with the objective criterion. +3.1. Contextualisation +As previously hinted, this research work focuses on the +interesting real-case scenario of a producer/retailer whose +production portfolio is composed of an important share of +renewable energy sources such as wind turbines and photo- +voltaic panels. Because of the substantial intermittency of +these generation assets, a strong connection to the energy +markets is required in order to fully satisfy its customers +regardless of the weather. Nevertheless, the consumers are +assumed to be well informed and willing to adapt their be- +haviour in order to consume renewable energy rather than +electricity purchased on the market whose origin may be +unknown. Within this particular context, the benefits of +the dynamic pricing approach taking advantage of the con- +sumers’ flexibility are maximised. Indeed, the insignificant +marginal cost associated with these intermittent renew- +able energy sources coupled with their low carbon foot- +print make this innovative approach interesting from an +economical perspective for both supply and demand sides, +but also in terms of ecology. Moreover, the autonomy of +the producer/retailer is expected to be reinforced by low- +ering its dependence on the energy markets. At the same +time, dependence on fossil fuels may be reduced as well. +In this research work, the predicted difference between +power production and consumption is assumed to be fully +secured in the day-ahead electricity market. Also called +spot market, the day-ahead market has an hourly resolu- +tion and is operated once a day for all hours of the follow- +ing day via a single-blind auction. In other words, trading +power for hour H of day D has to be performed ahead on +day D − 1 between 00:00 AM (market opening) and 12:00 +AM (market closure). +Therefore, the energy is at best +purchased 12 hours (00:00 AM of day D) up to 35 hours +(11:00 PM of day D) before the actual delivery of power. +Apart from the day-ahead electricity market, it is assumed +that there are no trading activities on the future/forward +nor intraday markets. Nevertheless, if there remains an +eventual mismatch between production and consumption +at the time of power delivery, the producer/retailer would +be exposed to the imbalance market. In this case, the so- +called imbalance price has to be inevitably paid as com- +pensation for pushing the power grid off balance. +3.2. Decision-making process overview +The decision-making problem studied in this research +work is characterised by a particularity: a variable time lag +between the moment a decision is made and the moment it +becomes effective. As previously explained, any remaining +difference between production and consumption after de- +mand response has to ideally be traded on the day-ahead +market. The purpose of this assumption is to limit the ex- +posure of the producer/retailer to the imbalance market. +For this reason, the price signal sent to the consumer on +day D has to be generated before the closing of the day- +ahead market on day D − 1. Additionally, it is assumed +that the price signal cannot be refreshed afterwards. +Basically, the decision-making problem at hand can be +formalised as follows. The core objective is to determine +a decision-making policy, denoted Π, mapping at time τ +input information of diverse nature Iτ to the electricity +price signal Sτ to be sent to the consumers over a future +time horizon well-defined: +Sτ = Π(Iτ), +(1) +where: +• Iτ represents the information vector gathering all the +available information (of diverse nature) at time τ +which may be helpful to make a relevant dynamic +pricing decision, +• Sτ represents a set of electricity prices generated at +time τ and shaping the dynamic price signal over a +well-defined future time horizon. +The dynamic pricing approach from the perspective of +the supply side belongs to a particular class of decision- +making problems: +automated planning and scheduling. +Contrarily to conventional decision-making outputting one +action at a time, planning decision-making is concerned +with the generation of a sequence of actions. +In other +words, a planning decision-making problem requires to +synthesise in advance a strategy or plan of actions over +a certain time horizon. Formally, the decision-making has +to be performed at a specific time τ about a control vari- +able over a future time horizon beginning at time τi > τ +and ending at time τf > τi. In this case, the decision- +making is assumed to be performed just before the closing +of the day-ahead market at 12:00 AM to determine the +price signal to be sent to the consumers throughout the +entire following day (from 00:00 AM to 11:59 PM). +In the next sections, a more accurate and thorough +mathematical formalisation of the dynamic pricing prob- +lem from the perspective of the supply side is presented. +Moreover, the planning problem previously introduced is +cast into a sequential decision-making problem. Indeed, +this research paper intends to focus on a decision-making +policy outputting a single price from the signal Sτ at a +time based on a subset of the information vector Iτ. +3 + +00:00 +12:00 +𝑦� 𝑦� +𝑦�� +Closing of the +day-ahead market +𝑥� +𝑥� +𝑥� +𝑥�� +Dynamic pricing +policy π +𝑦� +Forecasts +𝑝� +� +𝑐� +� +𝜆� +� +𝑐� +� +Time +Power +Time +Time +Time +Power +Power +Price +Demand +reponse +model +Figure 2: Illustration of the formalised decision-making problem related to dynamic pricing from the perspective of the supply side. The +notations xt and yt represent the inputs and outputs of a dynamic pricing policy π, which are not shown concurrent on the timeline since +the decision-making occurs multiple hours before the application of the dynamic pricing signal. The time axis of the four plots represents the +complete following day for which the dynamic prices are generated. The mathematical notations pF +t , cF +t and λF +t respectively represent the +forecast production, consumption and day-ahead market price for the time step t. Finally, the quantity c′ +t is the predicted consumption at +time step t after taking into consideration the dynamic pricing signal. +3.3. Timeline discretisation +Theoretically, the dynamic electricity price signal sent +to the consumer could be continuously changing over time. +More realistically, this research work adopts a discretisa- +tion of the continuous timeline so that this power price +is adapted at regular intervals. +Formally, this timeline +is discretised into a number of time steps t spaced by a +constant duration ∆t. If the duration ∆t is too large, the +synchronisation improvement between supply and demand +will probably be of poor quality. Conversely, lowering the +value of the duration ∆t increases the complexity of the +decision-making process, and a too high update frequency +may even confuse the consumer. There is a trade-off to +be found concerning this important parameter. In this re- +search work, the dynamic price signal is assumed to change +once per hour, meaning that ∆t is equal to one hour. This +choice is motivated by the hourly resolution of the day- +ahead market, which has proven to be an appropriate com- +promise over the years for matching power production and +consumption. Another relevant discretisation choice could +be to have a price signal which is updated every quarter of +an hour. In the rest of this research paper, the increment +(decrement) operations t + 1 (t − 1) are used to model the +discrete transition from time step t to time step t + ∆t +(t − ∆t), for the sake of clarity. +3.4. Dynamic pricing policy +Within the context previously described, a dynamic +pricing planning policy Π consists of the set of rules used +to make a decision regarding the future price signal sent to +the consumers over the next day. This planning policy can +be decomposed into a set of 24 dynamic pricing decision- +making policies π outputting a single electricity price for +one hour of the following day. +Mathematically, such a +dynamic pricing strategy can be defined as a programmed +policy π : X → Y, either deterministic or stochastic, which +outputs a decision yt ∈ Y for time step t based on some +input information xt ∈ X so as to maximise an objective +criterion. The input xt is derived from the information vec- +tor Iτ associated with the decision-making for time step t, +after potential preprocessing operations. The price signal +Sτ is composed of 24 dynamic pricing policy outputs yt. +In the rest of this research work, the time at which +the decision-making does occur should not be confused +with the time at which the dynamic price signal is active +(charging for energy consumption). The proposed formal- +isation assumes that the time step t refers to the time at +which the dynamic price is active, not decided. Therefore, +the decision-making of the dynamic pricing policy for time +step t (yt = π(xt)) is in fact performed hours in advance +of time step t. This complexity is illustrated in Figure 2 +describing the formalised decision-making problem. +4 + +I.3.5. Input of a dynamic pricing policy +The input space X of a dynamic pricing policy π com- +prises all the available information which may help to make +a relevant decision about future electricity prices so that +an appropriate demand response is induced. +Since the +decision-making occurs 12 up to 35 hours in advance of the +price signal delivery, this information mainly consists of +forecasts and estimations that are subject to uncertainty. +As depicted in Figure 2, the dynamic pricing policy input +xt ∈ X refers to the decision-making occurring at time +τ = t − h with h ∈ [12, 35] about the dynamic pricing +signal delivered to the consumer at time step t. In fact, +the quantity Iτ may be seen as the information contained +in the 24 inputs xt for t ∈ {τ + 12, ..., τ + 35}. Formally, +the input xt ∈ X is decided to be defined as follows: +xt = {P F +t , CF +t , ΛF +t , Yt, M}, +(2) +where: +• P F +t += {pF +t+ϵ ∈ R+ | ϵ = −k, ..., k} represents a set +of forecasts for the power production within a time +window centred around time step t and of size k, +• CF +t = {cF +t+ϵ ∈ R+ | ϵ = −k, ..., k} represents a set of +forecasts for the power consumption within a time +window centred around time step t and of size k, +• ΛF +t = {λF +t+ϵ ∈ R | ϵ = −k, ..., k} represents a set of +forecasts for the day-ahead market prices within a +window centred around time step t and of size k, +• Yt = {yt−ϵ ∈ R | ϵ = 1, ..., k} represents the series of +k previous values for the dynamic price signal sent +to the final consumer, +• M is a mathematical model of the demand response +to be expected from the consumption portfolio, with +the required input information. +The different forecasting models and the challenging +modelling of the consumption portfolio demand response +are discussed in more details in Section 4. +3.6. Output of a dynamic pricing policy +The output space Y of a dynamic pricing policy π only +includes the future price signal to be sent to the consumer. +Formally, the dynamic pricing policy output yt ∈ Y, which +represents the electricity price to be paid by the consumer +for its power consumption at time step t, is mathematically +defined as follows: +yt = et, +(3) +where et ∈ R represents the dynamic electricity price to +be paid by the demand side for its power consumption +at time step t. +Out of the scope of this research work +is the presentation of this price signal so that the impact +on the final consumer is maximised. Indeed, the way of +communicating the output of the dynamic pricing policy +has to be adapted to the audience, be it humans with +different levels of electricity market expertise or algorithms +(energy management systems). +3.7. Objective criterion +The dynamic pricing approach can provide multiple +benefits, in terms of economy, ecology but also autonomy. +Consequently, the objective criterion to be maximised by +a dynamic pricing policy π is not trivially determined. In +fact, several core objectives can be clearly identified: +• maximising the match between supply and demand, +• minimising the carbon footprint of power generation, +• minimising the electricity costs for the consumer, +• maximising the revenue of the producer/retailer. +Although some objectives overlap, these four criteria +are not completely compatible. For instance, maximising +the synchronisation between power supply and demand is +equivalent to minimising the carbon footprint associated +with the generation of electricity. Indeed, the production +portfolio of the producer/retailer being mainly composed +of intermittent renewable energy sources, its energy has a +reduced carbon footprint compared to the electricity that +can be purchased on the day-ahead market whose origin is +unknown. On the contrary, maximising the revenue of the +producer/retailer will obviously not lead to a minimised +electricity bill for the consumer. This research work makes +the choice to prioritise the maximisation of the synchroni- +sation between supply and demand, and equivalently the +minimisation of the carbon footprint, while translating the +other two core objectives into relevant constraints. Firstly, +the costs for the consumer have to be reduced with respect +to the situation without dynamic pricing. Secondly, the +profitability of the producer/retailer has to be guaranteed. +Formally, the objective criterion to be optimised by a +dynamic pricing policy π can be mathematically defined +as the following. First of all, the main target to evaluate +is the synchronisation between supply and demand, which +can be quantitatively assessed through the deviation ∆T . +This quantity has to ideally be minimised, and can be +mathematically expressed as follows: +∆T = +T −1 +� +t=0 +|pt − ct|, +(4) +where: +• t = 0 corresponds to the first electricity delivery hour +of a new day (00:00 AM), +• T is the time horizon considered, which should be a +multiple of 24 to have full days, +• pt is the actual power production (not predicted) +from the supply side at time step t, +• ct is the actual power consumption (not predicted) +from the demand side at time step t. +5 + +Afterwards, the first constraint concerning the reduced +costs for the consumer has to be modelled mathematically. +This is achieved via the electricity bill BT paid by the +consumer over the time horizon T, which can be expressed +as the following: +BT = +T −1 +� +t=0 +ct yt . +(5) +As previously explained, the consumer power bill BT +should not exceed that obtained without dynamic pricing. +In that case, the consumer is assumed to pay a price et, +which can for instance be a fixed tariff or a price indexed +on the day-ahead market price. +The situation without +dynamic pricing is discussed in more details in Section 5. +Consequently, the first constraint can be mathematically +expressed as follows: +T −1 +� +t=0 +ct yt ≤ +T −1 +� +t=0 +ct et , +(6) +where ct is the power consumption from the demand side +at time step t without dynamic pricing. +Then, the second constraint is about the profitability +of the producer/retailer, which is achieved if its revenue +exceeds its costs. The revenue RT of the producer/retailer +over the time horizon T can be mathematically expressed +as the following: +RT = +T −1 +� +t=0 +� +ct yt − (c′ +t − pF +t ) λt − (ct − pt) it +� +, +(7) +where: +• λt is the actual power price (not predicted) on the +day-ahead market at time step t, +• it is the actual imbalance price (not predicted) on +the imbalance market at time step t, +• c′ +t is the predicted power consumption at time step t +after demand response to the dynamic prices, based +on the demand response mathematical model M. +The first term corresponds to the payment of the cus- +tomers for their electricity consumption. The second term +is the revenue or cost induced by the predicted mismatch +between supply and demand, which is traded on the day- +ahead market. The last term is the cost or revenue caused +by the remaining imbalance between supply and demand, +which has to be compensated in the imbalance market. +The total costs incurred by the producer/retailer at +each time step t can be decomposed into both fixed costs +FC and marginal costs MC. In this particular case, the +marginal costs of production are assumed to be negligible +since the production portfolio is composed of intermittent +renewable energy sources such as wind turbines and pho- +tovoltaic panels. Therefore, the second constraint can be +mathematically expressed as follows: +T −1 +� +t=0 +� +ct yt − (c′ +t − pF +t ) λt − (ct − pt) it +� +≥ FC T . +(8) +Finally, the complete objective criterion to be opti- +mised by a dynamic pricing policy can be mathematically +expressed as follows: +minimise +π +T −1 +� +t=0 +|pt − ct|, +subject to +RT ≥ FC T , +BT ≤ +T −1 +� +t=0 +ct et . +(9) +4. Algorithmic components discussion +This section presents a thorough discussion about the +different algorithmic modules required to efficiently design +a dynamic pricing policy from the perspective of the supply +side. Firstly, the different forecasting blocks are rigorously +analysed. Secondly, the modelling of the demand response +induced by dynamic prices is discussed. Lastly, the proper +management of uncertainty is considered. +In parallel, for the sake of clarity, Figure 3 highlights +the interconnections between the different algorithmic com- +ponents in the scope of a dynamic pricing policy from the +perspective of the supply side. +Moreover, Algorithm 1 +provides a thorough description of the complete decision- +making process for the dynamic pricing problem at hand. +The complexity of the variable time lag between decision- +making and application is highlighted. Assuming that the +decision-making occurs once a day at 12:00 AM just be- +fore the closing of the day-ahead market for all hours of +the following day, the dynamic price at time step t is de- +cided hours in advance at time step t − [12 + (t%24)] with +the symbol % representing the modulo operation. +4.1. Production forecasting +The first forecasting block to be discussed concerns the +production of intermittent renewable energy sources such +as wind turbines and photovoltaic panels. Indeed, having +access to accurate predictions about the future output of +the production portfolio is key to the performance of a +dynamic pricing policy from the perspective of the supply +side. As previously explained in Section 3.4, the forecasts +have to be available one day ahead before the closing of +the day-ahead electricity market for all hours of the fol- +lowing day. Naturally, the generation of such predictions +introduces uncertainty, a complexity that has to be taken +into account to design sound dynamic pricing policies. +6 + +𝑝� +� +𝑐� +� +𝜆� +� +Time +Time +Time +Power +Power +Price +𝑐� +� +Time +Power +𝑐� +� +Time +Power +𝑦� +Time +Price +Figure 3: Illustration of the complete decision-making process related to dynamic pricing from the perspective of the supply side, with the +connections between the different algorithmic components highlighted. +Algorithm 1 Dynamic pricing complete decision-making process +The decision-making occurs once per day before the closing of the day-ahead market at 12:00 AM for all hours of the next day. +The decision-making for the dynamic price of time step t occurs at time step t − [12 + (t%24)]. +for τ = −12 to T − 12 do +Check whether the time is 12:00 AM to proceed to the decision-making. +if (τ + 12)%24 = 0 then +for t = τ + 12 to τ + 36 do +Gather the available information for production forecasting xP +t = {W F +t , AF +t , IP +t }. +Gather the available information for consumption forecasting xC +t = {W F +t , Tt, IC +t }. +Gather the available information for day-ahead market price forecasting xM +t += {xP +t , xC +t , GF +t , Mt, IM +t }. +Forecast production at time step t: pF +t = FP +� +xP +t +� +. +Forecast consumption at time step t: cF +t = FC +� +xC +t +� +. +Forecast the day-ahead market price at time step t: λF +t = FM +� +xM +t +� +. +end for +for t = τ + 12 to τ + 36 do +Gather the input information for the dynamic pricing policy xt = {P F +t , CF +t , ΛF +t , Yt, M}. +Make a dynamic pricing decision for time step t: yt = π(xt). +end for +Announce the dynamic prices for all hours of the following day {yt | t = τ + 12, ..., τ + 35}. +end if +end for +7 + +Dynamic pricing policy TX +MProduction forecasting FjConsumption forecasting F'Market price forecasting FDemand +resnonse + model MFormally, the forecasting model associated with the +output of the production portfolio is denoted FP . Its input +space XP comprises every piece of information that may +potentially have an impact on the generation of electricity +from intermittent renewable energy sources such as wind +turbines and photovoltaic panels for a certain time period. +Its output space YP is composed of a forecast regarding the +power generation from the production portfolio for that +same time period. Mathematically, the forecasting model +input xP +t ∈ XP and output yP +t ∈ YP at time step t can be +expressed as follows: +xP +t = {W F +t , AF +t , IP +t }, +(10) +yP +t = pF +t , +(11) +where: +• W F +t +represents various weather forecasts related to +the power production of intermittent renewable en- +ergy sources such as wind turbines and photovoltaic +panels (wind speed/direction, solar irradiance, etc.) +at the time step t, +• AF +t represents predictions about the available capac- +ity of the production portfolio at time step t, which +may be impacted by scheduled maintenance, repairs, +or other similar constraints, +• IP +t +represents any additional information that may +help to accurately forecast the future power gener- +ation of the producer/retailer’s production portfolio +at time step t. +In the scientific literature, the current state of the art +for forecasting the power production of intermittent renew- +able energy sources is mainly based on deep learning tech- +niques together with some data cleansing processes and +data augmentation approaches. +The best architectures +are recurrent neural networks (RNN), convolutional neural +networks (CNN) and transformers [20, 21, 22, 23, 24]. +4.2. Consumption forecasting +The objective of the next important forecasting model +deserving a discussion is to accurately predict the future +power demand of the consumption portfolio before any +demand response phenomenon is induced. Since the main +goal of a dynamic pricing policy is to maximise the syn- +chronisation between supply and demand, electricity load +forecasts are of equal importance to electricity generation +predictions. Similarly to the latter, the portfolio consump- +tion forecasts are assumed to be generated one day ahead +just before the closing of the day-ahead market for all 24 +hours of the following day. Additionally, the uncertainty +associated with these predictions has to be seriously taken +into account for the success of the dynamic pricing policy. +From a more formal perspective, the forecasting model +responsible for predicting the future electricity load of the +consumption portfolio is denoted FC. Its input space XC +includes all the information that may have an influence on +the residential electricity consumption for a certain time +period. Its output space YC comprises a forecast of the +power used by the consumption portfolio for that same +time period. Mathematically, the consumption forecasting +model input xC +t ∈ XC and output yC +t ∈ YC at time step t +can be expressed as the following: +xC +t = {W F +t , Tt, IC +t }, +(12) +yC +t = cF +t , +(13) +where: +• W F +t +represents various weather forecasts related to +the residential electricity consumption (temperature, +hygrometry, etc.) at the time step t, +• Tt represents diverse characteristics related to the +time step t (hour, weekend, holiday, season, etc.), +• IC +t represents supplementary information that could +potentially have an influence on the residential power +consumption at time step t. +Similarly to renewable energy production forecasting, +the state-of-the-art approaches for predicting the residen- +tial electricity load in the short term are mostly related +to deep learning techniques with preprocessed augmented +data: RNN, CNN, and transformers [25, 26, 27, 28, 22]. +4.3. Market price forecasting +The last forecasting block to be discussed concerns +the future day-ahead electricity market prices. Contrarily +to the forecasting of power production and consumption, +these price predictions are not critical to the success of a +dynamic pricing policy from the perspective of the supply +side. Still, having access to quality forecasts for the fu- +ture day-ahead market prices remains important in order +to satisfy the constraints related to the profitability of the +producer/retailer as well as the reduced electricity costs for +the consumer. Once again, the predictions are assumed to +be made just before the closing of the day-ahead market. +Moreover, the uncertainty associated with these forecasts +has to be taken into consideration. +Formally, the forecasting model related to the future +day-ahead electricity market prices is denoted FM. +Its +input space XM includes every single piece of information +which may potentially explain the future electricity price +on the day-ahead market for a certain hour. Its output +space YM comprises a forecast of the day-ahead market +price for that same hour. Mathematically, both forecasting +8 + +model input xM +t +∈ XM and output yM +t +∈ YM at time step +t can be expressed as follows: +xM +t += {xP +t , xC +t , GF +t , Mt, IM +t }, +(14) +yM +t += λF +t , +(15) +where: +• GF +t represents forecasts about the state of the power +grid as a whole (available production capacity, trans- +mission lines, etc.) at the time step t, +• Mt represents diverse information in various markets +related to energy (power, carbon, oil, gas, coal, etc.) +in neighbouring geographical areas at time step t, +• IM +t +represents any extra piece of information that +may help to predict the future electricity price on +the day-ahead market at time step t. +Once again, the scientific literature reveals that the +state-of-the-art approaches for day-ahead power market +price forecasting are mostly based on innovative machine +learning techniques [29, 30, 31, 32, 33]. +4.4. Demand response modelling +Another essential algorithmic component is the math- +ematical modelling of the residential demand response to +dynamic prices. In order to make relevant dynamic pric- +ing decisions, an estimation of the impact of the electricity +price on the consumer’s behaviour is necessary. In fact, +two important characteristics have to be studied: +The residential power consumption elasticity. This +quantity measures the average percentage change of the +residential power consumption in response to a percentage +change in the electricity price. In other words, the elastic- +ity captures the willingness of the consumer to adapt its +behaviour when the price of electricity either increases or +decreases. This elasticity is critical to the dynamic pricing +approach, since it assesses the receptiveness of the con- +sumers to dynamic prices. In fact, the residential power +consumption elasticity can be considered as a quantitative +indicator of the potential of the dynamic pricing approach. +The electricity load temporal dependence. +Time +plays an important role in power consumption. +Firstly, +the consumer’s behaviour is highly dependent on the time +of the day. The tendency to adapt this behaviour is also +expected to be time-dependent. Therefore, the residential +power consumption elasticity has to be a function of the +time within a day, among other things. Secondly, a higher +electricity price does not simply reduce the demand as with +other commodities, but rather shifts part of the consump- +tion earlier and/or later in time. This phenomenon reflects +a complex temporal dependence for power consumption, +which has to be accurately modelled in order to design a +performing dynamic pricing policy. +Formally, the mathematical model of the residential +demand response is denoted M. +Its input space XD is +composed of the predicted power consumption before any +demand response and the dynamic prices to be sent to the +consumers for several hours before and after the time pe- +riod analysed, together with information about that time +period. Its output space YD comprises the predicted power +consumption after demand response to dynamic prices for +that same time period. Mathematically, both demand re- +sponse model input xD +t ∈ XD and output yD +t ∈ YD at time +step t can be expressed as the following: +xD +t = {CF +t , Y ′ +t , Tt}, +(16) +yD +t = c′ +t, +(17) +where Y ′ +t = {yt+ϵ ∈ R | ϵ = −k, ..., k} is the dynamic price +signal within a time window centred around time step t +and of size k from which the demand response is induced. +As far as the scientific literature about the modelling of +demand response to dynamic prices is concerned, this in- +teresting topic has not yet received a lot of attention from +the research community. Still, there exists a few sound +works presenting demand response models and assessing +the receptiveness of the consumers to dynamic power prices +[15, 16, 17, 18, 19], as explained in Section 2. +4.5. Uncertainty discussion +As previously hinted, a dynamic pricing policy has to +make its decisions based on imperfect information. Indeed, +multiple forecasts for the electricity price, production and +consumption have to be generated 12 up to 35 hours in +advance. Naturally, these predictions comes with a level +of uncertainty that should not be neglected. +Moreover, +accurately modelling the residential demand response to +dynamic prices is a particularly challenging task. Because +of both the random human nature and the difficulty to +fully capture the consumers’ behaviour within a mathe- +matical model, a notable level of uncertainty should also +be considered at this stage. Therefore, multiple sources of +uncertainty can be identified in the scope of the dynamic +pricing decision-making problem at hand, and a proper +management of this uncertainty is necessary. +A stochastic reasoning is recommended to make sound +dynamic pricing decisions despite this substantial level of +uncertainty. Instead of considering each uncertain variable +(production, consumption, price, demand response) with +a probability of 1, the full probability distribution behind +these quantities has to be estimated and exploited. Based +on this information, the risk associated with uncertainty +may be mitigated. Moreover, safety margins may also con- +tribute to reduce this risk, but potentially at the expense +of a lowered performance. In fact, there generally exists +9 + +a trade-off between performance and risk, in line with the +adage: with great risk comes great reward. +5. Performance assessment methodology +This section presents a methodology for quantitatively +assessing the performance of a dynamic pricing policy in +a comprehensive manner. +As explained in Section 3.7, +several disjoint objectives can be clearly identified. +For +the sake of completeness, this research work presents three +quantitative indicators, one for each objective. The rela- +tive importance of these indicators is left to the discretion +of the reader according to its main intention among the +different objectives previously defined. +The performance indicators proposed are based on the +comparison with the original situation without dynamic +pricing. In this case, the consumer is assumed to be fully +ignorant about the mismatch problem between supply and +demand. No information is provided to the customers of +the producer/retailer, which consequently have an unin- +fluenced consumption behaviour. The price of electricity +et is freely determined by the producer/retailer. It may +for instance be a fixed tariff, or a price indexed on the +day-ahead market price: +et = α λt + β , +(18) +where α and β are parameters to be set by the retailer. +Firstly, the impact of a dynamic pricing policy on the +synchronisation between power supply and demand can be +assessed through the performance indicator S quantifying +the relative evolution of the deviation ∆T . This quantity +is mathematically expressed as follows: +S = 100 ∆T − ∆T +∆T +, +(19) +∆T = +T −1 +� +t=0 +|pt − ct| , +(20) +where ∆T represents the lack of synchronisation between +supply and demand without dynamic pricing. Therefore, +the quantity S has ideally to be maximised, with a perfect +synchronisation between supply and demand leading to a +value of 100% reduction in deviation. +Secondly, the consequence for the consumer regarding +its electricity bill can be evaluated with the quantity B +which informs about the relative evolution of this power +bill. It can be mathematically computed as the following: +B = 100 BT − BT +BT +, +(21) +where BT = �T −1 +t=0 ct et represents the electricity bill paid +by the consumer without dynamic pricing. Since the per- +formance indicator B represents the percentage reduction +in costs, it has to ideally be maximised. +Lastly, the enhancement in terms of revenue for the +producer/retailer can be efficiently quantified thanks to +the performance indicator R. This quantity represents the +relative evolution of the producer/retailer revenue and can +be mathematically expressed as follows: +R = 100 RT − RT +RT +, +(22) +RT = +T −1 +� +t=0 +� +ct et − (cF +t − pF +t ) λt − (ct − pt) it +� +, +(23) +where RT represents the producer/retailer revenue with- +out dynamic pricing. Obviously, the performance indicator +R has to ideally be maximised. +6. Conclusion +This research paper presents a detailed formalisation of +the decision-making problem faced by a producer/retailer +willing to adopt a dynamic pricing approach, in order to +induce an appropriate residential demand response. Three +core challenges are highlighted by this formalisation work. +Firstly, the objective criterion maximised by a dynamic +pricing policy is not trivially defined, since different goals +that are not compatible can be clearly identified. Secondly, +several complex algorithmic components are necessary for +the development of a performing dynamic pricing policy. +One can for instance mention different forecasting blocks, +but also a mathematical model of the residential demand +response to dynamic prices. Thirdly, the dynamic pricing +decisions have to be made based on imperfect information, +because this particular decision-making problem is highly +conditioned by the actual uncertainty for the future. +Several avenues are proposed for future work. In fact, +the natural extension of the present research is to design +innovative dynamic pricing policies from the perspective +of the supply side based on the formalisation performed. +While the present research paper exclusively focuses on +the philosophy and conceptual analysis of the approach, +there remain practical concerns that need to be properly +addressed in order to achieve performing decision-making +policies. To achieve that, a deeper analysis of the scientific +literature about each algorithmic component discussed in +Section 4 is firstly welcomed, in order to identify and re- +produce the state-of-the-art techniques within the context +of interest. Then, different approaches have to be investi- +gated for the design of the dynamic pricing policy itself. +One can for instance mention, among others, the stochastic +optimisation and deep reinforcement learning techniques. +Finally, the dynamic pricing policies developed have to be +rigorously evaluated, analysed, and compared by taking +advantage of real-life experiments. +10 + +Acknowledgements +Thibaut Théate is a Research Fellow of the F.R.S.- +FNRS, of which he acknowledges the financial support. +References +[1] IPCC, Climate Change 2021: The Physical Science Basis. Con- +tribution of Working Group I to the Sixth Assessment Report +of the Intergovernmental Panel on Climate Change, Cambridge +University Press, Cambridge, United Kingdom and New York, +NY, USA, 2021. +[2] H. +Ritchie, +M. +Roser, +Energy, +Our +World +in +Data, +https://ourworldindata.org/energy (2020). +[3] S. Chatzivasileiadis, D. Ernst, G. 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Castilla, A novel electricity price forecasting +approach based on dimension reduction strategy and rough ar- +tificial neural networks, IEEE Transactions on Industrial Infor- +matics 16 (2020) 2369–2381. +11 + diff --git a/7NFJT4oBgHgl3EQfmSym/content/tmp_files/load_file.txt b/7NFJT4oBgHgl3EQfmSym/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b2efae342bb1a1d060b8d8e9d29aaf0d1aa072a --- /dev/null +++ b/7NFJT4oBgHgl3EQfmSym/content/tmp_files/load_file.txt @@ -0,0 +1,532 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf,len=531 +page_content='Matching of Everyday Power Supply and Demand with Dynamic Pricing: Problem Formalisation and Conceptual Analysis Thibaut Théatea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Antonio Suterab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Damien Ernsta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='c aDepartment of Electrical Engineering and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' University of Liège,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Liège,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Belgium bHaulogy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Intelligent Systems Solutions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Braine-le-Comte,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Belgium cInformation Processing and Communications Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Institut Polytechnique de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' France Abstract The energy transition is expected to significantly increase the share of renewable energy sources whose production is intermittent in the electricity mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Apart from key benefits, this development has the major drawback of generating a mismatch between power supply and demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The innovative dynamic pricing approach may significantly contribute to mitigating that critical problem by taking advantage of the flexibility offered by the demand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' At its core, this idea consists in providing the consumer with a price signal which is evolving over time, in order to influence its consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This novel approach involves a challenging decision-making problem that can be summarised as follows: how to determine a price signal maximising the synchronisation between power supply and demand under the constraints of maintaining the producer/retailer’s profitability and benefiting the final consumer at the same time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' As a contribution, this research work presents a detailed formalisation of this particular decision-making problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Moreover, the paper discusses the diverse algorithmic components required to efficiently design a dynamic pricing policy: different forecasting models together with an accurate statistical modelling of the demand response to dynamic prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Keywords: Matching of supply and demand, dynamic pricing, demand response, power producer/retailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Introduction Climate change is undeniably a major challenge facing humanity in the 21st century [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' An ambitious trans- formation is required in all sectors to significantly lower their respective carbon footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Electricity generation is no exception, with the burning of fossil fuels, mainly coal and gas, being by far the dominant power source in the world today [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This sector has to undergo an important transformation of the global electricity mix by promoting power sources with a significantly lower carbon footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Belonging to that category are nuclear power, hydroelec- tricity, biomass or geothermal energy which are relatively controllable, but also the energy directly extracted from wind and sun which is conversely intermittent in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Since wind turbines and photovoltaic panels are expected to play a key role in the energy transition, solutions are required to address their variable production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Interesting technical avenues are the interconnection of power grids [3] and the development of storage capacities such as battery, pumped hydroelectricity or hydrogen [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Another promis- ing and innovative solution is to influence the behaviour of consumers through the use of dynamic pricing (DP), so that power supply and demand are better synchronised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Email addresses: thibaut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='theate@uliege.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='be (Thibaut Théate), dernst@uliege.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='be (Damien Ernst) The dynamic pricing approach consists in continuously adapting the electricity price that the final consumer has to pay in order to influence its consumption behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Basically, when demand exceeds supply, the power price would be increased in order to take down consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Conversely, a reduced price would be provided when there is excessive production compared to consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' From a graphical perspective, the objective is not only to shift the daily consumption curve but also to change its shape in order to better overlap with the intermittent production curve of renewable energy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This is illustrated in Figure 1 for a representative situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The innovative dynamic pricing approach relies on two important assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Firstly, the final consumer has to be equipped with a smart metering device to measure its production in real-time and with communication means for the price signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Secondly, the final consumer has to be able to provide a certain amount of flexibility regarding its power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Moreover, it has to be sufficiently receptive to the incentives offered to reduce its electricity bill in exchange for a behaviour change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' If these require- ments are met, the major strength of the dynamic pricing approach is its potential benefits for both the consumer and the producer/retailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Moreover, these benefits would not only be in terms of economy, but also potentially in terms of ecology and autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In fact, dynamic prices reward the flexibility of the demand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='11587v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='TR] 27 Jan 2023 12:00 06:00 18:00 12:00 06:00 18:00 Supply Demand Time Time Power Power Figure 1: Illustration of the dynamic pricing approach’s potential to shift and change the shape of a typical daily consumption curve (blue) so that there is a better synchronisation with the daily inter- mittent production curve of renewable energy sources (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The contributions of this research work are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Firstly, the complex decision-making problem faced by a producer/retailer willing to develop a dynamic pricing strategy is presented and rigorously formalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Secondly, the diverse algorithmic components required to efficiently design a dynamic pricing policy are thoroughly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' To the authors’ knowledge, demand response via dynamic pricing has received considerable attention from the re- search community, but from the perspective of the demand side alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Therefore, the present research may be consid- ered as a pioneer work studying dynamic pricing from the perspective of the supply side for taking advantage of the flexibility of the power consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This research paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' First of all, the scientific literature about both dynamic pricing and demand response is concisely reviewed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Then, Section 3 presents a detailed formalisation of the decision-making problem behind the novel dynamic pric- ing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Afterwards, Section 4 analyses the algorith- mic components necessary for the development of dynamic pricing policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Subsequently, a fair performance assess- ment methodology is introduced in Section 5 to quanti- tatively evaluate the performance of a dynamic pricing policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' To end this paper, Section 6 discusses interesting research avenues for future work and draws conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Literature review Over the last decade, the management of the demand side in the scope of the energy transition has received in- creasing attention from the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In fact, there exist multiple generic approaches when it comes to demand response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Without getting into too many details, the scientific literature includes some surveys summarising and discussing the different techniques available together with their associated challenges and benefits [5, 6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In this research work, the focus is exclusively set on the demand response induced by dynamic power prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' As previously mentioned, the scientific literature about demand response via dynamic pricing is primarily focused on the perspective of the demand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Multiple techniques have already been proposed to help the consumer provide flexibility and take advantage of behavioural changes to lower its electricity bill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' For instance, [10] presents a power scheduling method based on a genetic algorithm to opti- mise residential demand response via an energy manage- ment system, so that the electricity cost is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In [11], a technique based on dynamic programming is introduced for determining the optimal schedule of residential con- trollable appliances in the context of time-varying power pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' One can also mention [12] that proposes an energy sharing model with price-based demand response for mi- crogrids of peer-to-peer prosumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The approach is based on a distributed iterative algorithm and has been shown to lower the prosumers’ costs and improve the sharing of photovoltaic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' More recently, (deep) reinforcement learning techniques have been proven to be particularly relevant for controlling the residential demand response in the context of dynamic power prices [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' On the contrary, the question of inducing a residential demand response based on a dynamic pricing approach from the perspective of the supply side has not received a lot of attention from the research community yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Still, there are a few works in the scientific literature about the mathematical modelling of the demand response caused by dynamic power prices, which is a key element in achieving that objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' To begin with, [15] presents a simulation model highlighting the evolution of electricity consump- tion profiles when shifting from a fixed tariff to dynamic power prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The same objective is pursued by [16] which introduces a fully data-driven approach relying on the data collected by smart meters and exogenous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The resulting simulation model is based on consumption pro- files clustering and conditional variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Alternatively, [17] presents a functional model of residen- tial power consumption elasticity under dynamic pricing to assess the impact of different electricity price levels, based on a Bayesian probabilistic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In addition to these mathematical models, one can also mention some real-life experiments conducted to assess the responsiveness of res- idential electricity demand to dynamic pricing [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Problem formalisation This section presents a mathematical formalisation of the challenging sequential decision-making problem related to the dynamic pricing approach for inducing a residential demand response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' To begin with, the contextualisation considered for studying this particular problem is briefly described, followed by an overview of the decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Then, a discretisation of the continuous timeline is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Subsequently, the formal definition of a dy- namic pricing policy is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Lastly, the input and output spaces of a dynamic pricing policy are described, together with the objective criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Contextualisation As previously hinted, this research work focuses on the interesting real-case scenario of a producer/retailer whose production portfolio is composed of an important share of renewable energy sources such as wind turbines and photo- voltaic panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Because of the substantial intermittency of these generation assets, a strong connection to the energy markets is required in order to fully satisfy its customers regardless of the weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Nevertheless, the consumers are assumed to be well informed and willing to adapt their be- haviour in order to consume renewable energy rather than electricity purchased on the market whose origin may be unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Within this particular context, the benefits of the dynamic pricing approach taking advantage of the con- sumers’ flexibility are maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Indeed, the insignificant marginal cost associated with these intermittent renew- able energy sources coupled with their low carbon foot- print make this innovative approach interesting from an economical perspective for both supply and demand sides, but also in terms of ecology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Moreover, the autonomy of the producer/retailer is expected to be reinforced by low- ering its dependence on the energy markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' At the same time, dependence on fossil fuels may be reduced as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In this research work, the predicted difference between power production and consumption is assumed to be fully secured in the day-ahead electricity market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Also called spot market, the day-ahead market has an hourly resolu- tion and is operated once a day for all hours of the follow- ing day via a single-blind auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In other words, trading power for hour H of day D has to be performed ahead on day D − 1 between 00:00 AM (market opening) and 12:00 AM (market closure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Therefore, the energy is at best purchased 12 hours (00:00 AM of day D) up to 35 hours (11:00 PM of day D) before the actual delivery of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Apart from the day-ahead electricity market, it is assumed that there are no trading activities on the future/forward nor intraday markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Nevertheless, if there remains an eventual mismatch between production and consumption at the time of power delivery, the producer/retailer would be exposed to the imbalance market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In this case, the so- called imbalance price has to be inevitably paid as com- pensation for pushing the power grid off balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Decision-making process overview The decision-making problem studied in this research work is characterised by a particularity: a variable time lag between the moment a decision is made and the moment it becomes effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' As previously explained, any remaining difference between production and consumption after de- mand response has to ideally be traded on the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The purpose of this assumption is to limit the ex- posure of the producer/retailer to the imbalance market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' For this reason, the price signal sent to the consumer on day D has to be generated before the closing of the day- ahead market on day D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Additionally, it is assumed that the price signal cannot be refreshed afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Basically, the decision-making problem at hand can be formalised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The core objective is to determine a decision-making policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' denoted Π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' mapping at time τ input information of diverse nature Iτ to the electricity price signal Sτ to be sent to the consumers over a future time horizon well-defined: Sτ = Π(Iτ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' (1) where: Iτ represents the information vector gathering all the available information (of diverse nature) at time τ which may be helpful to make a relevant dynamic pricing decision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Sτ represents a set of electricity prices generated at time τ and shaping the dynamic price signal over a well-defined future time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The dynamic pricing approach from the perspective of the supply side belongs to a particular class of decision- making problems: automated planning and scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Contrarily to conventional decision-making outputting one action at a time, planning decision-making is concerned with the generation of a sequence of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In other words, a planning decision-making problem requires to synthesise in advance a strategy or plan of actions over a certain time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Formally, the decision-making has to be performed at a specific time τ about a control vari- able over a future time horizon beginning at time τi > τ and ending at time τf > τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In this case, the decision- making is assumed to be performed just before the closing of the day-ahead market at 12:00 AM to determine the price signal to be sent to the consumers throughout the entire following day (from 00:00 AM to 11:59 PM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In the next sections, a more accurate and thorough mathematical formalisation of the dynamic pricing prob- lem from the perspective of the supply side is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Moreover, the planning problem previously introduced is cast into a sequential decision-making problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Indeed, this research paper intends to focus on a decision-making policy outputting a single price from the signal Sτ at a time based on a subset of the information vector Iτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 3 00:00 12:00 𝑦� 𝑦� 𝑦�� Closing of the day-ahead market 𝑥� 𝑥� 𝑥� 𝑥�� Dynamic pricing policy π 𝑦� Forecasts 𝑝� � 𝑐� � 𝜆� � 𝑐� � Time Power Time Time Time Power Power Price Demand reponse model Figure 2: Illustration of the formalised decision-making problem related to dynamic pricing from the perspective of the supply side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The notations xt and yt represent the inputs and outputs of a dynamic pricing policy π, which are not shown concurrent on the timeline since the decision-making occurs multiple hours before the application of the dynamic pricing signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The time axis of the four plots represents the complete following day for which the dynamic prices are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The mathematical notations pF t , cF t and λF t respectively represent the forecast production, consumption and day-ahead market price for the time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Finally, the quantity c′ t is the predicted consumption at time step t after taking into consideration the dynamic pricing signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Timeline discretisation Theoretically, the dynamic electricity price signal sent to the consumer could be continuously changing over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' More realistically, this research work adopts a discretisa- tion of the continuous timeline so that this power price is adapted at regular intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Formally, this timeline is discretised into a number of time steps t spaced by a constant duration ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' If the duration ∆t is too large, the synchronisation improvement between supply and demand will probably be of poor quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Conversely, lowering the value of the duration ∆t increases the complexity of the decision-making process, and a too high update frequency may even confuse the consumer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' There is a trade-off to be found concerning this important parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In this re- search work, the dynamic price signal is assumed to change once per hour, meaning that ∆t is equal to one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This choice is motivated by the hourly resolution of the day- ahead market, which has proven to be an appropriate com- promise over the years for matching power production and consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Another relevant discretisation choice could be to have a price signal which is updated every quarter of an hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In the rest of this research paper, the increment (decrement) operations t + 1 (t − 1) are used to model the discrete transition from time step t to time step t + ∆t (t − ∆t), for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Dynamic pricing policy Within the context previously described, a dynamic pricing planning policy Π consists of the set of rules used to make a decision regarding the future price signal sent to the consumers over the next day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This planning policy can be decomposed into a set of 24 dynamic pricing decision- making policies π outputting a single electricity price for one hour of the following day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Mathematically, such a dynamic pricing strategy can be defined as a programmed policy π : X → Y, either deterministic or stochastic, which outputs a decision yt ∈ Y for time step t based on some input information xt ∈ X so as to maximise an objective criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The input xt is derived from the information vec- tor Iτ associated with the decision-making for time step t, after potential preprocessing operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The price signal Sτ is composed of 24 dynamic pricing policy outputs yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In the rest of this research work, the time at which the decision-making does occur should not be confused with the time at which the dynamic price signal is active (charging for energy consumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The proposed formal- isation assumes that the time step t refers to the time at which the dynamic price is active, not decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Therefore, the decision-making of the dynamic pricing policy for time step t (yt = π(xt)) is in fact performed hours in advance of time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This complexity is illustrated in Figure 2 describing the formalised decision-making problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 4 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Input of a dynamic pricing policy The input space X of a dynamic pricing policy π com- prises all the available information which may help to make a relevant decision about future electricity prices so that an appropriate demand response is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Since the decision-making occurs 12 up to 35 hours in advance of the price signal delivery, this information mainly consists of forecasts and estimations that are subject to uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' As depicted in Figure 2, the dynamic pricing policy input xt ∈ X refers to the decision-making occurring at time τ = t − h with h ∈ [12, 35] about the dynamic pricing signal delivered to the consumer at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In fact, the quantity Iτ may be seen as the information contained in the 24 inputs xt for t ∈ {τ + 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=', τ + 35}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Formally, the input xt ∈ X is decided to be defined as follows: xt = {P F t , CF t , ΛF t , Yt, M}, (2) where: P F t = {pF t+ϵ ∈ R+ | ϵ = −k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=', k} represents a set of forecasts for the power production within a time window centred around time step t and of size k, CF t = {cF t+ϵ ∈ R+ | ϵ = −k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=', k} represents a set of forecasts for the power consumption within a time window centred around time step t and of size k, ΛF t = {λF t+ϵ ∈ R | ϵ = −k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=', k} represents a set of forecasts for the day-ahead market prices within a window centred around time step t and of size k, Yt = {yt−ϵ ∈ R | ϵ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=', k} represents the series of k previous values for the dynamic price signal sent to the final consumer, M is a mathematical model of the demand response to be expected from the consumption portfolio, with the required input information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The different forecasting models and the challenging modelling of the consumption portfolio demand response are discussed in more details in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Output of a dynamic pricing policy The output space Y of a dynamic pricing policy π only includes the future price signal to be sent to the consumer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Formally, the dynamic pricing policy output yt ∈ Y, which represents the electricity price to be paid by the consumer for its power consumption at time step t, is mathematically defined as follows: yt = et, (3) where et ∈ R represents the dynamic electricity price to be paid by the demand side for its power consumption at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Out of the scope of this research work is the presentation of this price signal so that the impact on the final consumer is maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Indeed, the way of communicating the output of the dynamic pricing policy has to be adapted to the audience, be it humans with different levels of electricity market expertise or algorithms (energy management systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Objective criterion The dynamic pricing approach can provide multiple benefits, in terms of economy, ecology but also autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Consequently, the objective criterion to be maximised by a dynamic pricing policy π is not trivially determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In fact, several core objectives can be clearly identified: maximising the match between supply and demand, minimising the carbon footprint of power generation, minimising the electricity costs for the consumer, maximising the revenue of the producer/retailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Although some objectives overlap, these four criteria are not completely compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' For instance, maximising the synchronisation between power supply and demand is equivalent to minimising the carbon footprint associated with the generation of electricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Indeed, the production portfolio of the producer/retailer being mainly composed of intermittent renewable energy sources, its energy has a reduced carbon footprint compared to the electricity that can be purchased on the day-ahead market whose origin is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' On the contrary, maximising the revenue of the producer/retailer will obviously not lead to a minimised electricity bill for the consumer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This research work makes the choice to prioritise the maximisation of the synchroni- sation between supply and demand, and equivalently the minimisation of the carbon footprint, while translating the other two core objectives into relevant constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Firstly, the costs for the consumer have to be reduced with respect to the situation without dynamic pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Secondly, the profitability of the producer/retailer has to be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Formally, the objective criterion to be optimised by a dynamic pricing policy π can be mathematically defined as the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' First of all, the main target to evaluate is the synchronisation between supply and demand, which can be quantitatively assessed through the deviation ∆T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This quantity has to ideally be minimised, and can be mathematically expressed as follows: ∆T = T −1 � t=0 |pt − ct|, (4) where: t = 0 corresponds to the first electricity delivery hour of a new day (00:00 AM), T is the time horizon considered, which should be a multiple of 24 to have full days, pt is the actual power production (not predicted) from the supply side at time step t, ct is the actual power consumption (not predicted) from the demand side at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 5 Afterwards, the first constraint concerning the reduced costs for the consumer has to be modelled mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This is achieved via the electricity bill BT paid by the consumer over the time horizon T, which can be expressed as the following: BT = T −1 � t=0 ct yt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' (5) As previously explained, the consumer power bill BT should not exceed that obtained without dynamic pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In that case, the consumer is assumed to pay a price et, which can for instance be a fixed tariff or a price indexed on the day-ahead market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The situation without dynamic pricing is discussed in more details in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Consequently, the first constraint can be mathematically expressed as follows: T −1 � t=0 ct yt ≤ T −1 � t=0 ct et , (6) where ct is the power consumption from the demand side at time step t without dynamic pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Then, the second constraint is about the profitability of the producer/retailer, which is achieved if its revenue exceeds its costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The revenue RT of the producer/retailer over the time horizon T can be mathematically expressed as the following: RT = T −1 � t=0 � ct yt − (c′ t − pF t ) λt − (ct − pt) it � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' (7) where: λt is the actual power price (not predicted) on the day-ahead market at time step t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' it is the actual imbalance price (not predicted) on the imbalance market at time step t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' c′ t is the predicted power consumption at time step t after demand response to the dynamic prices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' based on the demand response mathematical model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The first term corresponds to the payment of the cus- tomers for their electricity consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The second term is the revenue or cost induced by the predicted mismatch between supply and demand, which is traded on the day- ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The last term is the cost or revenue caused by the remaining imbalance between supply and demand, which has to be compensated in the imbalance market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The total costs incurred by the producer/retailer at each time step t can be decomposed into both fixed costs FC and marginal costs MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In this particular case, the marginal costs of production are assumed to be negligible since the production portfolio is composed of intermittent renewable energy sources such as wind turbines and pho- tovoltaic panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Therefore, the second constraint can be mathematically expressed as follows: T −1 � t=0 � ct yt − (c′ t − pF t ) λt − (ct − pt) it � ≥ FC T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' (8) Finally, the complete objective criterion to be opti- mised by a dynamic pricing policy can be mathematically expressed as follows: minimise π T −1 � t=0 |pt − ct|, subject to RT ≥ FC T , BT ≤ T −1 � t=0 ct et .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' (9) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Algorithmic components discussion This section presents a thorough discussion about the different algorithmic modules required to efficiently design a dynamic pricing policy from the perspective of the supply side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Firstly, the different forecasting blocks are rigorously analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Secondly, the modelling of the demand response induced by dynamic prices is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Lastly, the proper management of uncertainty is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In parallel, for the sake of clarity, Figure 3 highlights the interconnections between the different algorithmic com- ponents in the scope of a dynamic pricing policy from the perspective of the supply side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Moreover, Algorithm 1 provides a thorough description of the complete decision- making process for the dynamic pricing problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The complexity of the variable time lag between decision- making and application is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Assuming that the decision-making occurs once a day at 12:00 AM just be- fore the closing of the day-ahead market for all hours of the following day, the dynamic price at time step t is de- cided hours in advance at time step t − [12 + (t%24)] with the symbol % representing the modulo operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Production forecasting The first forecasting block to be discussed concerns the production of intermittent renewable energy sources such as wind turbines and photovoltaic panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Indeed, having access to accurate predictions about the future output of the production portfolio is key to the performance of a dynamic pricing policy from the perspective of the supply side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' As previously explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='4, the forecasts have to be available one day ahead before the closing of the day-ahead electricity market for all hours of the fol- lowing day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Naturally, the generation of such predictions introduces uncertainty, a complexity that has to be taken into account to design sound dynamic pricing policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 6 𝑝� � 𝑐� � 𝜆� � Time Time Time Power Power Price 𝑐� � Time Power 𝑐� � Time Power 𝑦� Time Price Figure 3: Illustration of the complete decision-making process related to dynamic pricing from the perspective of the supply side, with the connections between the different algorithmic components highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Algorithm 1 Dynamic pricing complete decision-making process The decision-making occurs once per day before the closing of the day-ahead market at 12:00 AM for all hours of the next day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The decision-making for the dynamic price of time step t occurs at time step t − [12 + (t%24)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' for τ = −12 to T − 12 do Check whether the time is 12:00 AM to proceed to the decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' if (τ + 12)%24 = 0 then for t = τ + 12 to τ + 36 do Gather the available information for production forecasting xP t = {W F t , AF t , IP t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Gather the available information for consumption forecasting xC t = {W F t , Tt, IC t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Gather the available information for day-ahead market price forecasting xM t = {xP t , xC t , GF t , Mt, IM t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Forecast production at time step t: pF t = FP � xP t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Forecast consumption at time step t: cF t = FC � xC t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Forecast the day-ahead market price at time step t: λF t = FM � xM t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' end for for t = τ + 12 to τ + 36 do Gather the input information for the dynamic pricing policy xt = {P F t , CF t , ΛF t , Yt, M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Make a dynamic pricing decision for time step t: yt = π(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' end for Announce the dynamic prices for all hours of the following day {yt | t = τ + 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=', τ + 35}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=" end if end for 7 Dynamic pricing policy TX MProduction forecasting FjConsumption forecasting F'Market price forecasting FDemand resnonse model MFormally, the forecasting model associated with the output of the production portfolio is denoted FP ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Its input space XP comprises every piece of information that may potentially have an impact on the generation of electricity from intermittent renewable energy sources such as wind turbines and photovoltaic panels for a certain time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Its output space YP is composed of a forecast regarding the power generation from the production portfolio for that same time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Mathematically, the forecasting model input xP t ∈ XP and output yP t ∈ YP at time step t can be expressed as follows: xP t = {W F t , AF t , IP t }, (10) yP t = pF t , (11) where: W F t represents various weather forecasts related to the power production of intermittent renewable en- ergy sources such as wind turbines and photovoltaic panels (wind speed/direction, solar irradiance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=') at the time step t, AF t represents predictions about the available capac- ity of the production portfolio at time step t, which may be impacted by scheduled maintenance, repairs, or other similar constraints, IP t represents any additional information that may help to accurately forecast the future power gener- ation of the producer/retailer’s production portfolio at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In the scientific literature, the current state of the art for forecasting the power production of intermittent renew- able energy sources is mainly based on deep learning tech- niques together with some data cleansing processes and data augmentation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The best architectures are recurrent neural networks (RNN), convolutional neural networks (CNN) and transformers [20, 21, 22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Consumption forecasting The objective of the next important forecasting model deserving a discussion is to accurately predict the future power demand of the consumption portfolio before any demand response phenomenon is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Since the main goal of a dynamic pricing policy is to maximise the syn- chronisation between supply and demand, electricity load forecasts are of equal importance to electricity generation predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Similarly to the latter, the portfolio consump- tion forecasts are assumed to be generated one day ahead just before the closing of the day-ahead market for all 24 hours of the following day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Additionally, the uncertainty associated with these predictions has to be seriously taken into account for the success of the dynamic pricing policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' From a more formal perspective, the forecasting model responsible for predicting the future electricity load of the consumption portfolio is denoted FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Its input space XC includes all the information that may have an influence on the residential electricity consumption for a certain time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Its output space YC comprises a forecast of the power used by the consumption portfolio for that same time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Mathematically, the consumption forecasting model input xC t ∈ XC and output yC t ∈ YC at time step t can be expressed as the following: xC t = {W F t , Tt, IC t }, (12) yC t = cF t , (13) where: W F t represents various weather forecasts related to the residential electricity consumption (temperature, hygrometry, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=') at the time step t, Tt represents diverse characteristics related to the time step t (hour, weekend, holiday, season, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' ), IC t represents supplementary information that could potentially have an influence on the residential power consumption at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Similarly to renewable energy production forecasting, the state-of-the-art approaches for predicting the residen- tial electricity load in the short term are mostly related to deep learning techniques with preprocessed augmented data: RNN, CNN, and transformers [25, 26, 27, 28, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Market price forecasting The last forecasting block to be discussed concerns the future day-ahead electricity market prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Contrarily to the forecasting of power production and consumption, these price predictions are not critical to the success of a dynamic pricing policy from the perspective of the supply side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Still, having access to quality forecasts for the fu- ture day-ahead market prices remains important in order to satisfy the constraints related to the profitability of the producer/retailer as well as the reduced electricity costs for the consumer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Once again, the predictions are assumed to be made just before the closing of the day-ahead market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Moreover, the uncertainty associated with these forecasts has to be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Formally, the forecasting model related to the future day-ahead electricity market prices is denoted FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Its input space XM includes every single piece of information which may potentially explain the future electricity price on the day-ahead market for a certain hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Its output space YM comprises a forecast of the day-ahead market price for that same hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Mathematically, both forecasting 8 model input xM t ∈ XM and output yM t ∈ YM at time step t can be expressed as follows: xM t = {xP t , xC t , GF t , Mt, IM t }, (14) yM t = λF t , (15) where: GF t represents forecasts about the state of the power grid as a whole (available production capacity, trans- mission lines, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=') at the time step t, Mt represents diverse information in various markets related to energy (power, carbon, oil, gas, coal, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=') in neighbouring geographical areas at time step t, IM t represents any extra piece of information that may help to predict the future electricity price on the day-ahead market at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Once again, the scientific literature reveals that the state-of-the-art approaches for day-ahead power market price forecasting are mostly based on innovative machine learning techniques [29, 30, 31, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Demand response modelling Another essential algorithmic component is the math- ematical modelling of the residential demand response to dynamic prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In order to make relevant dynamic pric- ing decisions, an estimation of the impact of the electricity price on the consumer’s behaviour is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In fact, two important characteristics have to be studied: The residential power consumption elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This quantity measures the average percentage change of the residential power consumption in response to a percentage change in the electricity price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In other words, the elastic- ity captures the willingness of the consumer to adapt its behaviour when the price of electricity either increases or decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This elasticity is critical to the dynamic pricing approach, since it assesses the receptiveness of the con- sumers to dynamic prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In fact, the residential power consumption elasticity can be considered as a quantitative indicator of the potential of the dynamic pricing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The electricity load temporal dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Time plays an important role in power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Firstly, the consumer’s behaviour is highly dependent on the time of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The tendency to adapt this behaviour is also expected to be time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Therefore, the residential power consumption elasticity has to be a function of the time within a day, among other things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Secondly, a higher electricity price does not simply reduce the demand as with other commodities, but rather shifts part of the consump- tion earlier and/or later in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This phenomenon reflects a complex temporal dependence for power consumption, which has to be accurately modelled in order to design a performing dynamic pricing policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Formally, the mathematical model of the residential demand response is denoted M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Its input space XD is composed of the predicted power consumption before any demand response and the dynamic prices to be sent to the consumers for several hours before and after the time pe- riod analysed, together with information about that time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Its output space YD comprises the predicted power consumption after demand response to dynamic prices for that same time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Mathematically, both demand re- sponse model input xD t ∈ XD and output yD t ∈ YD at time step t can be expressed as the following: xD t = {CF t , Y ′ t , Tt}, (16) yD t = c′ t, (17) where Y ′ t = {yt+ϵ ∈ R | ϵ = −k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=', k} is the dynamic price signal within a time window centred around time step t and of size k from which the demand response is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' As far as the scientific literature about the modelling of demand response to dynamic prices is concerned, this in- teresting topic has not yet received a lot of attention from the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Still, there exists a few sound works presenting demand response models and assessing the receptiveness of the consumers to dynamic power prices [15, 16, 17, 18, 19], as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Uncertainty discussion As previously hinted, a dynamic pricing policy has to make its decisions based on imperfect information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Indeed, multiple forecasts for the electricity price, production and consumption have to be generated 12 up to 35 hours in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Naturally, these predictions comes with a level of uncertainty that should not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Moreover, accurately modelling the residential demand response to dynamic prices is a particularly challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Because of both the random human nature and the difficulty to fully capture the consumers’ behaviour within a mathe- matical model, a notable level of uncertainty should also be considered at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Therefore, multiple sources of uncertainty can be identified in the scope of the dynamic pricing decision-making problem at hand, and a proper management of this uncertainty is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' A stochastic reasoning is recommended to make sound dynamic pricing decisions despite this substantial level of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Instead of considering each uncertain variable (production, consumption, price, demand response) with a probability of 1, the full probability distribution behind these quantities has to be estimated and exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Based on this information, the risk associated with uncertainty may be mitigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Moreover, safety margins may also con- tribute to reduce this risk, but potentially at the expense of a lowered performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In fact, there generally exists 9 a trade-off between performance and risk, in line with the adage: with great risk comes great reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Performance assessment methodology This section presents a methodology for quantitatively assessing the performance of a dynamic pricing policy in a comprehensive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' As explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content='7, several disjoint objectives can be clearly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' For the sake of completeness, this research work presents three quantitative indicators, one for each objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The rela- tive importance of these indicators is left to the discretion of the reader according to its main intention among the different objectives previously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The performance indicators proposed are based on the comparison with the original situation without dynamic pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In this case, the consumer is assumed to be fully ignorant about the mismatch problem between supply and demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' No information is provided to the customers of the producer/retailer, which consequently have an unin- fluenced consumption behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' The price of electricity et is freely determined by the producer/retailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' It may for instance be a fixed tariff, or a price indexed on the day-ahead market price: et = α λt + β , (18) where α and β are parameters to be set by the retailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Firstly, the impact of a dynamic pricing policy on the synchronisation between power supply and demand can be assessed through the performance indicator S quantifying the relative evolution of the deviation ∆T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This quantity is mathematically expressed as follows: S = 100 ∆T − ∆T ∆T , (19) ∆T = T −1 � t=0 |pt − ct| , (20) where ∆T represents the lack of synchronisation between supply and demand without dynamic pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Therefore, the quantity S has ideally to be maximised, with a perfect synchronisation between supply and demand leading to a value of 100% reduction in deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Secondly, the consequence for the consumer regarding its electricity bill can be evaluated with the quantity B which informs about the relative evolution of this power bill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' It can be mathematically computed as the following: B = 100 BT − BT BT , (21) where BT = �T −1 t=0 ct et represents the electricity bill paid by the consumer without dynamic pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Since the per- formance indicator B represents the percentage reduction in costs, it has to ideally be maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Lastly, the enhancement in terms of revenue for the producer/retailer can be efficiently quantified thanks to the performance indicator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' This quantity represents the relative evolution of the producer/retailer revenue and can be mathematically expressed as follows: R = 100 RT − RT RT , (22) RT = T −1 � t=0 � ct et − (cF t − pF t ) λt − (ct − pt) it � , (23) where RT represents the producer/retailer revenue with- out dynamic pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Obviously, the performance indicator R has to ideally be maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Conclusion This research paper presents a detailed formalisation of the decision-making problem faced by a producer/retailer willing to adopt a dynamic pricing approach, in order to induce an appropriate residential demand response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Three core challenges are highlighted by this formalisation work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Firstly, the objective criterion maximised by a dynamic pricing policy is not trivially defined, since different goals that are not compatible can be clearly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Secondly, several complex algorithmic components are necessary for the development of a performing dynamic pricing policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' One can for instance mention different forecasting blocks, but also a mathematical model of the residential demand response to dynamic prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Thirdly, the dynamic pricing decisions have to be made based on imperfect information, because this particular decision-making problem is highly conditioned by the actual uncertainty for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Several avenues are proposed for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' In fact, the natural extension of the present research is to design innovative dynamic pricing policies from the perspective of the supply side based on the formalisation performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' While the present research paper exclusively focuses on the philosophy and conceptual analysis of the approach, there remain practical concerns that need to be properly addressed in order to achieve performing decision-making policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' To achieve that, a deeper analysis of the scientific literature about each algorithmic component discussed in Section 4 is firstly welcomed, in order to identify and re- produce the state-of-the-art techniques within the context of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Then, different approaches have to be investi- gated for the design of the dynamic pricing policy itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' One can for instance mention, among others, the stochastic optimisation and deep reinforcement learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' Finally, the dynamic pricing policies developed have to be rigorously evaluated, analysed, and compared by taking advantage of real-life experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFJT4oBgHgl3EQfmSym/content/2301.11587v1.pdf'} +page_content=' 10 Acknowledgements Thibaut Théate is a Research 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Diffusion Models for Generalized Talking Head Synthesis +Shuai Shen1 +Wenliang Zhao1 +Zibin Meng1 +Wanhua Li1 +Zheng Zhu2 +Jie Zhou1 +Jiwen Lu1 +1Tsinghua University +2PhiGent Robotics +… +Figure 1. We present a crafted conditional Diffusion model for generalized Talking head synthesis (DiffTalk). Given a driven audio, the +DiffTalk is capable of synthesizing high-fidelity and synchronized talking videos for multiple identities without further fine-tuning. +Abstract +Talking head synthesis is a promising approach for the +video production industry. +Recently, a lot of effort has +been devoted in this research area to improve the gener- +ation quality or enhance the model generalization. How- +ever, there are few works able to address both issues simul- +taneously, which is essential for practical applications. To +this end, in this paper, we turn attention to the emerging +powerful Latent Diffusion Models, and model the Talking +head generation as an audio-driven temporally coherent +denoising process (DiffTalk). +More specifically, instead +of employing audio signals as the single driving factor, +we investigate the control mechanism of the talking face, +and incorporate reference face images and landmarks as +conditions for personality-aware generalized synthesis. In +this way, the proposed DiffTalk is capable of producing +high-quality talking head videos in synchronization with the +source audio, and more importantly, it can be naturally gen- +eralized across different identities without any further fine- +tuning. +Additionally, our DiffTalk can be gracefully tai- +lored for higher-resolution synthesis with negligible extra +computational cost. Extensive experiments show that the +proposed DiffTalk efficiently synthesizes high-fidelity audio- +driven talking head videos for generalized novel identi- +ties. For more video results, please refer to this demon- +stration https://cloud.tsinghua.edu.cn/f/ +e13f5aad2f4c4f898ae7/. +1. Introduction +Talking head synthesis is a challenging and promising re- +search topic, which aims to synthesize a talking video with +given audio. This technique is widely applied in various +practical scenarios including animation, virtual avatars, on- +line education, and video conferencing [4,44,47,50,52]. +Recently a lot of effort has been devoted to this re- +search area to improve the generation quality or enhance +the model generalization. +Among these existing main- +stream talking head generation approaches, the 2D-based +methods usually depend on generative adversarial networks +(GANs) [6, 10, 16, 22, 28] for audio-to-lip mapping, and +1 +arXiv:2301.03786v1 [cs.CV] 10 Jan 2023 + +most of them perform competently on model generalization. +However, since GANs need to simultaneously optimize a +generator and a discriminator, the training process lacks sta- +bility and is prone to mode collapse [11]. Due to this re- +striction, the generated talking videos are of limited image +quality, and difficult to scale to higher resolutions. By con- +trast, 3D-based methods [2,17,42,46,53] perform better in +synthesizing higher-quality talking videos. Whereas, they +highly rely on identity-specific training, and thus cannot +generalize across different persons. Such identity-specific +training also brings heavy resource consumption and is not +friendly to practical applications. Most recently, there are +some 3D-based works [36] that take a step towards improv- +ing the generalization of the model. However, further fine- +tuning on specific identities is still inevitable. +Generation quality and model generalization are two es- +sential factors for better deployment of the talking head syn- +thesis technique to real-world applications. However, few +existing works are able to address both issues well. In this +paper, we propose a crafted conditional Diffusion model for +generalized Talking head synthesis (DiffTalk), that aims to +tackle these two challenges simultaneously. Specifically, to +avoid the unstable training of GANs, we turn attention to +the recently developed generative technology Latent Dif- +fusion Models [30], and model the talking head synthe- +sis as an audio-driven temporally coherent denoising pro- +cess. On this basis, instead of utilizing audio signals as +the single driving factor to learn the audio-to-lip transla- +tion, we further incorporate reference face images and land- +marks as supplementary conditions to guide the face iden- +tity and head pose for personality-aware video synthesis. +Under these designs, the talking head generation process +is more controllable, which enables the learned model to +naturally generalize across different identities without fur- +ther fine-tuning. As shown in Figure 1, with a sequence +of driven audio, our DiffTalk is capable of producing natu- +ral talking videos of different identities based on the corre- +sponding reference videos. Moreover, benefiting from the +latent space learning mode, our DiffTalk can be gracefully +tailored for higher-resolution synthesis with negligible ex- +tra computational cost, which is meaningful for improving +the generation quality. +Extensive experiments show that our DiffTalk can syn- +thesize high-fidelity talking videos for novel identities with- +out any further fine-tuning. Figure 1 shows the generated +talking sequences with one driven audio across three differ- +ent identities. Comprehensive method comparisons show +the superiority of the proposed DiffTalk, which provides a +strong baseline for the high-performance talking head syn- +thesis. To summarize, we make the following contributions: +• We propose a crafted conditional diffusion model for +high-quality and generalized talking head synthesis. By +introducing smooth audio signals as a condition, we +model the generation as an audio-driven temporally co- +herent denoising process. +• For personality-aware generalized synthesis, we further +incorporate dual reference images as conditions. In this +way, the trained model can be generalized across different +identities without further fine-tuning. +• The proposed DiffTalk can generate high-fidelity and +vivid talking videos for generalized identities. In exper- +iment, our DiffTalk significantly outperforms 2D-based +methods in the generated image quality, while surpassing +3D-based works in the model generalization ability. +2. Related Work +Audio-driven Talking Head Synthesis. +The talking +head synthesis aims to generate talking videos with lip +movements synchronized with the driving audio [14, 40]. +In terms of the modeling approach, we roughly divide the +existing methods into 2D-based and 3D-based ones. +In +the 2D-based methods, GANs [6, 10, 16, 28] are usually +employed as the core technologies for learning the audio- +to-lip translation. +Zhou et al. [52] introduce a speaker- +aware audio encoder for personalized head motion model- +ing. Prajwal et al. [28] boost the lip-visual synchroniza- +tion with a well-trained Lip-Sync expert [8]. +However, +since the training process of GANs lacks stability and is +prone to mode collapse [11], the generated talking videos +are always of limited image quality, and difficult to scale +to higher resolutions. Recently a series of 3D-based meth- +ods [4,20,39–41] have been developed. [39–41] utilize 3D +Morphable Models [2] for parametric control of the talk- +ing face. +More recently, the emerging Neural radiance +fields [26] provide a new solution for 3D-aware talking head +synthesis [3, 17, 24, 36]. However, most of these 3D-based +works highly rely on identity-specific training, and thus +cannot generalize across different identities. Shen et al. [36] +have tried to improve the generalization of the model, how- +ever, further fine-tuning on specific identities is still in- +evitable. In this work, we propose a brand-new diffusion +model-based framework for high-fidelity and generalized +talking head synthesis. +Latent Diffusion Models. Diffusion Probabilistic Mod- +els (DM) [37] have shown strong ability in various im- +age generation tasks [11, 19, 29]. However, due to pixel +space-based training [30,32], very high computational costs +are inevitable. +More recently, Rombach et al. [30] pro- +pose the Latent Diffusion Models (LDMs), and transfer the +training and inference processes of DM to a compressed +lower-dimension latent space for more efficient comput- +ing [13, 49]. With the democratizing of this technology, it +has been successfully employed in a series of works, in- +cluding text-to-image translation [21, 31, 33], super resolu- +tion [7, 12, 27], image inpainting [23, 25], motion genera- +tion [35,48], 3D-aware prediction [1,34,43]. In this work, +2 + +Att +Att +Att +Att +Att +Att +Att +Att +���0 +������ +… +������−1 +���1 +��� +������ +������ +������ +Reference +Audio +Landmark +������ +concatenate +concatenate +��� +������ +������−1 +������ +… +… +Conditions +0 +Diffusion Process +Denoising Process +������ +������ +��� +��� +Figure 2. Overview of the proposed DiffTalk for generalized talking head video synthesis. Apart from the audio signal condition to drive +the lip motions, we further incorporate reference images and facial landmarks as extra driving factors for personalized facial modeling. +In this way, the talking head generation process is more controllable, which enables the learned model to generalize across different +identities without further fine-tuning. Furthermore, benefiting from the latent space learning mode, we can graceful improve our DiffTalk +for higher-resolution synthesis with slight extra computational cost. +drawing on these successful practices, we model the talk- +ing head synthesis as an audio-driven temporally coherent +denoising process and achieve superior generation results. +3. Methodology +3.1. Overview +To tackle the challenges of generation quality and model +generalization for better real-world deployment, we model +the talking head synthesis as an audio-driven temporally co- +herent denoising process, and term the proposed method as +DiffTalk. An overview of the proposed DiffTalk is shown in +Figure 2. By introducing smooth audio features as a condi- +tion, we improve the diffusion model for temporally coher- +ent facial motion modeling. For further personalized facial +modeling, we incorporate reference face images and facial +landmarks as extra driving factors. In this way, the talking +head generation process is more controllable, which enables +the learned model to generalize across different identities +without any further fine-tuning. Moreover, benefiting from +the latent space learning mode, we can graceful improve +our DiffTalk for higher-resolution synthesis with negligible +extra computational cost, which contributes to improving +the generation quality. In the following, we will detail the +proposed conditional Diffusion Models for high-fidelity and +generalized talking head generation in Section 3.2. In Sec- +tion 3.3, the progressive inference stage is introduced for +better inter-frame consistency. +3.2. Conditional Diffusion Model for Talking Head +The emergence of Latent Diffusion Models (LDMs) [19, +30] provides a straightforward and effective way for high- +fidelity image synthesis. To inherit its excellent properties, +we adopt this advanced technology as the foundation of our +method and explore its potential in modeling the dynamic +talking head. With a pair of well-trained image encoder EI +and decoder DI which are frozen in training [13], the in- +put face image x ∈ RH×W ×3 can be encoded into a latent +space z0 = EI(x) ∈ Rh×w×3, where H/h = W/w = f, +H, W are the height and width of the original image and +f is the downsampling factor. In this way, the learning is +transferred to a lower-dimensional latent space, which is +more efficient with fewer train resources. On this basis, the +standard LDMs are modeled as a time-conditional UNet- +based [32] denoising network M, which learns the reverse +process of a Markov Chain [15] of length T. The corre- +sponding objective can be formulated as: +LLDM := Ez,ϵ∼N (0,1),t +� +∥ϵ − M (zt, t)∥2 +2 +� +, +(1) +where t ∈ [1, · · · , T] and zt is obtained through the forward +diffusion process from z0. ˜zt−1 = zt − M(zt, t) is the +denoising result of zt at time step t. The final denoised +result ˜z0 is then upsampled to the pixel space with the pre- +trained image decoder ˜x = DI(˜z0), where ˜x ∈ RH×W ×3 +is the reconstructed face image. +Given a source identity and driven audio, our goal is to +train a model for generating a natural target talking video in +3 + +Audio Stream +16 time intervals +DeepSpeech RNN +Feature Map +Feature +Extractor +Temporal +Filtering +16 windown size +Figure 3. Visualization of the smooth audio feature extractor. For +better temporal coherence, two-stage smoothing operations are in- +volved in this module. +synchronization with the audio condition while maintaining +the original identity information. Furthermore, the trained +model also needs to work for novel identities during infer- +ence. To this end, the audio signal is introduced as a basic +condition to guide the direction of the denoising process for +modeling the audio-to-lip translation. +Smooth Audio Feature Extraction. To better incorpo- +rate temporal information, we involve two-stage smoothing +operations in the audio encoder EA, as shown in Figure 3. +Firstly, following the practice in VOCA [9], we reorganize +the raw audio signal into overlapped windows of size 16 +time intervals (corresponding to audio clips of 20ms), where +each window is centered on the corresponding video frame. +A pre-trained RNN-based DeepSpeech [18] module is then +leveraged to extract the per-frame audio feature map F. For +better inter-frame consistency, we further introduce a learn- +able temporal filtering [41]. It receives a sequence of adja- +cent audio features [Fi−w, . . . , Fi, . . . , Fi+w] with w = 8 +as input, and computes the final smoothed audio feature for +the i-th frame as a ∈ RDA in a self-attention-based learn- +ing manner, where DA denotes the audio feature dimension. +By encoding the audio information, we bridge the modality +gap between the audio signals and the visual information. +Introducing such smooth audio features as a condition, we +extend the diffusion model for temporal coherence-aware +modeling of face dynamics when talking. The objective is +then formulated as: +LA := Ez,ϵ∼N (0,1),a,t +� +∥ϵ − M (zt, t, a)∥2 +2 +� +. +(2) +Identity-Preserving Model Generalization. In addi- +tion to learning the audio-to-lip translation, another essen- +tial task is to realize the model generalization while pre- +serving complete identity information in the source image. +Generalized identity information includes face appearance, +head pose, and image background. To this end, a reference +mechanism is designed to empower our model to general- +ize to new individuals unseen in training, as shown in Fig- +ure 2. Specifically, a random face image xr of the source +identity is chosen as a reference condition, which contains +appearance and background information. To prevent train- +ing shortcuts, we limit the selection of xr to 60 frames be- +yond the target image. +However, since the ground-truth +face image has a completely different pose from xr, the +model is expected to transfer the pose of xr to the target +face without any prior information. This is somehow an +ill-posed problem with no unique solution. For this rea- +son, we further incorporate the masked ground-truth im- +age xm as another reference condition to provide the target +head pose guidance. The mouth region of xm is completely +masked to ensure that the ground truth lip movements are +not visible to the network. In this way, the reference xr fo- +cuses on affording mouth appearance information, which +additionally reduces the training difficulty. +Before serv- +ing as conditions, xr and xm are also encoded into the la- +tent space through the trained image encoder, and we have +zr = DI(xr) ∈ Rh×w×3, zm = DI(xm) ∈ Rh×w×3. On +this basis, an auxiliary facial landmarks condition is also in- +cluded for better control of the face outline. Similarly, land- +marks in the mouth area are masked to avoid shortcuts. The +landmark feature l ∈ RDL is obtained with an MLP-based +encoder EL, where DL is the landmark feature dimension. +In this way, combining these conditions with audio feature +a, we realize the precise control over all key elements of +a dynamic talking face. With C = {a, zr, zm, l} denoting +the condition set, the talking head synthesis is finally mod- +eled as a conditional denoising process optimized with the +following objective: +L := Ez,ϵ∼N (0,1),C,t +� +∥ϵ − M (zt, t, C)∥2 +2 +� +, +(3) +where the network parameters of M, EA and EL are jointly +optimized via this equation. +Conditioning Mechanisms. Based on the modeling of +the conditional denoising process in Eq. 3, we pass these +conditions C to the network in the manner shown in Fig- +ure 2. Specifically, following [30], we implement the UNet- +based backbone M with the cross-attention mechanism for +better multimodality learning. The spatially aligned refer- +ences zr and zm are concatenated channel-wise with the +noisy map zT to produce a joint visual condition Cv = +[zT ; zm; zr] ∈ Rh×w×9. Cv is fed to the first layer of the +network to directly guide the output face in an image-to- +image translation fashion. Additionally, the driven-audio +feature a and the landmark representation l are concatenated +into a latent condition Cl = [a; l] ∈ RDA+DL, which serves +as the key and value for the intermediate cross-attention +layers of M. +To this extent, all condition information +C = {Cv, Cl} are properly integrated into the denoising +network M to guide the talking head generation process. +4 + +DDIM-based Denoising +������,1 +Random ������,1 +���1 +������,2 +������,2 +���2 +������ +DDIM-based Denoising +DDIM-based Denoising +������,��� +… +������,��� +Figure 4. Illustration of the designed progressive inference strat- +egy. For the first frame, the setting of the visual condition Cv +remains the same as for training, where xr,1 is a random face im- +age from the target identity. Subsequently, the synthetic image ˜xi +is employed as the reference condition xr,i+1 for the next frame +to enhance the temporal coherence of the generated video. +Higher-Resolution Talking Head Synthesis Our pro- +posed DiffTalk can also be gracefully extended for higher- +resolution talking head synthesis with negligible extra com- +putational cost and faithful reconstruction effects. Specif- +ically, considering the trade-off between the perceptual +loss and the compression rate, for training images of size +256 × 256 × 3, we set the downsampling factor as f = 4 +and obtain the latent space of 64 × 64 × 3. Furthermore, +for higher-resolution generation of 512 × 512 × 3, we just +need to adjust the paired image encoder EI and decoder DI +with a bigger downsampling factor f = 8. Then the trained +encoder is frozen and employed to transfer the training pro- +cess to a 64 × 64 × 3 latent space as well. This helps to +relieve the pressure on insufficient resources, and therefore +our model can be gracefully improved for higher-resolution +talking head video synthesis. +3.3. Progressive Inference +We perform inference with Denoising Diffusion Implicit +Model-based (DDIM) [38] iterative denoising steps. DDIM +is a variant of the standard DM to accelerate sampling for +more efficient synthesis. To further boost the coherence of +the generated talking videos, we develop a progressive ref- +erence strategy in the reference process as shown in Fig- +ure 4. Specifically, when rendering a talking video sequence +with the trained model, for the first frame, the setting of the +visual condition Cv remains the same as for training, where +xr,1 is a random face image from the target identity. Sub- +sequently, this synthetic face image is exploited as the xr +for the next frame. In this way, image details between adja- +cent frames remain consistent, resulting in a smoother tran- +sition between frames. It is worth noting that this strategy +is not used for training. Since the difference between adja- +cent frames is small, we need to eliminate such references +to avoid learning shortcuts. +0 +100 +0 +100 +GT +w.o. Smooth +w. Smooth +Figure 5. Ablation study on the audio smoothing operation. We +show the differences between adjacent frames as heatmaps for bet- +ter visualization. The results without audio filtering present obvi- +ous high heat values in the mouth region, which indicates the jitters +in this area. By contrast, with smooth audio as the condition, the +generated video frames show smoother transitions. +4. Experiments +4.1. Experimental Settings +Dataset. To train the audio-driven diffusion model, an +audio-visual dataset HDTF [51] is used. +It contains 16 +hours of talking videos in 720P or 1080P from more than +300 identities. +We randomly select 100 videos with the +length of about 5 hours for training, while the remaining +data serve as the test set. Apart from this public dataset, we +also use some other videos for cross-dataset evaluation. +Metric. We evaluate our proposed method through vi- +sual results coupled with quantitative indicators. +PSNR +(↑), SSIM (↑) [45] and LPIPS (↓) [49] are three metrics +for assessing image quality. The LPIPS is a learning-based +perceptual similarity measure that is more in line with hu- +man perception, we therefore recommend this metric as a +more objective indicator. +The SyncNet score (Offset↓ / +Confidence↑) [8] checks the audio-visual synchronization +quality, which is important for the audio-driven talking head +generation task. +(‘↓’ indicates that the lower the better, +while ‘↑’ means that the higher the better.) +Implementation Details. We resize the input image to +256 × 256 for experiments. The downsampling factor f is +set as 4, so the latent space is 64 × 64 × 3. For training the +model for higher resolution synthesis, the input is resized to +512 × 512 with f = 8 to keep the same size of latent space. +The length of the denoising step T is set as 200 for both the +5 + +Ground Truth +A +A + L +A + L + R +A + M +A + L + M + R +ID 1 +ID 2 +Figure 6. Ablation study on the design of the conditions. The marks above these images refer to the following meanings, ‘A’: Audio; +‘L’: Landmark; ‘R’: Random reference image; ‘M’: Masked ground-truth image. We show the generated results under different condition +settings on two test sets, and demonstrate the effectiveness of our final design, i.e. A+L+M+R. +Method +PSNR↑ SSIM↑ LPIPS↓ SyncNet↓↑ +Test Set A +GT +- +- +- +0/9.610 +w/o +33.67 +0.944 +0.024 +1/5.484 +w +34.17 +0.946 +0.024 +1/6.287 +Test Set B +GT +- +- +- +0/9.553 +w/o +32.70 +0.924 +0.031 +1/5.197 +w +32.73 +0.925 +0.031 +1/5.387 +Table 1. Ablation study to investigate the contribution of the audio +smoothing operation. ‘w’ indicates the model is trained with the +audio features after temporal filtering and vice versa. +training and inference process. The feature dimensions are +DA = DL = 64. Our model takes about 15 hours to train +on 8 NVIDIA 3090Ti GPUs. +4.2. Ablation Study +Effect of the Smooth Audio. In this subsection, we in- +vestigate the effect of the audio smooth operations. Quanti- +tative results in Table 1 show that the model equipped with +the audio temporal filtering module outperforms the one +without smooth audio, especially in the SyncNet score. We +further visualize the differences between adjacent frames as +the heatmaps shown in Figure 5. The results without audio +filtering present obvious high heat values in the mouth re- +gion, which indicates the jitters in this area. By contrast, +with smooth audio as the condition, the generated video +frames show smoother transitions, which are reflected in the +soft differences of adjacent frames. +Design of the Conditions. A major contribution of this +work is the ingenious design of the conditions for general +and high-fidelity talking head synthesis. In Figure 6, we +show the generated results under different condition settings +step by step, to demonstrate the superiority of our design. +Method +PSNR↑ SSIM↑ LPIPS↓ SyncNet↓↑ +Test Set A +GT +- +- +- +4/7.762 +w/o +34.17 +0.946 +0.024 +1/6.287 +w +33.95 +0.946 +0.023 +-1/6.662 +Test Set B +GT +- +- +- +3/8.947 +w/o +32.73 +0.925 +0.031 +1/5.387 +w +33.02 +0.925 +0.030 +1/5.999 +Table 2. Ablation study on the effect of the progressive inference +strategy. ‘w/o’ indicates that a random reference image is em- +ployed as the condition, and ‘w’ means that the reference is the +generated result of the previous frame. +With pure audio as the condition, the model fails to gener- +alize to new identities, and the faces are not aligned with the +background in the inpainting-based inference. Adding land- +marks as another condition tackles the misalignment prob- +lem. A random reference image is further introduced try- +ing to provide the identity information. Whereas, since the +ground-truth face image has a different pose from this ran- +dom reference, the model is expected to transfer the pose of +reference to the target face. This greatly increases the diffi- +culty of training, leading to hard network convergence, and +the identity information is not well learned. Using the au- +dio and masked ground-truth images as driving factors mit- +igates the identity inconsistency and misalignment issues, +however the appearance of the mouth can not be learned +since this information is not visible to the network. For +this reason, we employ the random reference face and the +masked ground-truth image together for dual driving, where +the random reference provides the lip appearance message +and the masked ground-truth controls the head pose and +identity. Facial landmarks are also incorporated as a con- +dition that helps to model the facial contour better. Results +6 + +GT +ATVG +MakeItTalk +Wav2Lip +Ours +DFRF +AD-NeRF +3D-based Methods +2D-based Methods +Figure 7. Visual comparison with some representative 2D-based talking head generation methods ATVGnet [5], MakeitTalk [52] and +Wav2Lip [28], and with some recent 3D-based ones AD-NeRF [17] and DFRF [36]. The results of DFRF are synthesized with the base +model without fine-tuning for fair comparisons. AD-NeRF is trained on these two identities respectively to produce the results. +in Figure 6 show the effectiveness of such design in synthe- +sizing realism and controllable face images. +Impact of the Progressive Inference. Temporal corre- +lation inference is developed in this work through the pro- +gressive reference strategy. We conduct an ablation study +in Table 2 to investigate the impact of this design. ‘w/o’ in- +dicates that a random reference image xr is employed, and +‘w’ means that the generated result of the previous frame +is chosen as the reference condition. With such progressive +inference, the SyncNet scores are further boosted, since the +temporal correlation is better modeled and the talking style +becomes more coherent. The LPIPS indicator is also en- +hanced with this improvement. PSNR tends to give higher +scores to blurry images [49], so we recommend LPIPS as a +more representative metric for visual quality. +4.3. Method Comparison +Comparison with 2D-based Methods. In this section, +we perform method comparisons with some representative +2D-based talking head generation approaches including the +ATVGnet [5], MakeitTalk [52] and Wav2Lip [28]. Figure 7 +visualizes the generated frames of these methods. It can +be seen that the ATVGnet performs generation based on +cropped faces with limited image quality. The MakeItTalk +synthesizes plausible talking frames, however the back- +ground is wrongly wrapped with the mouth movements. +This phenomenon is more observable in the video form +result, and greatly affects the visual experience. +Gener- +ated talking faces of Wav2Lip appear artifacts in the square +boundary centered on the mouth, since the synthesized area +7 + +Method +Test Set A +Test Set B +General +PSNR↑ +SSIM↑ +LPIPS↓ +SyncNet↓↑ +PSNR↑ +SSIM↑ +LPIPS↓ +SyncNet↓↑ +Method +GT +- +- +- +-1/8.979 +- +- +- +-2/7.924 +- +MakeItTalk [52] +18.77 +0.544 +0.19 +-4/3.936 +17.70 +0.648 +0.129 +-3/3.416 +✓ +Wav2Lip [28] +25.50 +0.761 +0.140 +-2/8.936 +33.38 +0.942 +0.027 +-3/9.385 +✓ +AD-NeRF [17] +27.89 +0.885 +0.072 +-2/5.639 +30.14 +0.947 +0.023 +-3/4.246 + +DFRF [36] +28.60 +0.892 +0.068 +-1/5.999 +33.57 +0.949 +0.025 +-2/4.432 +FT Req. +Ours +34.54 +0.950 +0.024 +-1/6.381 +34.01 +0.950 +0.020 +-1/5.639 +✓ +Table 3. Comparison with some representative talking head synthesis methods on two test sets as in Figure 7. The best performance is +highlighted in red (1st best) and blue (2nd best). Our DiffTalk obtains the best PSNR, SSIM, and LPIPS values, and comparable SyncNet +scores simultaneously. It is worth noting that the DFRF is fine-tuned on the specific identity to obtain these results, while our method can +directly be utilized for generation without further fine-tuning. (‘FT Req.’ means that fine-tuning operation is required for the DFRF.) +and the original image are not well blended. By contrast, +the proposed DiffTalk generates natural and realistic talk- +ing videos with accurate audio-lip synchronization, owing +to the crafted conditioning mechanism and stable training +process. For more objective comparisons, we further eval- +uate the quantitative results in Table 3. Our DiffTalk far +surpasses [28] and [52] in all image quality metrics. For +the audio-visual synchronization metric SyncNet, the pro- +posed method reaches a high level and is superior than +MakeItTalk. Although the DiffTalk is slightly inferior to +Wav2Lip on the SyncNet score, it is far better than Wav2Lip +in terms of image quality. In conclusion, our method outper- +forms these 2D-based methods under comprehensive con- +sideration of the qualitative and quantitative results. +Comparison with 3D-based Methods. For more com- +prehensive evaluations, we further compare with some +recent high-performance 3D-based works including AD- +NeRF [17] and DFRF [36]. They realize implicitly 3D head +modeling with the NeRF technology, so we treat them as +generalized 3D-based methods. The visualization results +are shown in Figure 7. AD-NeRF models the head and torso +parts separately, resulting in misalignment in the neck re- +gion. More importantly, it is worth noting that AD-NeRF +is a non-general method. In contrast, our method is able to +handle unseen identities without further fine-tuning, which +is more in line with the practical application scenarios. The +DFRF relies heavily on the fine-tuning operation for model +generalization, and the generated talking faces with only +the base model are far from satisfactory as shown in Fig- +ure 7. More quantitative results in Table 3 also show that our +method surpasses [17, 36] on the image quality and audio- +visual synchronization indicators. +4.4. Expand to Higher Resolution +In this section, we perform experiments to demonstrate +the capacity of our method on generating higher-resolution +images. In Figure 8, we show the synthesis frames of two +models (a) and (b). (a) is trained on 256 × 256 images with +the downsampling factor f = 4, so the latent space is of +(a) Resolution: 256 × 256, ���=4 +(b) Resolution: 512 × 512, ���=8 +Figure 8. Generated results with higher resolution. +size 64 × 64 × 3. For (b), 512 × 512 images with f = +8 are used for training the model. Since both models are +trained based on a compressed 64 × 63 × 3 latent space, +the pressure of insufficient computing resources is relieved. +We can therefore comfortably expand our model for higher- +resolution generation just as shown in Figure 8, where the +synthesis quality in (b) significantly outperforms that in (a). +5. Conclusion and Discussion +In this paper, we have proposed a generalized and high- +fidelity talking head synthesis method based on a crafted +conditional diffusion model. Apart from the audio signal +condition to drive the lip motions, we further incorporate +reference images as driving factors to model the personal- +ized appearance, which enables the learned model to com- +fortably generalize across different identities without any +further fine-tuning. Furthermore, our proposed DiffTalk can +be gracefully tailored for higher-resolution synthesis with +negligible extra computational cost. +Limitations. The proposed method models talking head +generation as an iterative denoising process, which needs +more time to synthesize a frame compared with most GAN- +based approaches. This is also a common problem of LDM- +based works which warrants further research. Nonetheless, +we have a large speed advantage over most 3D-based meth- +ods. Since talking head technology may raise potential mis- +use issues, we are committed to combating these malicious +behaviors and advocate positive applications. Additionally, +researchers who want to use our code will be required to get +authorization and add watermarks to the generated videos. +8 + +References +[1] Miguel Angel Bautista, Pengsheng Guo, Samira Abnar, Wal- +ter Talbott, Alexander Toshev, Zhuoyuan Chen, Laurent +Dinh, Shuangfei Zhai, Hanlin Goh, Daniel Ulbricht, et al. +Gaudi: A neural architect for immersive 3d scene genera- +tion. arXiv, 2022. 2 +[2] Volker Blanz and Thomas Vetter. A morphable model for the +synthesis of 3d faces. In SIGGRAPH, 1999. 2 +[3] Eric R Chan, Marco Monteiro, Petr Kellnhofer, Jiajun Wu, +and Gordon Wetzstein. +pi-gan: Periodic implicit genera- +tive adversarial networks for 3d-aware image synthesis. In +CVPR, 2021. 2 +[4] Lele Chen, Guofeng Cui, Celong Liu, Zhong Li, Ziyi Kou, Yi +Xu, and Chenliang Xu. Talking-head generation with rhyth- +mic head motion. In ECCV, 2020. 1, 2 +[5] Lele Chen, Ross K Maddox, Zhiyao Duan, and Chenliang +Xu. 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In Computer Graphics Forum, 2018. 2 +10 + diff --git a/CdE2T4oBgHgl3EQfSAcB/content/tmp_files/load_file.txt b/CdE2T4oBgHgl3EQfSAcB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2bb91a5b4e09871a9682fd0934a60d3ca64d766b --- /dev/null +++ b/CdE2T4oBgHgl3EQfSAcB/content/tmp_files/load_file.txt @@ -0,0 +1,532 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf,len=531 +page_content='DiffTalk: Crafting Diffusion Models for Generalized Talking Head Synthesis Shuai Shen1 Wenliang Zhao1 Zibin Meng1 Wanhua Li1 Zheng Zhu2 Jie Zhou1 Jiwen Lu1 1Tsinghua University 2PhiGent Robotics … Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' We present a crafted conditional Diffusion model for generalized Talking head synthesis (DiffTalk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Given a driven audio, the DiffTalk is capable of synthesizing high-fidelity and synchronized talking videos for multiple identities without further fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Abstract Talking head synthesis is a promising approach for the video production industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Recently, a lot of effort has been devoted in this research area to improve the gener- ation quality or enhance the model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' How- ever, there are few works able to address both issues simul- taneously, which is essential for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To this end, in this paper, we turn attention to the emerging powerful Latent Diffusion Models, and model the Talking head generation as an audio-driven temporally coherent denoising process (DiffTalk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' More specifically, instead of employing audio signals as the single driving factor, we investigate the control mechanism of the talking face, and incorporate reference face images and landmarks as conditions for personality-aware generalized synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this way, the proposed DiffTalk is capable of producing high-quality talking head videos in synchronization with the source audio, and more importantly, it can be naturally gen- eralized across different identities without any further fine- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Additionally, our DiffTalk can be gracefully tai- lored for higher-resolution synthesis with negligible extra computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Extensive experiments show that the proposed DiffTalk efficiently synthesizes high-fidelity audio- driven talking head videos for generalized novel identi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For more video results, please refer to this demon- stration https://cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='cn/f/ e13f5aad2f4c4f898ae7/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Introduction Talking head synthesis is a challenging and promising re- search topic, which aims to synthesize a talking video with given audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' This technique is widely applied in various practical scenarios including animation, virtual avatars, on- line education, and video conferencing [4,44,47,50,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Recently a lot of effort has been devoted to this re- search area to improve the generation quality or enhance the model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Among these existing main- stream talking head generation approaches, the 2D-based methods usually depend on generative adversarial networks (GANs) [6, 10, 16, 22, 28] for audio-to-lip mapping, and 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='03786v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='CV] 10 Jan 2023 most of them perform competently on model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' However, since GANs need to simultaneously optimize a generator and a discriminator, the training process lacks sta- bility and is prone to mode collapse [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Due to this re- striction, the generated talking videos are of limited image quality, and difficult to scale to higher resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' By con- trast, 3D-based methods [2,17,42,46,53] perform better in synthesizing higher-quality talking videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Whereas, they highly rely on identity-specific training, and thus cannot generalize across different persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Such identity-specific training also brings heavy resource consumption and is not friendly to practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Most recently, there are some 3D-based works [36] that take a step towards improv- ing the generalization of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' However, further fine- tuning on specific identities is still inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Generation quality and model generalization are two es- sential factors for better deployment of the talking head syn- thesis technique to real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' However, few existing works are able to address both issues well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this paper, we propose a crafted conditional Diffusion model for generalized Talking head synthesis (DiffTalk), that aims to tackle these two challenges simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Specifically, to avoid the unstable training of GANs, we turn attention to the recently developed generative technology Latent Dif- fusion Models [30], and model the talking head synthe- sis as an audio-driven temporally coherent denoising pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' On this basis, instead of utilizing audio signals as the single driving factor to learn the audio-to-lip transla- tion, we further incorporate reference face images and land- marks as supplementary conditions to guide the face iden- tity and head pose for personality-aware video synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Under these designs, the talking head generation process is more controllable, which enables the learned model to naturally generalize across different identities without fur- ther fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' As shown in Figure 1, with a sequence of driven audio, our DiffTalk is capable of producing natu- ral talking videos of different identities based on the corre- sponding reference videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Moreover, benefiting from the latent space learning mode, our DiffTalk can be gracefully tailored for higher-resolution synthesis with negligible ex- tra computational cost, which is meaningful for improving the generation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Extensive experiments show that our DiffTalk can syn- thesize high-fidelity talking videos for novel identities with- out any further fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Figure 1 shows the generated talking sequences with one driven audio across three differ- ent identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Comprehensive method comparisons show the superiority of the proposed DiffTalk, which provides a strong baseline for the high-performance talking head syn- thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To summarize, we make the following contributions: We propose a crafted conditional diffusion model for high-quality and generalized talking head synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' By introducing smooth audio signals as a condition, we model the generation as an audio-driven temporally co- herent denoising process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For personality-aware generalized synthesis, we further incorporate dual reference images as conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this way, the trained model can be generalized across different identities without further fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The proposed DiffTalk can generate high-fidelity and vivid talking videos for generalized identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In exper- iment, our DiffTalk significantly outperforms 2D-based methods in the generated image quality, while surpassing 3D-based works in the model generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Related Work Audio-driven Talking Head Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The talking head synthesis aims to generate talking videos with lip movements synchronized with the driving audio [14, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In terms of the modeling approach, we roughly divide the existing methods into 2D-based and 3D-based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In the 2D-based methods, GANs [6, 10, 16, 28] are usually employed as the core technologies for learning the audio- to-lip translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' [52] introduce a speaker- aware audio encoder for personalized head motion model- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Prajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' [28] boost the lip-visual synchroniza- tion with a well-trained Lip-Sync expert [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' However, since the training process of GANs lacks stability and is prone to mode collapse [11], the generated talking videos are always of limited image quality, and difficult to scale to higher resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Recently a series of 3D-based meth- ods [4,20,39–41] have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' [39–41] utilize 3D Morphable Models [2] for parametric control of the talk- ing face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' More recently, the emerging Neural radiance fields [26] provide a new solution for 3D-aware talking head synthesis [3, 17, 24, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' However, most of these 3D-based works highly rely on identity-specific training, and thus cannot generalize across different identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' [36] have tried to improve the generalization of the model, how- ever, further fine-tuning on specific identities is still in- evitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this work, we propose a brand-new diffusion model-based framework for high-fidelity and generalized talking head synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Latent Diffusion Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Diffusion Probabilistic Mod- els (DM) [37] have shown strong ability in various im- age generation tasks [11, 19, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' However, due to pixel space-based training [30,32], very high computational costs are inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' More recently, Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' [30] pro- pose the Latent Diffusion Models (LDMs), and transfer the training and inference processes of DM to a compressed lower-dimension latent space for more efficient comput- ing [13, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' With the democratizing of this technology, it has been successfully employed in a series of works, in- cluding text-to-image translation [21, 31, 33], super resolu- tion [7, 12, 27], image inpainting [23, 25], motion genera- tion [35,48], 3D-aware prediction [1,34,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this work, 2 Att Att Att Att Att Att Att Att ���0 ������ … ������−1 ���1 ��� ������ ������ ������ Reference Audio Landmark ������ concatenate concatenate ��� ������ ������−1 ������ … … Conditions 0 Diffusion Process Denoising Process ������ ������ ��� ��� Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Overview of the proposed DiffTalk for generalized talking head video synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Apart from the audio signal condition to drive the lip motions, we further incorporate reference images and facial landmarks as extra driving factors for personalized facial modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this way, the talking head generation process is more controllable, which enables the learned model to generalize across different identities without further fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Furthermore, benefiting from the latent space learning mode, we can graceful improve our DiffTalk for higher-resolution synthesis with slight extra computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' drawing on these successful practices, we model the talk- ing head synthesis as an audio-driven temporally coherent denoising process and achieve superior generation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Overview To tackle the challenges of generation quality and model generalization for better real-world deployment, we model the talking head synthesis as an audio-driven temporally co- herent denoising process, and term the proposed method as DiffTalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' An overview of the proposed DiffTalk is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' By introducing smooth audio features as a condi- tion, we improve the diffusion model for temporally coher- ent facial motion modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For further personalized facial modeling, we incorporate reference face images and facial landmarks as extra driving factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this way, the talking head generation process is more controllable, which enables the learned model to generalize across different identities without any further fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Moreover, benefiting from the latent space learning mode, we can graceful improve our DiffTalk for higher-resolution synthesis with negligible extra computational cost, which contributes to improving the generation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In the following, we will detail the proposed conditional Diffusion Models for high-fidelity and generalized talking head generation in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='3, the progressive inference stage is introduced for better inter-frame consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Conditional Diffusion Model for Talking Head The emergence of Latent Diffusion Models (LDMs) [19, 30] provides a straightforward and effective way for high- fidelity image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To inherit its excellent properties, we adopt this advanced technology as the foundation of our method and explore its potential in modeling the dynamic talking head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' With a pair of well-trained image encoder EI and decoder DI which are frozen in training [13], the in- put face image x ∈ RH×W ×3 can be encoded into a latent space z0 = EI(x) ∈ Rh×w×3, where H/h = W/w = f, H, W are the height and width of the original image and f is the downsampling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this way, the learning is transferred to a lower-dimensional latent space, which is more efficient with fewer train resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' On this basis, the standard LDMs are modeled as a time-conditional UNet- based [32] denoising network M, which learns the reverse process of a Markov Chain [15] of length T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The corre- sponding objective can be formulated as: LLDM := Ez,ϵ∼N (0,1),t � ∥ϵ − M (zt, t)∥2 2 � , (1) where t ∈ [1, · · · , T] and zt is obtained through the forward diffusion process from z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' ˜zt−1 = zt − M(zt, t) is the denoising result of zt at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The final denoised result ˜z0 is then upsampled to the pixel space with the pre- trained image decoder ˜x = DI(˜z0), where ˜x ∈ RH×W ×3 is the reconstructed face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Given a source identity and driven audio, our goal is to train a model for generating a natural target talking video in 3 Audio Stream 16 time intervals DeepSpeech RNN Feature Map Feature Extractor Temporal Filtering 16 windown size Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Visualization of the smooth audio feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For better temporal coherence, two-stage smoothing operations are in- volved in this module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' synchronization with the audio condition while maintaining the original identity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Furthermore, the trained model also needs to work for novel identities during infer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To this end, the audio signal is introduced as a basic condition to guide the direction of the denoising process for modeling the audio-to-lip translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Smooth Audio Feature Extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To better incorpo- rate temporal information, we involve two-stage smoothing operations in the audio encoder EA, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Firstly, following the practice in VOCA [9], we reorganize the raw audio signal into overlapped windows of size 16 time intervals (corresponding to audio clips of 20ms), where each window is centered on the corresponding video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' A pre-trained RNN-based DeepSpeech [18] module is then leveraged to extract the per-frame audio feature map F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For better inter-frame consistency, we further introduce a learn- able temporal filtering [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' It receives a sequence of adja- cent audio features [Fi−w, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' , Fi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' , Fi+w] with w = 8 as input, and computes the final smoothed audio feature for the i-th frame as a ∈ RDA in a self-attention-based learn- ing manner, where DA denotes the audio feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' By encoding the audio information, we bridge the modality gap between the audio signals and the visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Introducing such smooth audio features as a condition, we extend the diffusion model for temporal coherence-aware modeling of face dynamics when talking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The objective is then formulated as: LA := Ez,ϵ∼N (0,1),a,t � ∥ϵ − M (zt, t, a)∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' (2) Identity-Preserving Model Generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In addi- tion to learning the audio-to-lip translation, another essen- tial task is to realize the model generalization while pre- serving complete identity information in the source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Generalized identity information includes face appearance, head pose, and image background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To this end, a reference mechanism is designed to empower our model to general- ize to new individuals unseen in training, as shown in Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Specifically, a random face image xr of the source identity is chosen as a reference condition, which contains appearance and background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To prevent train- ing shortcuts, we limit the selection of xr to 60 frames be- yond the target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' However, since the ground-truth face image has a completely different pose from xr, the model is expected to transfer the pose of xr to the target face without any prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' This is somehow an ill-posed problem with no unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For this rea- son, we further incorporate the masked ground-truth im- age xm as another reference condition to provide the target head pose guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The mouth region of xm is completely masked to ensure that the ground truth lip movements are not visible to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this way, the reference xr fo- cuses on affording mouth appearance information, which additionally reduces the training difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Before serv- ing as conditions, xr and xm are also encoded into the la- tent space through the trained image encoder, and we have zr = DI(xr) ∈ Rh×w×3, zm = DI(xm) ∈ Rh×w×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' On this basis, an auxiliary facial landmarks condition is also in- cluded for better control of the face outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Similarly, land- marks in the mouth area are masked to avoid shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The landmark feature l ∈ RDL is obtained with an MLP-based encoder EL, where DL is the landmark feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this way, combining these conditions with audio feature a, we realize the precise control over all key elements of a dynamic talking face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' With C = {a, zr, zm, l} denoting the condition set, the talking head synthesis is finally mod- eled as a conditional denoising process optimized with the following objective: L := Ez,ϵ∼N (0,1),C,t � ∥ϵ − M (zt, t, C)∥2 2 � , (3) where the network parameters of M, EA and EL are jointly optimized via this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Conditioning Mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Based on the modeling of the conditional denoising process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 3, we pass these conditions C to the network in the manner shown in Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Specifically, following [30], we implement the UNet- based backbone M with the cross-attention mechanism for better multimodality learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The spatially aligned refer- ences zr and zm are concatenated channel-wise with the noisy map zT to produce a joint visual condition Cv = [zT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' zm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' zr] ∈ Rh×w×9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Cv is fed to the first layer of the network to directly guide the output face in an image-to- image translation fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Additionally, the driven-audio feature a and the landmark representation l are concatenated into a latent condition Cl = [a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' l] ∈ RDA+DL, which serves as the key and value for the intermediate cross-attention layers of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To this extent, all condition information C = {Cv, Cl} are properly integrated into the denoising network M to guide the talking head generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 4 DDIM-based Denoising ������,1 Random ������,1 ���1 ������,2 ������,2 ���2 ������ DDIM-based Denoising DDIM-based Denoising ������,��� … ������,��� Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Illustration of the designed progressive inference strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For the first frame, the setting of the visual condition Cv remains the same as for training, where xr,1 is a random face im- age from the target identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Subsequently, the synthetic image ˜xi is employed as the reference condition xr,i+1 for the next frame to enhance the temporal coherence of the generated video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Higher-Resolution Talking Head Synthesis Our pro- posed DiffTalk can also be gracefully extended for higher- resolution talking head synthesis with negligible extra com- putational cost and faithful reconstruction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Specif- ically, considering the trade-off between the perceptual loss and the compression rate, for training images of size 256 × 256 × 3, we set the downsampling factor as f = 4 and obtain the latent space of 64 × 64 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Furthermore, for higher-resolution generation of 512 × 512 × 3, we just need to adjust the paired image encoder EI and decoder DI with a bigger downsampling factor f = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Then the trained encoder is frozen and employed to transfer the training pro- cess to a 64 × 64 × 3 latent space as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' This helps to relieve the pressure on insufficient resources, and therefore our model can be gracefully improved for higher-resolution talking head video synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Progressive Inference We perform inference with Denoising Diffusion Implicit Model-based (DDIM) [38] iterative denoising steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' DDIM is a variant of the standard DM to accelerate sampling for more efficient synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To further boost the coherence of the generated talking videos, we develop a progressive ref- erence strategy in the reference process as shown in Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Specifically, when rendering a talking video sequence with the trained model, for the first frame, the setting of the visual condition Cv remains the same as for training, where xr,1 is a random face image from the target identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Sub- sequently, this synthetic face image is exploited as the xr for the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this way, image details between adja- cent frames remain consistent, resulting in a smoother tran- sition between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' It is worth noting that this strategy is not used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Since the difference between adja- cent frames is small, we need to eliminate such references to avoid learning shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 0 100 0 100 GT w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Smooth w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Smooth Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Ablation study on the audio smoothing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' We show the differences between adjacent frames as heatmaps for bet- ter visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The results without audio filtering present obvi- ous high heat values in the mouth region, which indicates the jitters in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' By contrast, with smooth audio as the condition, the generated video frames show smoother transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Experimental Settings Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' To train the audio-driven diffusion model, an audio-visual dataset HDTF [51] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' It contains 16 hours of talking videos in 720P or 1080P from more than 300 identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' We randomly select 100 videos with the length of about 5 hours for training, while the remaining data serve as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Apart from this public dataset, we also use some other videos for cross-dataset evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' We evaluate our proposed method through vi- sual results coupled with quantitative indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' PSNR (↑), SSIM (↑) [45] and LPIPS (↓) [49] are three metrics for assessing image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The LPIPS is a learning-based perceptual similarity measure that is more in line with hu- man perception, we therefore recommend this metric as a more objective indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The SyncNet score (Offset↓ / Confidence↑) [8] checks the audio-visual synchronization quality, which is important for the audio-driven talking head generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' (‘↓’ indicates that the lower the better, while ‘↑’ means that the higher the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=') Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' We resize the input image to 256 × 256 for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The downsampling factor f is set as 4, so the latent space is 64 × 64 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For training the model for higher resolution synthesis, the input is resized to 512 × 512 with f = 8 to keep the same size of latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The length of the denoising step T is set as 200 for both the 5 Ground Truth A A + L A + L + R A + M A + L + M + R ID 1 ID 2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Ablation study on the design of the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The marks above these images refer to the following meanings, ‘A’: Audio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' ‘L’: Landmark;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' ‘R’: Random reference image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' ‘M’: Masked ground-truth image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' We show the generated results under different condition settings on two test sets, and demonstrate the effectiveness of our final design, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' A+L+M+R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Method PSNR↑ SSIM↑ LPIPS↓ SyncNet↓↑ Test Set A GT 0/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='610 w/o 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='024 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='484 w 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='946 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='024 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='287 Test Set B GT 0/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='553 w/o 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='031 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='197 w 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='031 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='387 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Ablation study to investigate the contribution of the audio smoothing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' ‘w’ indicates the model is trained with the audio features after temporal filtering and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' training and inference process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The feature dimensions are DA = DL = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Our model takes about 15 hours to train on 8 NVIDIA 3090Ti GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Ablation Study Effect of the Smooth Audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this subsection, we in- vestigate the effect of the audio smooth operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Quanti- tative results in Table 1 show that the model equipped with the audio temporal filtering module outperforms the one without smooth audio, especially in the SyncNet score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' We further visualize the differences between adjacent frames as the heatmaps shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The results without audio filtering present obvious high heat values in the mouth re- gion, which indicates the jitters in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' By contrast, with smooth audio as the condition, the generated video frames show smoother transitions, which are reflected in the soft differences of adjacent frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Design of the Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' A major contribution of this work is the ingenious design of the conditions for general and high-fidelity talking head synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In Figure 6, we show the generated results under different condition settings step by step, to demonstrate the superiority of our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Method PSNR↑ SSIM↑ LPIPS↓ SyncNet↓↑ Test Set A GT 4/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='762 w/o 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='946 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='024 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='287 w 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='946 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='023 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='662 Test Set B GT 3/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='947 w/o 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='031 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='387 w 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='030 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='999 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Ablation study on the effect of the progressive inference strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' ‘w/o’ indicates that a random reference image is em- ployed as the condition, and ‘w’ means that the reference is the generated result of the previous frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' With pure audio as the condition, the model fails to gener- alize to new identities, and the faces are not aligned with the background in the inpainting-based inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Adding land- marks as another condition tackles the misalignment prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' A random reference image is further introduced try- ing to provide the identity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Whereas, since the ground-truth face image has a different pose from this ran- dom reference, the model is expected to transfer the pose of reference to the target face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' This greatly increases the diffi- culty of training, leading to hard network convergence, and the identity information is not well learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Using the au- dio and masked ground-truth images as driving factors mit- igates the identity inconsistency and misalignment issues, however the appearance of the mouth can not be learned since this information is not visible to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For this reason, we employ the random reference face and the masked ground-truth image together for dual driving, where the random reference provides the lip appearance message and the masked ground-truth controls the head pose and identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Facial landmarks are also incorporated as a con- dition that helps to model the facial contour better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Results 6 GT ATVG MakeItTalk Wav2Lip Ours DFRF AD-NeRF 3D-based Methods 2D-based Methods Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Visual comparison with some representative 2D-based talking head generation methods ATVGnet [5], MakeitTalk [52] and Wav2Lip [28], and with some recent 3D-based ones AD-NeRF [17] and DFRF [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The results of DFRF are synthesized with the base model without fine-tuning for fair comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' AD-NeRF is trained on these two identities respectively to produce the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' in Figure 6 show the effectiveness of such design in synthe- sizing realism and controllable face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Impact of the Progressive Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Temporal corre- lation inference is developed in this work through the pro- gressive reference strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' We conduct an ablation study in Table 2 to investigate the impact of this design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' ‘w/o’ in- dicates that a random reference image xr is employed, and ‘w’ means that the generated result of the previous frame is chosen as the reference condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' With such progressive inference, the SyncNet scores are further boosted, since the temporal correlation is better modeled and the talking style becomes more coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The LPIPS indicator is also en- hanced with this improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' PSNR tends to give higher scores to blurry images [49], so we recommend LPIPS as a more representative metric for visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Method Comparison Comparison with 2D-based Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In this section, we perform method comparisons with some representative 2D-based talking head generation approaches including the ATVGnet [5], MakeitTalk [52] and Wav2Lip [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Figure 7 visualizes the generated frames of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' It can be seen that the ATVGnet performs generation based on cropped faces with limited image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The MakeItTalk synthesizes plausible talking frames, however the back- ground is wrongly wrapped with the mouth movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' This phenomenon is more observable in the video form result, and greatly affects the visual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Gener- ated talking faces of Wav2Lip appear artifacts in the square boundary centered on the mouth, since the synthesized area 7 Method Test Set A Test Set B General PSNR↑ SSIM↑ LPIPS↓ SyncNet↓↑ PSNR↑ SSIM↑ LPIPS↓ SyncNet↓↑ Method GT 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='979 2/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='924 MakeItTalk [52] 18.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='023 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='246 \x15 DFRF [36] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='068 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='999 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='025 2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='432 FT Req.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Ours 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='024 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='381 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='020 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='639 ✓ Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Comparison with some representative talking head synthesis methods on two test sets as in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The best performance is highlighted in red (1st best) and blue (2nd best).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Our DiffTalk obtains the best PSNR, SSIM, and LPIPS values, and comparable SyncNet scores simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' It is worth noting that the DFRF is fine-tuned on the specific identity to obtain these results, while our method can directly be utilized for generation without further fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' (‘FT Req.’ means that fine-tuning operation is required for the DFRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=') and the original image are not well blended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' By contrast, the proposed DiffTalk generates natural and realistic talk- ing videos with accurate audio-lip synchronization, owing to the crafted conditioning mechanism and stable training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For more objective comparisons, we further eval- uate the quantitative results in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Our DiffTalk far surpasses [28] and [52] in all image quality metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For the audio-visual synchronization metric SyncNet, the pro- posed method reaches a high level and is superior than MakeItTalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Although the DiffTalk is slightly inferior to Wav2Lip on the SyncNet score, it is far better than Wav2Lip in terms of image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In conclusion, our method outper- forms these 2D-based methods under comprehensive con- sideration of the qualitative and quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Comparison with 3D-based Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For more com- prehensive evaluations, we further compare with some recent high-performance 3D-based works including AD- NeRF [17] and DFRF [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' They realize implicitly 3D head modeling with the NeRF technology, so we treat them as generalized 3D-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The visualization results are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' AD-NeRF models the head and torso parts separately, resulting in misalignment in the neck re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' More importantly, it is worth noting that AD-NeRF is a non-general method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In contrast, our method is able to handle unseen identities without further fine-tuning, which is more in line with the practical application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The DFRF relies heavily on the fine-tuning operation for model generalization, and the generated talking faces with only the base model are far from satisfactory as shown in Fig- ure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' More quantitative results in Table 3 also show that our method surpasses [17, 36] on the image quality and audio- visual synchronization indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Expand to Higher Resolution In this section, we perform experiments to demonstrate the capacity of our method on generating higher-resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In Figure 8, we show the synthesis frames of two models (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' (a) is trained on 256 × 256 images with the downsampling factor f = 4, so the latent space is of (a) Resolution: 256 × 256, ���=4 (b) Resolution: 512 × 512, ���=8 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Generated results with higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' size 64 × 64 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' For (b), 512 × 512 images with f = 8 are used for training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Since both models are trained based on a compressed 64 × 63 × 3 latent space, the pressure of insufficient computing resources is relieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' We can therefore comfortably expand our model for higher- resolution generation just as shown in Figure 8, where the synthesis quality in (b) significantly outperforms that in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Conclusion and Discussion In this paper, we have proposed a generalized and high- fidelity talking head synthesis method based on a crafted conditional diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Apart from the audio signal condition to drive the lip motions, we further incorporate reference images as driving factors to model the personal- ized appearance, which enables the learned model to com- fortably generalize across different identities without any further fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Furthermore, our proposed DiffTalk can be gracefully tailored for higher-resolution synthesis with negligible extra computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The proposed method models talking head generation as an iterative denoising process, which needs more time to synthesize a frame compared with most GAN- based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' This is also a common problem of LDM- based works which warrants further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Nonetheless, we have a large speed advantage over most 3D-based meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Since talking head technology may raise potential mis- use issues, we are committed to combating these malicious behaviors and advocate positive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Additionally, researchers who want to use our code will be required to get authorization and add watermarks to the generated videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' Motiondif- fuse: Text-driven human motion generation with diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' arXiv, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 2 [49] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' The unreasonable effectiveness of deep features as a perceptual metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In CVPR, 2018.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' State of the art on monocular 3d face reconstruction, tracking, and applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' In Computer Graphics Forum, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} +page_content=' 2 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfSAcB/content/2301.03786v1.pdf'} diff --git a/DNE0T4oBgHgl3EQfggEA/vector_store/index.faiss b/DNE0T4oBgHgl3EQfggEA/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..592811bf5ccb4b22d8f88faae6fc31c2a34e0eaf --- /dev/null +++ b/DNE0T4oBgHgl3EQfggEA/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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velocity of different arm tracers, taken near the tangent to a spiral arm. A slight +difference is predicted by the density wave theory, given the shock predicted at the entrance to the inner spiral arm. In many of +these spiral arms, the observed velocity offset confirms the prediction of the density wave theory (with a separation between the +maser velocity and the CO gas peak velocity, of about 20 km/s) - when the observed offset is bigger than the error estimates. + +1. +Introduction. + +How to get the precise location of a spiral arm? We could draw a spiral fit through parallax-distance data of +interferometric-based radio masers or optical Gaia DR3 young stars in spiral arms, and a fit to an arm model. Or we could find +the tangents from the sun to tracers inside a spiral arm (galactic longitudes), and a fit to an arm model. Both approaches give the +same locations for spiral arms. Here we will use this second approach. +The arm tangent to a spiral arm is a line from the Sun to that spiral arm, being tangent to the arm (not crossing the +arm). It should be mentioned that the tangent from the Sun to a spiral arm, done several times in different arm tracers, provides a +precise galactic longitude on which to fit an arm model. Such a catalog of over 200 observed arm tangents has been published +[1],[2],[3]. It was found observationally that each arm tracer was offset from each other arm tracer [4]: radio masers near the +inner arm edge, but broad diffuse CO gas peaking at the outer arm edge, and many other tracers peaking in between. +For stars and gas in Galactic quadrants I and IV, one can look tangentially to a spiral arm. Using one arm tracer, a +telescope scan shows a consistent value in Galactic longitude, from one telescope to the next. +While making a telescope drift in galactic longitude, along the disk of the Milky Way galaxy, we can record the intensity +of an arm in a tracer (maser, HII regions, broad diffuse CO gas peaks, etc). When the telescope sweeps across a spiral arm +width, there will be a galactic longitude where the intensity of that arm tracer increases, peaks, and then decreases; so we will +record the galactic longitude where the peak intensity was located in that arm tracer. If possible, the observers also recorded the +radial velocity as observed at that peak. +We published a 4-arm spiral model as fitted to the tangent in broad diffuse CO gas peaking in each spiral arm (see +[5],[6], for the basic equations). A later fit [7] was done, with more data and with improved Galactic parameters: 8.15 kpc for the +Sun’s distance to the Galactic Center [8]. Other arm parameters are the arm pitch angle = 13.1O, and the arms start at 2.2 kpc +from the Galactic Center. Fitting uncertainties have been explained in [7]. The start of the Norma spiral arm in Galactic quadrant I +is thus at ro= 2.2 kpc and at an angle -30o below the horizontal line at the Galactic Center (perpendicular to the sun-to-Galactic +Center line-of-sight).. +This global arm pitch angle was found earlier using a fit of arm segments well over both Galactic quadrants I and IV, +enabling better precision (Table 1 in [9]; Tables 1 and 2 in [10]; Fig. 4 in [7]); small localised pitch deviations along the Galactic +radius are thus smoothed out (see Fig. 1 in [11]). Velocity wise, we took 233 km/s for the circular orbital velocity of the Local +Standard of Rest around the Galactic Center [12]. +For a more complete overview of the arm tangents as related to the Milky Way disk structure, see [6]. +From published arm tracer separations and ages, the relative speed of the gas away from the arm shock front is +estimated near 81 km/s –see [13]; [1]; [7]. In addition, a superposition of the known Galactic magnetic field can be made over the +model spiral arm above, indicating that a counter-clockwise magnetic field covers the Sagittarius arm in Galactic Quadrant I and +the Crux-Centaurus arm in Galactic Quadrant IV; the other arm segments and all other arms have a clockwise Galactic magnetic +field [14]. + + +In this paper, we check the predictions of the density wave theory regarding the speed of some tracers, relative to each +another tracer. Section 2 deals with different radial velocities for different arm tracers. Section 3 deals with Galactic dynamics +and the density wave theory. Section 4 compares the locations of both approaches (parallax, arm tangent). Section 5 shows a +concluding discussion. + +2. +New results. + + +Kinematic velocity of the galactic disk. At arm tangent points, one can observe the radial velocity of specific arm +tracers, such as the broad diffuse CO gas or of the radio masers. Being taken at separate galactic longitudes, their velocities +must differ somewhat. Having gone through a shock front at different times in the density wave, they must respond differently. +We adopt the version of the density wave theory with shocks, in which the gas flow enters the arm at a supersonic +velocity and creates a shock, and later the gas leaves the arm at a subsonic velocity (see fig. 3 in [15]). Going from one arm to +the next, the gas orbit looks like a pointed oval streamline with a sharp bend at each shock location (see fig. 3 in [16]. The orbit +is thus not quite circular around the Galactic Center, as typical excursions in azimuthal and radial velocities are about 20 km/s +(Fig. 12 and Fig.13 in [16]. +Table 1 assembles the line of sight radial velocity values, as observed in some tangents to spiral arms in the Milky +Way galaxy [1]. Also, mean results are shown in Table 1 and Figure 2. The label C is for the broad diffuse CO gas near the +Potential Minimum of the density wave (and other blue tracers nearby, on the outer arm side). The label M is for the Maser data +located near the shock of the density wave (and other orange tracers nearby, on the inner arm side). Both means for label C and +label M are always within 20 km/s of each other. Typical errors bars are ±5 km/s. + + + + +Table 1 – Mean radial velocity of each arm tracer, at each arm tangent (a) + + +Mean tangent Longit.: 283o +310o +328o +338o +346o + +018o +030o +050o + +At Gal. radius (kpc): +8.0 +6.3 +4.5 +3.2 +2.5 + +2.8 +4.2 +6.3 +----------------------------------------------------------------------------------------------------------------------------------- +Chemical + +Vrad in Vrad in Vrad in Vrad in Vrad in +Vrad in Vrad in Vrad in + +Tracer: + +Carina Crux- +Norma Start +Start + +Start +Scutum Sagit- + + + + +arm +Cen- +arm +of +of + +of +arm +tarius + + + + + +taurus +Perseus Sagit- +Norma +arm + + + + +arm + +arm +tarius +arm + + + + + + + +arm + + + +(km/s) (km/s) (km/s) (km/s) (km/s) +(km/s) (km/s) (km/s) +----------------------------------------------------------------------------------------------------------------------------------- +blue group: +12CO at 8’ + +-8.8 +-46.6 +-97.6 +-126.7 -136 + ++125 ++95.0 +55.3 +[CII] at 80’’ + +- +- +-106(d) -120(e) - + +- +- +- + +[CII] at 12” + +- +- +- +- +- + +- ++114(b) - +HI atom + +-9 +-44 +-79 +- +- + +- +- +- + +HII complex + + - +- +- +- +- + +- ++100.0 +61 + +13CO + + +- +-35 +-85 +-115 +- + +- ++95.0 +60 + +--------------------------------------------------------------------------------------------------------------------------------------- +Blue mean radial vel.: +-9±5 -42±5 -92±5 -121±5 +-136±5 +125±5 +101±5 +59±5 +--------------------------------------------------------------------------------------------------------------------------------------- +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + +orange group: +[CII] at 80’’ + +- +- +-99(f) +-127(g - + +- +- +- +[Cii] at 12’’ + +- +- +- +- +- + +- ++115(c) - + +Warm 12CO cores +- +- +- +- +- + +- ++95 ++60 + +Masers + ++10 +-56.5 +-102 +-106.7 -120 + ++105 ++100.8 +65.5 + +--------------------------------------------------------------------------------------------------------------------------------------- +Orange mean radial vel.: ++10±5 -56±5 -101±5 -117±5 -120±5 +105 ±5 +104 ±5 +63 ±5 +--------------------------------------------------------------------------------------------------------------------------------------- +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +--------------------------------------------------------------------------------------------------------------------------------------- +Orange - Blue radial vel.: +19±5 -14 ±5 -9 ±5 +4 ±5 +16±5 -20±5 +3±5 +4 ±5 +Masers – 12CO radial vel ++19±5 -10 ±5 -4 ±5 +20 ±5 +16±5 -20±5 +6 ±5 +10 ±5 +--------------------------------------------------------------------------------------------------------------------------------------- +Notes: +(a): Some data are referenced. All other data from Table 1 in [1], +(b): Table 1 in Velusamy et al [17] for longitudes l=28o to 30o. +(c): Table 1 in Velusamy et al [17] for longitudes l=31o-33o. +(d): Fig.7b in Velusamy et al [18] – ll= 327o – 329o. +(e): Fig.8c in Velusamy et al [18] – l= 334o–336o. +(f): Fig.7a in Velusamy et al [18] – l=329o-331o. +(g): Fig.8b in Velusamy et al [18] – l=336o-338o. + + +Figure 1 shows the results. Arm tangents (a line from the Sun tangentially to the arm) to the spiral arms are observed +in Galactic quadrant I at the Sagittarius arm (near l=050o), at the Scutum arm (near l=030o), at the Norma arm (near l=18o ), and +in Galactic quadrant IV at the Carina arm (near l= 283o), at the Crux-Centaurus arm (near l=310o), at the Norma arm (near +l=328o), at the Perseus start arm (near l=338o), and at the Sagittarius start arm (near l=346o). + + +Figure 1. The view of the Galactic disk, seen from above. Galactic quadrants I, II, III, and IV are indicated. The Sun is +indicated at 8.15 kpc from the Galactic Center. Four spiral arms are shown, each in a different color. In this rendering, +the arm pitch angle = 13.1O, and each arm starts at 2.2 kpc from the Galactic Center. + +Figure 2 shows the results. In Galactic quadrant I, where all tracers should have a positive velocity, the slower M label +should have a slightly smaller value than the faster C label, giving a negative value in the last row of Table 1. The M label is +close to the shock front in the inner arm side, having a general slowing down + + +It +kSun +ca +5 +kiloparsecs +Gal.Ctr. +5 +- +0 +1 +1 +-10 +-5 +0 +5 +10 +kiloparsecsFigure 2. Radial velocity of each spiral arm. The actual speed (vertical axis; in km/s) is given at each Galactic +longitude (horizontal axis, in degrees), while the number shown near each arm is the actual distance (kpc) from the +Sun. The arm tangents are shown by vertical dashed lines (from the bottom or the top). The actual location (radial +velocity, Galactic longitude) of the radio masers (M) or the broad diffuse CO (C) at the arm tangent can be seen. The +orbital circular speed used around the Galactic Center = 233 km/sec. As before, the arm pitch angle = 13.1O, while the +arm starts at 2.2 kpc from the Galactic Center, and the Sun is distanced from the Galactic Center by 8.15 kpc. + +[CII] gas. In addition to masers and diffuse CO gas, we can add the [CII] line observed at a wavelength of 158 +microns: Velusamy et al [17] at 12” resolution around the Scutum spiral arm near l= 30o ; Velusamy et al [18] at 80” resolution +around the Norma arm near l= 328o and the Perseus arm near l= 338o. In these [CII] data, both the on-tangent (near the +maser tangent and the shock) and an off-tangent (near the diffuse CO tangent and the outer arm side) were measured. These +data are also reported in Table 1. + +The [CII] at 80’’ are taken at different galactic longitudes, thus some of which being ‘on-tangent’ and some being ‘off- +tangent’ in their observations - see notes at bottom of table (matching the longitudes of other arm tracers). +Prediction: The density wave theory would predict in Galactic Quadrant IV that the orbiting CO gas (negative speed) +would win over the slower maser (M) speed (as observed in Table 1 for the Carina arm at 283o, the Perseus start arm at 338o, +the Sagittarius start arm at 346o), and predict in Galactic quadrant I that the orbiting CO gas (positive speed) would win over the +slower maser speed (as observed in Table 1 for the Norma arm at 018o). Consequently, requiring all arms with an offset larger +than 3 times the velocity error, then all 4 remaining arms satisfy the prediction of the density wave theory. +It does not seem the case for 4 arms: the Sagittarius arm at l=50o, the Scutum arm at =30o, the Norma arm at 328o, +the Crux-Centaurus arm at 310o; for these 4 arms, the velocity offset is smaller than 3 times the velocity error, so the offset is not +significant. + +3. +Galactic dynamics. + + +Sun (km/s) +Q +Q +V +18 +34 +LSR radial velocity seen at S +115 +M +24 +16 +2 +13 +16 +15 +12 +19 +10 +8 +16 +24 +SagittariusCarina +M +1 +ScutumCrux-Centaurus +N +C +33 +NormaCygnus +7 +23 +Perseus +7 +50 +O +-50It has been observed that each arm tracer is separated from other arm tracers (see [4], [3]), as the orbiting gas flows +through a spiral arm, entering at a shock in the inner arm side, then forming protostars, masers, proto-HII regions, and exiting on +the outer arm side near the Potential Minimum of the density wave [16]. +Recent measured values have been made for the distance of the Sun to the Galactic Center (8.15 kpc; [8]) and the +mean speed of the Local Center of Rest near the Sun (233 km/s; [12]). + +Observed speed for new stars to flee the shock front/inner arm edge. A recent compilation of arm tangents, using +many arm tracers, was provided by [1]; that paper computed an age gradient of 11.3 ± 2 Myr/kpc, or a relative speed of arm +tracers of 87 ± 10 km/s away from the shock front (inner arm edge). A very similar result from two different methods gave 12.0 ± +2 Myr/kpc and 81 ± 10 km/s [13], and 12.9 ± 2 Myr/kpc and 76 ± 10 km/s [7]. Taking here a statistical mean value would then +give 12.1 ± 1 Myr/kpc and 81.3 ± 5 km/s. +This relative speed value would mostly apply at a Galactic radius near 7 kpc, where most masers are located - near the +Sagittarius arm in Galactic quadrant I (see masers in Fig. 1 in [19]). +At the masers’ orbit near 7 kpc of galactic radius, the linear density wave’s pattern speed (shock) would be 233 – +81.3 = 151.7 km/s; thus the angular pattern speed at 7 kpc is 151.7 / 7.0 = 21.7 km/s/kpc. The co-rotation radius would then be +233/ 21.7 = 10.7 ±1 kpc, just beyond the Perseus arm along the Galactic Meridian (Sun to Galactic Center line). The co-rotation +radius is where the gas going at orbital speed equals the linear value of the angular pattern speed. +Figure 3 shows the speeds mentioned for the density wave’s arm pattern, the new starforming masers, and the +orbiting gas around the Galactic Center. + +Figure 3. A typical cross-section of a spiral arm, running over 350 parsecs from the inner arm edge (at right) to the +Potential Minimum (at left). Each arm tracer is separated - see [4], [13]. The Gas flow and old stars, orbiting around a circular +orbit around the Galactic Center at 233 km/s, come from the right of the spiral arm in this rendering. The dust and shock at the +inner arm edge (red zone) helps some of the gas to contract under gravity to form embedded protostars and masers (orange +zone), within 0.5 Myrs of the shock front. These protostars evolve to become newly-formed stars and young HII regions (green +zone). Old broad diffuse CO gas congregates in regions near the density wave’s Potential Minimum (blue zone). The masers +and gas inside the spiral arm goes at a relative speed near 81 km/s away from the density wave’s pattern speed (shock front and +arm pattern speed going at 152 km/s). The sum of these speeds (152 and 81) equals the absolute speed noted earlier (233). + + +The relative speed of the density wave (151.7 km/s) and that of the masers relative to that (an extra 81.3 km/s) means +that the masers are going above the arm’s inner edge by 233/151.7 = 1.53 times faster (Mach 1.5). At this speed, the masers + +Masers racing across a spiral arm +caldifuse120gas +electrons +electrons +to +Cal. +Center +recombination +hot dustlane [MIR] +hot dust lane [NIR] +d stars [NIR] +synchr. +free +气 +thermal +Hativ. +arbitrary units +三 +interarm +imterarm +potential min-> +inner arm edge-> +-200 +233 km/s +81 km/s +152 +km/s +233 +km/s +PIO +stars Flow +Masers +Flow +Arm +pattern +Gas +Flow +-400 +spiral +armwidth +009- +-200 +-100 +100 +200 +300 +400 +500 +Peak position of each spiral arm tracer (po)and young stars would cross the spiral arm, from the Potential Minimum and broad diffuse CO down to the shock and dust lane +(separated by about 350 pc) in a time of 350 pc/ 81.3 km/s = 4.3 Myrs. + +Time to reach the next spiral arm. At the Sun’s orbit near 8.15 kpc of galactic radius, to cover a quarter of a circle +at the mean relative speed from the shock front, one needs 0.25x2x3.14x8150 pc =12 802 pc; at the mean relative speed of +81.3 km/s = 157.5 ±10 Myrs; that is the time for the Earth to experience its passage from one spiral arm to the next arm in the +rotating frame of the spiral arm. +Implications for the density wave. In a relative frame, stars and gas speed away from the inner arm’s shock front at +about 81 km/s [13], [1],[ 7]. +In an absolute frame, stars and gas orbit around the Galactic Center at 233 km/s; so the velocity difference is the linear +pattern speed, being 233 – 81 = 152 km/s. This value of 152 km/s is the linear pattern speed of the density wave, which applies +at the solar orbit, so the required angular pattern speed is 152/ 8.1 = 18.8 km/s/kpc.. + +Near a Galactic radius of 8.1 kpc, that angular pattern speed value near 19 km/s/kpc is in the range suggested by +various authors, e.g. [20], and implies a galactic co-rotation radius of 233/18.8 = 12.4 ±1 kpc. + +To go from one spiral arm to the next (2 x 3.15 x 8.15kpc / 4), at such a relative speed (81 km/s), requires a time period +of 12800 pc / 81 km/s = 158 Myrs (in the rotating frame of the spiral arms, orbiting around the Galactic Center). This would also +be the mean time between two major extinctions on Earth - a difficult number to get (even with a large error bar). During that +time, the arms would have turned 1.87 times in their orbit around the Galactic Center (152 km/s x158 Myrs). If the Sun’s orbit is +not strictly circular but an ellipse, this time value may change somewhat by the amount of the non-zero orbital eccentricity. + +Error bars for these numerical deductions are large and not easily ascertained, but the numerical values from these +recent extinctions here are not far off the numerical values obtained from the Galactic dynamics of the previous section. +A recent statistical analysis of Earth extinctions obtained a period near 176-188 Myrs, arguably due to successive +passages of the Sun and Earth through a Galactic spiral arm ([21]; [22],[23]). Others may differ (± 20 Myrs) - successive +transits of the Earth through successive spiral arms may have seed crust production every 170-200 Myrs [24] through the Oort’s +Cloud around the solar system being perturbed by nearby stars (shooting icy constituents down toward the Sun).. + +4. +Locations of spiral arms – complementarity of arm tangents with radio masers and optical Gaia parallaxes + +The precise determinations of the locations of each spiral arm in our Milky Way galaxy could be done using +precise distance measurement (parallax of objects inside spiral arms) and also by finding the location in galactic longitude of the +tangents from the Sun to tracers inside these spiral arms). These two methods should yield the same results for the locations of +spiral arms, being complementary in essence. +Catalogs of precise parallax measurements of radio masers with precise distances have been published, as well +as a map of arm locations (Fig. 1 in [25]). +Catalogs of precise arm tangents in Galactic longitudes have been published and such arm locations inferred +(Vallée [1] with 205 tangents; Vallée [2] with 107 tangents). They led to drawing the map of arm location (Fig.1 in [1] for nearby +arms; [14] for distant arms). A comparison between the locations of observed arm tangents and a fitted model of the locations of +arm tangents showed a very good fit, as well as a comparison of the arm models from the parallax of radio masers and the arm +models model from the arm tangents (Fig.3 in [7]). +The new Gaia DR3 map (Fig. 14 in [5]) shows young open star clusters at optical wavelengths near spiral arms, +within 4 kpc of the Sun’s location, with the arm locations copied from the locations of radio masers (from Fig.1 in [25]). But these +DR3 young open star clusters are not well aligned with radio masers: thus (1) no new arm fit was done; (2) their distribution is not +continuous as there are observed gaps and discontinuities along an arm, (3) there appear to be different optical widths across a +radio arm; (4) the young open cluster stars in front of the radio masers are not expected there in some theories. + + There is a good complementarity among the locations of spiral arms. Thus all around the Sun we get the same +locations of the spiral arms, either through radio masers (distance) or through arm tangents (longitudes). In addition, the Gaia +DR3 optical parallaxes of young star clusters give the same locations of spiral arms as radio masers, albeit with a much larger +arm width (larger distance errors). +The question of the possible bending of the Perseus arm is mentioned (Section 4 in [26] suggested +2 arms; in contrario, [27] suggested an interarm island near the Perseus arm). Also,the width of each spiral arm is thus not well +defined in Gaia DR3; in contrario, a new multi-tracer approach for defining the spiral arm width was advocated (masers near the +Shock front/inner arm versus broad diffuse broad CO gas near the Potential Minimum/outer arm – see [28]). + + + Close to the Galactic Center, each model must start the spiral arms. Our tangent model starts each spiral arm +near 2.2 kpc away from the Galactic Center (see Section 1 and Figure 1), and this start value may differ from the parallax models +(having larger errors with larger distances). + +5. +Conclusion + +We employed the arm model found recently, as fitted to the arm tangents in galactic longitudes, with 8.1 kpc as the +Sun to Galactic Center distance (Figure 1). +We made an examination of the observed radial velocity of some arm tracers, at the tangent from the Sun to the spiral +arm, yielding the folllowing. Comparing the radial velocity of the Masers at the tangent points, to the same from the broad diffuse +CO gas near the Potential Minimum, the prediction of the density wave theory with shocks seems to be valid (Figure 2; Table 1). + +A similar comparison, this time taking the [CII] observations near the arm tangent (on-site, and slightly off-site) does +not change the statistical results. + + +Our modeling (Figures 1 and 2) can be employed with current estimates of the passage of Earth through a spiral arm. +The ensuing implications are computed for the Galactic angular spiral arm speed (near 22 km/s/kpc) and for the Galactic co- +rotation radius (nearer 11 – 12 kpc). The time to reach the next spiral arm is near 158 Myrs. (Section 3). + + +Acknowledgements. The figure production used the PGPLOT software at the NRC Canada in Victoria. + +Data Availability. All data underlying this article are available in the article (references given below), and / or will be +shared on reasonable request to the Corresponding author. + +Funding. No funds or grants were received during the preparation of this paper. I used the facilities at NRC HAA DAO. + +Declarations: Competing interests. None. + +References +[1] Vallée. J.P. 2022a. Catalog of spiral arm tangents (Galactic longitudes) in the Milky Way, and the age gradient based on various arm tracers +New Astron, 97, 101896 (1-14). +[2] Vallée, J.P. 2014b. Catalog of observed tangents to the spiral arms in the MIlky Wsy galaxy. ApJ Suppl. Ser., 215, 1 (1-9). +[3] Vallée, J.P. 2016a, A substructure inside spiral arms, and a mirror image across the galactic meridian. Astrophys J., +vol.821, art.53, pp.1-12. +[4] Vallée, J.P. 2014a. The spiral arms of the Milky Way: the relative location of each different arm tracer, within a typical spiral arm width. +Astron J., vol. 148, art.5, pp.1-9. +[5] Vallée, J.P. 2008, New velocimetry & revised cartography of the spiral arms in the Milky Way – a consistent symbiosis. Astron.J., v135, p1301-1310. +[6] Vallée, J.P. 2017a. A guided map to the spiral arms in the galactic disk of the Milky Way. Astronomical Review, vol.13, pp.113-146. +[7] Vallée, J.P. 2022b. The observed age gradient in the Milky Way – as a test of spiral arm structure. Ap Space Sci., 367, 26 (1-10). +[8] Abuter, R., Amorim, A. et al 2019. A geometric distance measurement to the Galactic Center black hole with 0.3% uncertainty. A & A , 625, L10. +[9] Vallée, J.P. 2017b. The Norma spiral arm: large-scale pitch angle. Ap Sp. Sci., 362, 173 (5pp). +[10] Vallée, J.P. 2015. Different studies of the global pitch angle of the Milky Way’s spiral arms. MNRAS, 450, 4277-4284. +[11] Vallée, J.P. 2016b. The start of the Sagittarius spiral arm (Sagittarius origin) and the start of the Norma spiral arm (Norma origin): model-computed and +observed arm tangents at Galactic longitudes -20o zh . +(3.23) +7 + +4. Configuration entropy +The configuration entropy (CE) definition is motivated by information theory. In particular in the Shannon +information entropy [58] that for a discrete random variable with probabilities pn of assuming one of n +possible values is defined as: +S = − +� +n +pn log pn . +(4.1) +The Shannon entropy represents a measure of the information content of the random variable. +A definition for the CE was proposed in references [25–27] as a continuous version of Eq. (4.1). For a +one-dimensional system it reads: +SC[f] = − +� +dk f(k) ln f(k) , +(4.2) +being f(k) the so-called modal fraction, usually defined, for a localized physical system, in terms of the +energy density in momentum space, ρ(k), namely +f(k) = +|ρ(k)|2 +|ρ(k)|2max +(4.3) +where |ρ(k)|2 +max is the maximum value assumed by |ρ(k)|2. Instead of the maximum value of the energy +density, one could eventually use +� |ρ(k)|2dk in the denominator of Eq. (4.3). In this alternative definition, +the modal fraction appears as a normalized function, which would be more similar to the Shannon entropy, +but in the continuous case it could lead to negative values for the CE. +It is now known through many examples, as those articles cited in the introduction, that the CE works +as a measure of the stability of physical systems. In a few words, the current interpretation states that +the CE increases as the instability of the system increases. +4.1. +CE definition for the rotating QGP +The Fourier transform of the BH energy density is given by: +�ρ(k, ω) = 1 +2π lim +ϵ→0 +� ∞ +ϵ +dz ρBH(z, ω) eikz , +(4.4) +with ρBH(z, ω) defined by Eq. (3.23), where one notices that the total energy density has two different +expressions, separated by the horizon position. The modal fraction, necessary for building the CE, is +defined in terms of the squared absolute value of �ρ(k, ω). Using (4.4), it can be written as +|�ρ(k, ω)|2 = +� 1 +2π lim +ϵ→0 +� ∞ +ϵ +ρBH(z, ω) cos(kz) dz +�2 ++ +� 1 +2π lim +ϵ→0 +� ∞ +ϵ +ρBH(z, ω) sin(kz) dz +�2 +. +(4.5) +Eq. (4.5) above does possess an analytical solution. So we apply numerical methods in order to determine +the CE. +Using numerical integration, we plot in Figure 1 |�ρ(k, ω)|2 at ¯T = 0.6, where ¯T = T/√c. The IR +parameter √c is determined by hadronic phenomenology – see, for instance, [21]. For other temperatures +the pattern of the curve is similar. +As one can see, |�ρ(k, ω)|2 first reaches the global maximum |�ρ(k)|2 +max, then the curve begins to decrease +as the momentum increases, oscillating smoothly and tending to zero as k → ∞. Moreover, |�ρ(k)|2 +max +gets larger as the rotational speed increases, with a small increase in the value of k where these maxima +8 + +Figure 1: Absolute value of the rotating BH energy density, |�ρ(k)|2, versus momentum at ¯T = 0.6 and +different rotational velocities: ωl = 0.1 (blue), ωl = 0.2 (orange); and ωl = 0.3 (green). +occur. These global maxima are well-defined, and can be easily computed by numerical methods. After +determining the maxima of the absolute value of the energy density in Fourier space (4.5), we are able to +evaluate the configuration entropy of the rotating plasma at different angular velocities and temperatures, +by using the definition (4.2) together with (4.3). +4.2. +Results obtained for the CE +Applying numerical methods, we computed the CE for different values of the dimensionless temperature ˜T +and of the rotational speed ωl. The values obtained are displayed in Table 1, in Appendix A. In order to +show the behaviour of the CE we plot, in Figure 2, the case of fixed temperature ¯T6 = 0.6 as a function of +the rotational speed ωl. One notices that the CE increases monotonically with the speed, indicating that, +for a fixed temperature, the larger is the rotational speed, the more unstable is the plasma. In particular, +one also notices that as ωl approaches the speed of light, ωl → 1, the CE diverges. This behaviour is +present for all the analyzed temperatures, as can be seen in Figure 3, where we plot the CE for four +different temperatures. +The asymptotic singular behaviour in the ωl → 1 limit can be understood in a simple way. Looking +at the expression (2.12) for the Hawking temperature of a rotating black hole, one notices that, for a +fixed temperature, as the rotational speed increases, the horizon position zh decreases. The limit ωl → 1 +corresponds to zh → 0. On the other hand, it is known that an increase in the CE is associated with an +increase in the instability of the physical system. In the present case the instability corresponds to the +contraction of the BH dimensions in the limit ωl → 1. +From the point of view of AdS/QCD duality, this contraction of the BH corresponds to the contraction +of the quark-gluon plasma. In the limit ωl → 1 the volume of plasma goes to zero. This is the reason why +the CE diverges. This result is consistent with a result found recently in [32], where it was shown that +9 + +[P(k)12 +80 +wl= 0.1 +wl= 0.2 +60 +wl= 0.3 +40 +20 +k +10 +20 +30 +40 +50 +60the CE of bottomonium quasi-states becomes singular when the temperature or the magnetic field reach +values such that the quasi-states completely dissociate in the medium. In other words, the disappearance +of the bottomnium in the medium (associated with the deconfinement of the quarks) is translated by the +CE as an infinite instability. +Figure 2: Configuration entropy of rotating QGP as a function of the rotational speed (ωl) at ¯T6 = 0.6. +In order to have a clear understanding about the variation of the CE with temperature and rotational +speed, we plot in Fig. 4 the CE as a function of ˜T, at three different values of ωl. One notices that for +a fixed rotational speed, the CE increases with the temperature. This increase in the CE is associated +to the increase in the thermodynamic instability caused by Hawking radiation. Black holes at higher +temperatures are subject to a stronger loss of energy as a consequence of Hawking radiation. These results +are consistent with the interpretation of the CE as an indicator of the stability of physical systems. +5. Conclusions +We investigated the dependence of the configuration entropy on the rotational speed for a quark-gluon +plasma with cylindrical symmetry using a holographic model. The plasma was represented by the grand +canonical ensemble (with null chemical potential). In this scenario, we obtained an expression for the +energy density of the rotating AdS black hole, dual to the plasma, that was applied to the calculation +of the CE. The dependence of this quantity on the rotational speed of the black hole was studied for +different temperatures using numerical methods. +The current interpretation of the configuration entropy states that it works as a measure of the stability +of physical systems. In short, the CE increases as the instability increases [25–27]. The result found in +section 4 – the CE increases with the rotational speed ωl – is consistent with this interpretation. Rotation +of the plasma implies a Lorentz type of contraction. So that the plasma becomes smaller in volume as ωl +increases. The limit ωl → 1 would correspond to a plasma of zero volume that consistently corresponds +10 + +CE(T6, wl) +150 +100 +50 +[m +0.2 +0.4 +0.6 +0.8 +1.0Figure 3: CE of rotating QGP as a function of ωl, at different temperatures: ¯T3 = 0.3 (blue), ¯T4 = 0.4 +(orange), ¯T6 = 0.6 (green), and ¯T8 = 0.8 (red). +Figure 4: Configuration entropy of rotating QGP as a function of ¯T, at different rotational velocities: +ωl = 0 (blue), ωl = 0.4 (orange); and ωl = 0.6 (green). +to a positive singularity in the CE indicating a “maximum instability”. +It is interesting to relate this result with a similar singular limit of the CE recently found in [32]. In +11 + +CE(T, wl) +300 +250 +200 +:0.3 += 0.4 +150 += 0.6 +100 += 0.8 +50 +[ +0.2 +0.4 +0.6 +0.8 +1.0CE(T) +80 +60 +wl= 0.0 +wl= 0.4 +40 +9'0 =1m +20 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9this article the CE for bottomonium quasi-states was calculated and it was found that it becomes singular +in the limits when the temperature or the magnetic field approach values such that the quasi-states +completely dissociate in the medium. The disappearance of the bottomonium quasi-states, associated with +the deconfinement of the heavy quarks is translated by the CE as an infinite instability, corresponding to +a positive infinite CE. Here we found the similar result that in the limit when the volume of the plasma +goes to zero, that would correspond to the disappearance of the plasma, the CE goes to (positive) infinity. +The combined effect of temperature and rotation was already investigated. It was found in [17] that, +for a non-rotating plasma, the CE increases with the temperature indicating the instability caused by the +evaporation of the black hole via Hawking radiation. Here we have shown that for a plasma with a fixed +non-vanishing rotational speed the CE also increases in a monotonic way with temperature. +12 + +Appendix +A. +CE(T, ω) +X +¯T3 = 0.3 +¯T4 = 0.4 +¯T5 = 0.5 +¯T6 = 0.6 +¯T7 = 0.7 +¯T8 = 0.8 +¯T9 = 0.9 +ωl = 0.0 +17.3177 +27.1699 +35.7645 +43.5432 +50.9320 +57.9978 +64.8159 +ωl = 0.1 +17.4737 +27.3514 +35.9698 +43.7748 +51.1908 +57.9978 +65.1263 +ωl = 0.2 +17.9544 +27.9103 +36.6029 +44.4887 +51.9886 +59.1616 +66.0833 +ωl = 0.3 +18.7997 +28.8953 +37.7188 +45.7479 +53.3956 +60.7113 +67.7703 +ωl = 0.4 +20.0886 +30.3998 +39.4261 +47.6759 +55.5495 +63.0780 +70.3519 +ωl = 0.5 +21.9659 +32.5914 +41.9220 +50.4976 +58.7014 +66.5529 +74.1494 +ωl = 0.6 +24.7137 +35.7812 +45.5766 +54.6347 +63.3233 +71.6411 +79.6887 +ωl = 0.7 +28.8948 +40.6155 +51.1591 +60.9631 +70.3972 +79.4415 +88.1499 +ωl = 0.8 +35.8003 +48.7337 +60.6088 +71.6941 +82.3997 +92.6117 +102.292 +ωl = 0.9 +50.3687 +66.3365 +81.2459 +95.0799 +108.213 +120.801 +133.186 +ωl = 0.91 +52.9314 +69.4661 +84.9202 +99.1757 +112.752 +125.940 +138.910 +ωl = 0.92 +55.9181 +73.1201 +89.2013 +103.917 +118.083 +131.964 +145.649 +ωl = 0.93 +59.4664 +77.4686 +94.2683 +109.542 +124.557 +139.232 +153.798 +ωl = 0.94 +63.7885 +82.7700 +100.376 +116.435 +132.489 +148.253 +163.931 +ωl = 0.95 +69.2319 +89.4363 +107.993 +125.228 +142.681 +159.877 +177.005 +ωl = 0.96 +76.4179 +98.1476 +118.090 +137.079 +156.494 +175.663 +194.774 +ωl = 0.97 +86.5942 +110.284 +132.700 +154.408 +176.772 +198.872 +220.913 +ωl = 0.98 +102.651 +129.994 +157.074 +237.976 +210.908 +237.976 +264.957 +ωl = 0.99 +136.231 +173.956 +212.166 +249.391 +288.362 +326.702 +364.847 +ωl = 0.995 +182.902 +236.174 +290.410 +342.816 +398.298 +452.576 +506.449 +Table 1: Configuration entropy of rotating QGP at different temperatures ( ¯T = T/√c) and rotational +velocities (ωl). +Acknowledgments: The authors are supported by FAPERJ — Funda¸c˜ao Carlos Chagas Filho de +Amparo `a Pesquisa do Estado do Rio de Janeiro, CNPq - Conselho Nacional de Desenvolvimento Cient´ıfico +e Tecnol´ogico. 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D 74 (2006) 015005. +17 + diff --git a/TdAzT4oBgHgl3EQfXvwu/content/tmp_files/load_file.txt b/TdAzT4oBgHgl3EQfXvwu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33dd894dfe68007708d754ef8e4793b38e707ea9 --- /dev/null +++ b/TdAzT4oBgHgl3EQfXvwu/content/tmp_files/load_file.txt @@ -0,0 +1,1080 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf,len=1079 +page_content='Configuration entropy of a rotating quark-gluon plasma from holography Nelson R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Bragaa∗, Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Ferreirab†, Octavio C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Junqueiraa‡ a UFRJ — Universidade Federal do Rio de Janeiro, Instituto de F´ısica, Caixa Postal 68528, Rio de Janeiro, Brasil b Instituto de F´ısica y Astronomia , Universidad de Valparaiso, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Gran Bretana 1111, Valparaiso, Chile The configuration entropy (CE) provides a measure of the stability of physical systems that are spatially localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' An increase in the CE is associated with an increase in the instability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In this work we apply a recently developed holographic description of a rotating plasma, in order to investigate the behaviour of the CE when the plasma has angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Considering the holographic dual to the plasma, namely a rotating AdS black hole, the CE is computed at different rotational speeds and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The result obtained shows not only an increase with the rotational speed v but, in particular, a divergence of the CE as v approaches the speed of light: v → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' We discuss the results obtained showing that they are consistent with the change in the geometry of the black hole caused by the rotation and the corresponding variation of the volume of the dual plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' We also connect the results found here with those obtained in a recent work, where it was shown that the complete dissociation of heavy mesons in a plasma is represented by a positive singularity in the CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Introduction In the last decades, one of the focus of the community devoted to the study of strong interactions physics is the search for understanding the Quark-Gluon Plasma (QGP), a state of matter formed by deconfined partons that interact strongly [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The QGP is formed experimentally through ultra-relativistic collisions of heavy nuclei produced in particle accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' There are many properties that affect the behaviour of QGP, like temperature, density and the presence of magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Another, less studied, property is the rotation of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' When the collision is non-central, the resulting system acquires not only a strong magnetic field but also angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' A fraction of this angular momentum can be transferred to the polarization of the strange quarks in the QGP due to spin-orbit interaction, as discussed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' ∗ braga@if.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='ufrj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='br † lffaulhaber@mgail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='com ‡ octavioj@pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='if.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='ufrj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='br 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='01322v1 [hep-th] 3 Jan 2023 Until recently, no experimental signals of this effect had been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Only in 2017, the global hyperon polarization has been observed in Au + Au collisions at RHIC [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This discovery opened a new window to study the properties of rotating Quantum Chromodynamics (QCD) matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In particular, the influence of the rotation on the QCD phase diagram have become a topic under active investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Some phenomenological models have recently been applied to the study of the effect of rotation in the QGP as, for example, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' [4–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In particular, using the AdS/QCD approach [11,17], it was found that the critical temperature of the confinement/deconfinement transition decreases with increasing angular velocity, which is in agreement with others phenomenological models as the Nambu-Jona Lasinio (NJL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In contrast, the simulations of relativistic rotation on the confinement/deconfinement phase transition in gluodynamics lattice [13,14] showed that the critical temperature increases with increasing angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Here we are interested in studying how rotation affects the stability of the quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' So, it is important to make it clear what does one mean by stability in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' There are two aspects to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The first is the fact that the plasma phase of QCD matter exists, or is stable, for temperatures above some critical value Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' For lower temperatures, QCD matter is in the confined phase [21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The other instability is related to the Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The higher is the temperature of the black hole dual to the plasma, the stronger is the emission of radiation and the corresponding loss of energy, resulting in an increase in the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' A rotating plasma with uniform rotational speed is described holographically by a rotating black hole with cylindrical symmetry, like those studied in [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The rotation is obtained by a coordinate transformation and the holographic model obtained predicts that plasma rotation affects the critical temperature of confinement/deconfinement transition [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This fact motivated the present study concerning the stability of the rotating plasma, but now considering the configuration entropy (CE) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The idea of using the CE as an indicator of stability of physical systems started in [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Afterwards many examples appeared in the literature where the CE plays the role of representing stability in different physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' For instance, in the context of AdS/QCD approach, the CE has provided new results involving thermal behaviour of quarkonium in a medium [28–32], the mass spectra of several particles [33–39], nuclear electromagnetic transitions [40] and in confinement/deconfinement transition [41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In particular, in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' [32] it was shown how does the CE marks the total dissociation of quasi-particles in a medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Namely, the CE diverges when the particles are completely dissociated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Other applications in astrophysics, cosmology, AdS/CFT correspondence, nuclear physics and field theory can be found, for example, in references [44–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The CE is based on the Shannon information entropy [58], in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' It is in general defined in terms of the energy density of the physical state in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Assuming that the black hole is represented by the grand canonical ensemble, which is consistent with the fact that the plasma is rotating, one can find an expression for the rotating black hole energy density as a function of the holographic coordinate, the temperature and the rotational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The Fourier transform does not possess an analytical solution, so that the results in this work were obtained through the application of numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The results obtained show an interesting nontrivial aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' For a fixed temperature, the CE diverges as the rotational speed of the black hole approaches the asymptotic limit corresponding to the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' As we will discuss, there is a consistent interpretation for this behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This work is organized as follows: in Section 2, we describe the geometry of a rotating cylindrical black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Section 3 is devoted to the construction of the rotating plasma energy density, in the soft wall AdS/QCD model, assuming that the system is represented by the grand canonical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In Section 4, 2 we compute the CE of the rotating plasma at different rotational velocities and temperatures, and relate the results with the stability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Section 5 contains our conclusions and the appendix shows a table with additional results for the CE as a function of the temperature and rotational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' AdS black holes with cylindrical symmetry At finite temperature, the anti-de Sitter space with radius L possesses two solutions given by the following metrics with compact time direction: ds2 = L2 z2 � dt2 + d−→x 2 + dz2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1) and ds2 = L2 z2 � f(z)dt2 + d−→x 2 + dz2 f(z) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2) with f(z) = 1 − z4/z4 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The first geometry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1) corresponds to the thermal AdS space, while the second one (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2) to the AdS black hole (BH) geometry, being zh the location of the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Both geometris are solutions of Einstein’s equations with negative cosmological constant Λ = −12/L2 and constant curvature R = −20/L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The time coordinate of the BH geometry has a period β, related to the horizon position and to the Hawking temperature, T = 1/β = 1/(πzh) [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Requiring that the asymptotic limits of the two geometries at z = ϵ, with ϵ → 0, are the same, one finds that the period of the time component of the thermal AdS space is β′ = πzh � f(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Rotating cylindrical black hole In order to represent a QGP rotating with homogeneous speed, we take the metrics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2) in cylindrical coordinates ds2 = L2 z2 � −dt2 + l2dφ2 + 2 � i=1 dx2 i + dz2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3) and ds2 = L2 z2 � −f(z)dt2 + l2dφ2 + 2 � i=1 dx2 i + dz2 f(z) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4) where l is the radius of a hyper-cylinder and 0 ≤ φ ≤ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Note that these new metrics have a topology that is different from those of metrics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2) since one of the spatial coordinates has been compactfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Then, we perform a coordinate transformation to an observer for which the angular coordinate is varying uniformly with time around a cylinder with radius l, namely, t → 1 √ 1 − l2ω2 � t + l2ωφ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5) φ → 1 √ 1 − l2ω2 (φ + ωt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6) After a straightforward calculation, one obtains the rotating cylindrical black hole metric in canonical form: ds2 = −N(z)dt2 + L2 z2 dz2 f(z) + R(z) (dφ + P(z)dt)2 + L2 z2 2 � i−1 dx2 i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7) 3 with N(z) = L2 z2 f(z)(1 − ω2l2) 1 − f(z)ω2l2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8) R(z) = L2 z2 � γ2l2 − f(z)γ2ω2l4� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9) P(z) = ω(1 − f(z)) 1 − f(z)ω2l2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='10) where ω is the angular velocity of the rotating cylindrical black hole, being γ the Lorentz factor, γ(ωl) = 1 √ 1 − l2ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='11) This metric represents, via holography, a plasma that rotates with the same angular velocity ω of the rotating black hole, around a cylinder with radius l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' As shown in [17], the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7) is a solution of the same Einstein equation satisfied by the AdS black hole metrics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' For the cylindrical thermal AdS space of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3), the rotating version of the metric has the same form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7) but using f(z) = 1 in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' It is important to remark that in the plasma formed in heavy ion collisions the rotational speed is not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The QGP formed in accelerators clearly does not have a cylindrical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' What we are doing here is considering a simpler situation where the speed is the same for all the parts of the plasma in order to understand the qualitative effects of the plasma rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This approach has some similarity with the one used for studying the effect of magnetic fields on the plasma, where in general it is considered that the field is uniform [32,60–70], although the actual fields acting on the QGP formed in non central heavy ion collisions are not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Then, defining h00 = −N(z), the Hawking temperature can be obtained from the surface gravity formula [12]: T = |κG 2π | = ���� limz→zh −1 2 � gzz −h00(z)h00,z 2π ���� = 1 πzh � 1 − ω2l2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='12) where κG is the surface gravity, and gzz the zz component of the inverse of the cylindrical black hole metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The expression of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='12) shows that the rotation affects the relation between the Hawking temperature T of the black hole and its horizon position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This property will be important to understand the relation between the dynamics of the rotating BH geometry, the stability of the plasma and its configuration entropy at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Energy density in the soft wall AdS/QCD model The soft wall holographic AdS/QCD model [71] is built from the introduction in the AdS geometry of a dilaton background Φ(z) = cz2, where c is a parameter with dimension of energy squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In the gauge theory side of the gauge/gravity duality √c plays the role of an infrared energy (IR) cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Regularized action density One can write the five-dimensional gravitational action, at zero temperature, in the general form [21,22] I = − 1 2κ2 � zf 0 dz � d4x√ge−Φ (R − Λ) = 4 L2κ2 � zf 0 dz � d4x√ge−cz2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1) 4 where κ is the gravitational coupling associated with the Newton constant, √c is the IR energy parameter and Λ the cosmological constant, related to the curvature as Λ = 3 5R = −12 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The integral in z has an upper limit, that we named as zf, that is equal to zh for the black hole case and to z → ∞ for the thermal AdS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In order to compute the total energy density in the grand canonical ensemble and perform the analysis of the Hawking-Page transition, we must determine the on-shell regularized action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' For the rotating metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7), the determinant of gµν is g = L10 z10 , so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1) becomes Ion-shell = 4L3 κ2 � zf 0 dz � d4xz−5e−cz2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2) Since the integration over the spatial bulk coordinates is trivial and only contributes with a volume factor, V3D, we now define an action density E = 1 V3D Ion-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' For a compact time direction in the Euclidean signature, we have 0 ≤ t < βs, where βs depends on the space considered and the general form of the action density is then given by the expression Es(ε) = 4L3 κ2 � βs 0 dt � zf ε dz z−5e−cz2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3) where ε is an ultraviolet (UV) regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' As mentioned before, for the black hole geometry, βBH = 1/T, while for the thermal AdS one, βAdS = � f(ϵ)βBH = � f(ϵ)/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' We must introduce the UV regulator because the black hole and thermal AdS space possesses infinite action densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In order to eliminate these divergencies, one defines the regularized free energy density of the rotating black hole as the difference between the energy densities of the two geometries, △ E(ε) = lim ε→0 [EBH(ε) − EAdS(ε)] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4) where EBH(ε) = 4L3 κ2 β � zh ε dz z−5e−cz2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5) EAdS(ε) = 4L3 κ2 � f(ϵ) β � ∞ ε dz z−5e−cz2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6) From the regularized action density (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4), one can determine the energy density of the rotating BH in the grand canonical ensemble, taking into account the contribution of the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' From gauge/gravity duality, it will correspond to the energy density of the rotating plasma with cylindrical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Total energy density in the grand canonical ensemble The first step to compute the configuration entropy of the plasma at different temperatures and angular velocities is to determine the total energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' First we consider the total energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Due to rotation, we should assume that the system is represented by the grand canonical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In this case, the total energy, with zero chemical potential, is defined by E = −∂ log Z ∂β + ωJ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7) 5 where Z is the partition function, and J, the angular momentum, which in turn is defined by J = 1 β ∂ log Z ∂ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8) In the semiclassical Hawking-Page approach, log Z = −I, such that E = ∂I ∂β + ωJ , with J = 1 β ∂I ∂ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9) As it was done in the previous subsection, we factorize the trivial three dimensional volume in the bulk spatial variables V3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This corresponds to replacing in the previous Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' the action I by ∆ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Explicitly: U ≡ E V3D = ∂ △ E ∂β + ω 1 β ∂ △ E ∂ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='10) Now we define an energy density in the holographic coordinate z , ρBH(z, T, ω), by: � ∞ 0 ρBH(z, T, ω) dz = U(T, ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='11) Thus, using equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6), together with β � f(ϵ) = β − π4ϵ4 2β3(1 − ω2l2)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='12) and introducing the coordinate u = z/zh, one can write △E = 4L3 κ2 � �β � 1 ϵπ/(β √ 1−ω2l2) du u−5 � β √ 1 − ω2l2 π �−4 e−c( uβ √ 1−ω2l2 π )2 − � β − π4ϵ4 2β3(1 − ω2l2)2 � � ∞ ϵπ/(β √ 1−ω2l2) du u−5 � β √ 1 − ω2l2 π �−4 e−c( uβ √ 1−ω2l2 π )2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='13) Taking the derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='13) with respect to β, and eliminating terms O(ϵ) by taking the ultraviolet limit ϵ → 0, the first term of the total energy, U1 = ∂ △ E/∂β in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='10), reads U1 = 4L3 κ2 � − 7ϵ4 2z8 h � 1 ϵπ/(β √ 1−ω2l2) du u−5e−c( uβ √ 1−ω2l2 π )2 + 1 2z4 h + 3 z4 h � ∞ 1 du u−5e−c( uβ √ 1−ω2l2 π )2 + 2c z2 h � ∞ 1 du u−3e−c( uβ √ 1−ω2l2 π )2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='14) The first two terms of the equation above can be rewritten in the form − 7ϵ4 2z8 h � 1 ϵπ/(β √ 1−ω2l2) du u−5e−c( uβ √ 1−ω2l2 π )2 + 1 2z4 h = − 3 8z4 h , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='15) then, using the following Gaussian integration, lim ϵ→0 � − 3√c 4z3 h √π � ∞ ϵzh e−c(uzh)2 du � = − 3 8z4 h , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='16) 6 and returning to the holographic z-variable, one obtains U1 = lim ϵ→0 �� zh ϵ ρ(1) 1 (z) dz + � ∞ zh ρ(2) 1 (z) dz � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='17) with ρ(1) 1 (z) = −4L3 κ2 3√c 4z4 h √πe−cz2 , if ϵ ≤ z ≤ zh , ρ(2) 1 (z) = 4L3 κ2 � − 3√c 4z4 h √π + 3 z5 + 2c z3 � e−cz2 , if z ≥ zh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='18) In order to compute the second term of the total energy, U2 = 1 β ∂△E ∂ω , we must work out the expression ∂ △ E ∂ω = −4L3 κ2 � 4ϵ4l2ω z8 h(1 − l2ω2) � 1 ϵ/zh du u−5e−c(uzh)2 − l2ω 2z4 h(1 − l2ω2) − 4l2ω z4 h(1 − l2ω2) � ∞ 1 du u−5e−c(uzh)2 − 2cl2ω z2 h(1 − l2ω2) � ∞ 1 du u−3e−c(uzh)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='19) Similarly to the previous calculation to determine U1, by using the expression l2ω 2z4 h(1 − l2ω2) = l2ω√c z3 h √π(1 − l2ω2) � ∞ ϵ/zh e−c(uzh)2du , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='20) and returning to z-variable, the contribution of the angular momentum to the rotating black hole total energy yields U2(ω) = lim ϵ→0 �� zh ϵ ρ(1) 2 (z, ω) dz + � ∞ zh ρ(2) 2 (z, ω) dz � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='21) whereby ρ(1) 2 (z, ω) = −4L3 κ2 l2ω2γ2√c z4 h √π e−cz2 , if ϵ ≤ z ≤ zh , ρ(2) 2 (z, ω) = 4L3 κ2 l2ω2γ2 � − √c z4 h √π + 4 z5 + 2c z3 � e−cz2 , if z ≥ zh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='22) Finally, one concludes that the total energy density of the cylindrical rotating BH in the grand canonical ensemble, obtained by adding the partial densities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='22) is given by the expression ρBH(z, ω) = � � � � � � � � � � � � � � � � � � � −4L3 κ2 √c z4 h √π � 3 4 + l2ω2γ2� e−cz2 , if ϵ ≤ z ≤ zh , 4L3 κ2 � − √c z4 h √π � 3 4 + l2ω2γ2� + 1 z5 �3 + 4l2ω2γ2� + 2c z3 �1 + l2ω2γ2� � e−cz2 , if z > zh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='23) 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Configuration entropy The configuration entropy (CE) definition is motivated by information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In particular in the Shannon information entropy [58] that for a discrete random variable with probabilities pn of assuming one of n possible values is defined as: S = − � n pn log pn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1) The Shannon entropy represents a measure of the information content of the random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' A definition for the CE was proposed in references [25–27] as a continuous version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' For a one-dimensional system it reads: SC[f] = − � dk f(k) ln f(k) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2) being f(k) the so-called modal fraction, usually defined, for a localized physical system, in terms of the energy density in momentum space, ρ(k), namely f(k) = |ρ(k)|2 |ρ(k)|2max (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3) where |ρ(k)|2 max is the maximum value assumed by |ρ(k)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Instead of the maximum value of the energy density, one could eventually use � |ρ(k)|2dk in the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In this alternative definition, the modal fraction appears as a normalized function, which would be more similar to the Shannon entropy, but in the continuous case it could lead to negative values for the CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' It is now known through many examples, as those articles cited in the introduction, that the CE works as a measure of the stability of physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In a few words, the current interpretation states that the CE increases as the instability of the system increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' CE definition for the rotating QGP The Fourier transform of the BH energy density is given by: �ρ(k, ω) = 1 2π lim ϵ→0 � ∞ ϵ dz ρBH(z, ω) eikz , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4) with ρBH(z, ω) defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='23), where one notices that the total energy density has two different expressions, separated by the horizon position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The modal fraction, necessary for building the CE, is defined in terms of the squared absolute value of �ρ(k, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4), it can be written as |�ρ(k, ω)|2 = � 1 2π lim ϵ→0 � ∞ ϵ ρBH(z, ω) cos(kz) dz �2 + � 1 2π lim ϵ→0 � ∞ ϵ ρBH(z, ω) sin(kz) dz �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5) above does possess an analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' So we apply numerical methods in order to determine the CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Using numerical integration, we plot in Figure 1 |�ρ(k, ω)|2 at ¯T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6, where ¯T = T/√c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The IR parameter √c is determined by hadronic phenomenology – see, for instance, [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' For other temperatures the pattern of the curve is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' As one can see, |�ρ(k, ω)|2 first reaches the global maximum |�ρ(k)|2 max, then the curve begins to decrease as the momentum increases, oscillating smoothly and tending to zero as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Moreover, |�ρ(k)|2 max gets larger as the rotational speed increases, with a small increase in the value of k where these maxima 8 Figure 1: Absolute value of the rotating BH energy density, |�ρ(k)|2, versus momentum at ¯T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 and different rotational velocities: ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1 (blue), ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2 (orange);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' and ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' These global maxima are well-defined, and can be easily computed by numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' After determining the maxima of the absolute value of the energy density in Fourier space (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5), we are able to evaluate the configuration entropy of the rotating plasma at different angular velocities and temperatures, by using the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2) together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Results obtained for the CE Applying numerical methods, we computed the CE for different values of the dimensionless temperature ˜T and of the rotational speed ωl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The values obtained are displayed in Table 1, in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In order to show the behaviour of the CE we plot, in Figure 2, the case of fixed temperature ¯T6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 as a function of the rotational speed ωl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' One notices that the CE increases monotonically with the speed, indicating that, for a fixed temperature, the larger is the rotational speed, the more unstable is the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In particular, one also notices that as ωl approaches the speed of light, ωl → 1, the CE diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This behaviour is present for all the analyzed temperatures, as can be seen in Figure 3, where we plot the CE for four different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The asymptotic singular behaviour in the ωl → 1 limit can be understood in a simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Looking at the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='12) for the Hawking temperature of a rotating black hole, one notices that, for a fixed temperature, as the rotational speed increases, the horizon position zh decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The limit ωl → 1 corresponds to zh → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' On the other hand, it is known that an increase in the CE is associated with an increase in the instability of the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In the present case the instability corresponds to the contraction of the BH dimensions in the limit ωl → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' From the point of view of AdS/QCD duality, this contraction of the BH corresponds to the contraction of the quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In the limit ωl → 1 the volume of plasma goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This is the reason why the CE diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This result is consistent with a result found recently in [32], where it was shown that 9 [P(k)12 80 wl= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1 wl= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2 60 wl= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3 40 20 k 10 20 30 40 50 60the CE of bottomonium quasi-states becomes singular when the temperature or the magnetic field reach values such that the quasi-states completely dissociate in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In other words, the disappearance of the bottomnium in the medium (associated with the deconfinement of the quarks) is translated by the CE as an infinite instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Figure 2: Configuration entropy of rotating QGP as a function of the rotational speed (ωl) at ¯T6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In order to have a clear understanding about the variation of the CE with temperature and rotational speed, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 4 the CE as a function of ˜T, at three different values of ωl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' One notices that for a fixed rotational speed, the CE increases with the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' This increase in the CE is associated to the increase in the thermodynamic instability caused by Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Black holes at higher temperatures are subject to a stronger loss of energy as a consequence of Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' These results are consistent with the interpretation of the CE as an indicator of the stability of physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Conclusions We investigated the dependence of the configuration entropy on the rotational speed for a quark-gluon plasma with cylindrical symmetry using a holographic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The plasma was represented by the grand canonical ensemble (with null chemical potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In this scenario, we obtained an expression for the energy density of the rotating AdS black hole, dual to the plasma, that was applied to the calculation of the CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The dependence of this quantity on the rotational speed of the black hole was studied for different temperatures using numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The current interpretation of the configuration entropy states that it works as a measure of the stability of physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In short, the CE increases as the instability increases [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The result found in section 4 – the CE increases with the rotational speed ωl – is consistent with this interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Rotation of the plasma implies a Lorentz type of contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' So that the plasma becomes smaller in volume as ωl increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The limit ωl → 1 would correspond to a plasma of zero volume that consistently corresponds 10 CE(T6, wl) 150 100 50 [m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='0Figure 3: CE of rotating QGP as a function of ωl, at different temperatures: ¯T3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3 (blue), ¯T4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4 (orange), ¯T6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 (green), and ¯T8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Figure 4: Configuration entropy of rotating QGP as a function of ¯T, at different rotational velocities: ωl = 0 (blue), ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4 (orange);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' and ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' to a positive singularity in the CE indicating a “maximum instability”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' It is interesting to relate this result with a similar singular limit of the CE recently found in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' In 11 CE(T, wl) 300 250 200 :0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4 150 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 100 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8 50 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='0CE(T) 80 60 wl= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='0 wl= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content="4 40 9'0 =1m 20 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9this article the CE for bottomonium quasi-states was calculated and it was found that it becomes singular in the limits when the temperature or the magnetic field approach values such that the quasi-states completely dissociate in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The disappearance of the bottomonium quasi-states, associated with the deconfinement of the heavy quarks is translated by the CE as an infinite instability, corresponding to a positive infinite CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Here we found the similar result that in the limit when the volume of the plasma goes to zero, that would correspond to the disappearance of the plasma, the CE goes to (positive) infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' The combined effect of temperature and rotation was already investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' It was found in [17] that, for a non-rotating plasma, the CE increases with the temperature indicating the instability caused by the evaporation of the black hole via Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Here we have shown that for a plasma with a fixed non-vanishing rotational speed the CE also increases in a monotonic way with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 12 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' CE(T, ω) X ¯T3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3 ¯T4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4 ¯T5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5 ¯T6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 ¯T7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7 ¯T8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8 ¯T9 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3177 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1699 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7645 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5432 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9320 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9978 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8159 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4737 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3514 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9698 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7748 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1908 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9978 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1263 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9544 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9103 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6029 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4887 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9886 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1616 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='0833 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7997 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8953 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7188 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7479 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3956 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7113 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7703 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='0886 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3998 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4261 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6759 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5495 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='0780 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3519 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9659 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5914 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9220 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4976 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7014 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5529 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1494 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7137 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7812 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5766 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6347 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3233 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6411 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6887 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8948 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6155 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1591 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9631 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3972 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4415 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1499 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='8003 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7337 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6088 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6941 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3997 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='6117 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='292 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3687 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='3365 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2459 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='0799 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='213 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='801 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='186 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='91 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9314 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4661 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9202 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1757 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='752 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='940 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='910 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='92 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='9181 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1201 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2013 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='917 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='083 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='964 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='649 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='93 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4664 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4686 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2683 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='542 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='557 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='232 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='798 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='94 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7885 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='7700 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='376 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='435 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='489 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='253 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='931 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='95 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='2319 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4363 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='993 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='228 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='681 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='877 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='005 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='96 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='4179 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='1476 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='090 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='079 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='494 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='663 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='774 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='97 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='5942 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='284 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='700 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='408 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='772 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='872 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='913 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='98 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='651 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='994 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='074 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='976 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='908 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='976 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='957 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='99 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='231 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='956 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='166 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='391 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='362 326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='702 364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='847 ωl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='995 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='902 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='174 290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='410 342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='816 398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='298 452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='576 506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content='449 Table 1: Configuration entropy of rotating QGP at different temperatures ( ¯T = T/√c) and rotational velocities (ωl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Acknowledgments: The authors are supported by FAPERJ — Funda¸c˜ao Carlos Chagas Filho de Amparo `a Pesquisa do Estado do Rio de Janeiro, CNPq - Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' da Rocha, “Tensor mesons, AdS/QCD and information”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' C 80 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQfXvwu/content/2301.01322v1.pdf'} +page_content=' 5, (2020) 375.' metadata={'source': 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b/UtE1T4oBgHgl3EQfIgOA/content/tmp_files/2301.02940v1.pdf.txt @@ -0,0 +1,1857 @@ +arXiv:2301.02940v1 [eess.SY] 7 Jan 2023 +GA-Aided Directivity in Volumetric and Planar +Massive-Antenna Array Design +Bruno Felipe Costa, Taufik Abrão +Londrina State University (UEL), +Department of Electrical Engineering, Londrina, 86057-970, Parana, Brazil +Abstract +The problem of directivity enhancement, leading to the increase in the directivity gain over a certain +desired angle of arrival/departure (AoA/AoD), is considered in this work. A new formulation of the +volumetric array directivity problem is proposed using the rectangular coordinates to describe each +antenna element, and the desired azimuth and elevation angles with a general element pattern. Such a +directivity problem is formulated to find the optimal minimum distance between the antenna elements +dmin aiming to achieve as high directivity gains as possible. An expedited implementation method is +developed to place the antenna elements in a distinctive plane dependent on (θ0; φ0). A novel concept +on optimizing directivity for the uniform planar array (OUPA) is introduced to find a quasi-optimal +solution for the non-convex optimization problem with low complexity. This solution is reached by +deploying the proposed successive evaluation and validation (SEV) method. Moreover, the genetic +algorithm (GA) method was deployed to find the directivity optimization solution expeditiously. For +a small number of antenna elements , typically N ∈ [4, . . . , 9], the achievable directivity by GA +optimization demonstrates gains of ∼ 3 dBi compared with the traditional beamforming technique, +using steering vector for uniform linear arrays (ULA) and uniform circular arrays (UCA), while gains +of ∼ 1.5 dBi is attained when compared with an improved UCA directivity method. For a larger +number of antenna elements , two improved GA procedures, namely GA-marginal and GA-stall, were +proposed and compared with the OUPA method. OUPA also indicates promising directivity gains +surpassing 30 dBi for massive MIMO scenarios. +Keywords: +Directivity, Antenna Array, 5G, Optimization, Omnidirectional, Genetic Algorithm +(GA), Optimal Uniform Planar Array (OUPA) +1. Introduction +Array antennas can provide many advantages in flexibility and scalability concerning conven- +tional antennas. This is related to the capacity of array pattern reconfiguration and adaptability. +Such adaptability enables various applications, including radar communications, satellite communi- +cations, wireless communications, radio-astronomy, remote sensing, and direction of arrival (DoA) +estimation. Optimally designed arrays are used to obtain high directivity patterns, improving the +array performance. +Uniform array configurations, such as uniform linear (ULA), planar (UPA), rectangular (URA), +circular or cylindrical (UCA) antenna arrays, have been deployed to improve the performance of +massive antenna-based 5G and B5G communication systems. Since planar antenna array can deploy +a higher number of antenna elements at a limited physical space and capture the beamspace in both +the horizontal and vertical directions in three-dimensional (3D) propagation, the development of +different planar array configurations, such as URA and UCA, have become a natural choice among +the different possible array geometries, representing a great interest in the mmWave massive MIMO +system designs. +Array directivity optimization remains an open issue, especially in cases where the antenna- +element position is considered as an optimization variable. Indeed, the choice of antenna-elements +Preprint submitted to Arxiv (https://arxiv.org/) +January 10, 2023 + +position as a parameter to be optimized can be justified by the concept of array virtualization, in +which antenna-elements cooperate among each other, forming a virtual antenna array (VAA) with +better directivity performance that the original physical antenna array. +Recent works on array optimization have designed linear sparse array antennas. For instance, +the algorithm proposed in [1] jointly and deterministically maximizes the aperture efficiency and +directivity. Besides, [2] proposes a planar circular sparse array antenna design, where the positions +and dimensions of the radiators are jointly optimized. +In [3] an optimal circular antenna array +for maximum sidelobe levels (SLLs) reduction was investigated using a metaheuristic approach called +player algorithm. An optimization of sidelobe level and aperture efficiency for aperiodic array antennas +was proposed in [4]. Multiobjective optimization (MOO) is deployed in [5] to efficiently and effectively +design subarrays in linear antenna array; the optimization was made using a memetic differential +evolution (mDE) algorithm. Following the same research direction, the authors in [6, 7] used MOO +and metaheuristic methods to optimize the beampattern in collaborative array beamforming design. +In [8], the MOO technique is deployed to discuss the planar antenna array design under different +optimization criteria, such as the side lobe, beam steering quality, minimum power beam width, +and maximum directivity, as a function of element lengths and distances between these elements. +A fundamental question is exploring the Pareto front modeling optimal trade-offs among multiple +conflicting objectives, including the structure size, its performance, and the minimization of cost. +The well-known heuristic algorithm nondominated sorting genetic algorithm II (NSGA-II) has been +used for this purpose. Many other works are deploying MOO and/or metaheuristics solutions, such +as [9], [10], [11], [12] and [13]. +In [14], the optimal radiation pattern is generated using a spherical phased array. In contrast, +in [15], a technique based on changes in the subspace is deployed to improve the directivity of the +omnidirectional UCA. Moreover, the optimum directive beamformer has been proposed in [16], by +deploying the concept of generalized directivity maximization; the concept of directivity is based +on the mean beam power pattern, which is crucial in assessing and optimizing the performance of +directivity arrays subjected to array imperfections and sensor mismatch. Recently, by adopting a 3D +wireless channel model, and taking into account both the azimuth and elevation angles of departure +(AoD), the authors of [17] derive the probability density function for the distances between randomly +distributed users and the BS antenna array, considering three URA, ULA and UCA uniform array +topologies. +Also, they derived closed-form expressions for the squared inner product of different +channel vectors, facilitating the interuser interference analysis. +Against this background, the current work proposes a new methodology to maximize the directiv- +ity of omnidirectional volumetric antenna arrays. By assuming a uniform planar array (UPA) confined +on a specific plane with a minimum distance between the antenna elements optimized, the proposed +method takes in hand the successive evaluation and validation (SEV) procedure. To demonstrate +effectiveness, the proposed method is implemented by deploying evolutionary heuristic optimization; +hence, to implement the directivity optimization methodology under a large number of antennas in +planar arrays, we suggested two variants of the genetic algorithm: GA-marginal and GA-stall, both +made the GA a promising optimization tool to solve UPA directivity problem in such large scale +antenna scenario. An exciting finding of the proposed methodology is that a plane space constraints +the antenna-element positioning solutions as a function of the desired elevation and azimuth angles. +The contribution of this work is fourfold: +a) formulation of optimization problem for directivity maximization; the novelty of this design +is brought by the optimization of each antenna-element position bounded by a volumetric +constraint on the search space, which is given by a parallelepiped with a central point at the +origin and all the points bounded; +b) a geometric interpretation for the solution in case of omnidirectional scenarios, more specifically, +this interpretation results in restraining the position of each antenna element to a plane equation +dependent on the desired angles (θ0, φ0); +2 + +c) using a novel concept based on the UPA geometry, an expedited implementation method is +developed consisting in placing the antenna elements in a distinctive plane dependent on θ0 and +φ0; hence, using the geometry properties of the UPA, a directivity optimization problem was +formulated to find the optimal minimum distance between the antenna-elements dmin aiming +to achieve as high directivity gains as possible; +d) to corroborate the effectiveness of the proposed method, a genetic algorithm (GA) is deployed +to implement the directivity enhancement method with low complexity; when compared with +a conventional geometric design, the proposed method has demonstrated a significant improve- +ment in directivity gains on the desired angle of departure (DoD). +The remainder of the paper is organized as follows. Section 2 describes the concept of the ra- +diation pattern, directivity optimization via position antenna-element placement, and the adopted +system model. Section 3 formulates the directivity optimization problem, with particular attention +on the optimization problem for the omnidirectional case; besides, geometric interpretation is evoked, +and then the problem is recast in a simplified way. For comparison purposes in terms of directivity, +different geometric array structures are considered. A novel directivity optimization method applied +to the UPA geometries is devised in Section 4 as an effective low-complexity optimization method. +Numerical results are developed in Section 5 by deploying heuristic GA as a practical implementa- +tion tool supporting our finding on the UPA-basis antenna array directivity optimization technique. +Concluding remarks are pointed out in Section 6. +2. Radiation Pattern and Directivity via Element Positioning +For non-isotropic antennas, the directivity is the level of radiated signal power in a specific di- +rection. This is especially important for applications such as beamforming in massive MIMO 5G +wireless communications. Fig. 1 depicts the general concept of directivity enhancement via an array +of antenna elements with low directivity; the constructive and destructive interference can be guided, +by changing the position and/or phase of each element, aiming to achieve a high directivity antenna +pattern [18]. In this work, the omnidirectional antenna-element configuration has been chosen to +implement the array directivity enhancement, which is a more realistic scenario compared to the +isotropic sources, while remaining a suitable configuration to be analytically analyzed due to its sim- +ple element factor, expressed by Υ(θ) = cos θ. Furthermore, the applicability of this configuration +is interesting in many fields. +Such configuration dramatically simplifies the analytical directivity +expression, while the non-convex directivity optimization problem remains a special issue. +2.1. Radiation Pattern of an Antenna Array +An antenna array is a grouping of N antennas whose geometry can assume many different forms, +such as uniform linear, circular, rectangular, non-uniform grouping, and so forth; moreover, all ele- +ments in the array work jointly to increase the directivity of the arrangement. The entire field of an +antenna array is obtained by multiplying the field of a single element Υe(θ, φ) and the array factor +Υa(θ, φ); this is called the radiation pattern of an antenna array and can be written as: +Υ(θ, φ) = Υe(θ, φ)Υa(θ, φ) +(1) +The element factor (EF) is the radiation pattern of a single-element antenna, and the model must +resemble the pattern of a real antenna. Using spherical coordinates, the EF is periodic in θ and φ +angles. Thus, the radiated power of an arbitrary antenna can be described using Fourier analysis, +i.e. it can be written using a linear combination of powers of cosine and sine functions. Hence, the +EF can be written: +Υe(θ) = sinu (θ) cosv (θ) +(2) +The array factor (AF) describes a combination of radiating elements in an array without con- +sidering the element radiation pattern. Thus, the array factor can be interpreted as the radiation +3 + +Figure 1: Directivity representation in a) isotropic source, b) omnidirectional source, and c) via constructive and +destructive interferences of low-directivity sources. +pattern by replacing the actual elements by the isotropic (point) sources, i.e., Υe(θ, φ) = 1. The AF +has a dependency in terms of position, relative phase and relative amplitude of each antenna element +and is given by: +Υa(θ, φ) = +N +� +n=1 +Anej(αn+kpn·ap) +(3) +where N is the number of antenna elements, An is the relative amplitude of n-th antenna element, +αn is the relative phase of n-th antenna element, pn is the position vector of n-th antenna element; k +is the wave number, and ap is the unit vector of observation point in spherical coordinates, given by: +ap = sin(θ) cos(φ)ˆi + sin(θ) sin(φ)ˆj + cos(θ)ˆk +(4) +The position vector can be written as a combination of the elements in the 3-D cartesian axis: +pn = xnˆi + ynˆj + znˆk +(5) +In this work, we have considered the position of each antenna element of the array as the variable to +be optimized, which the matrix can mathematically defined: +P = +� +p1 p2 . . . pN +�T +(6) +where the position vector pn defines the localization of the n-th antenna element in the 3-D space. +2.2. Directivity for Arbitrary Volumetric Antenna Arrays +Directivity can be defined as the level of irradiation signal power in a specific direction to the +detriment of others. +This is especially important for some applications, such as beamforming in +non-isotropic massive MIMO antennas in 5G wireless communications. Antenna directivity can be +written as the ratio between the radiation intensity in a desired angle and the sum of the radiation +intensity in all the other directions: +D(θ0, φ0) = +|Υ(θ0, φ0)|2 +1 +4π +2π +� +0 +π� +0 +|Υ(θ, φ)|2 sin (θ)dθdφ +(7) +4 + +High Directivity Patternenna +ray +Constructive and Destructive +InterferencesAnt +Al +n-th Antenna +Yncos.0 +nal SourceMe= 1 +Isotropic Source +Omnidirecti +Element Factor with low Directivitywhere |Υ(θ, φ)|2 is the radiation intensity. Besides, for an arbitrary array with the general element +pattern (cosv θ sinu θ, ∀ u, v), the directivity expression can be defined as [18]: +D(θ0, φ0, u, v) = f1 +f2 += +sin2u (θ0) cos2v (θ0) +N +� +m,n=1 +AnAmξmn cos [Ωmn] +In(u, v) + Im(u, v) +(8) +where +In(u, v) = +N +� +n=1 +A2 +n +�1 +8((−1)2v + 1)B(u + 1, v + 1 +2) +� +Im(u, v) = 2 (−1)(v+2u) +N +� +n,m=1 +m̸=n +n>m +u +� +κ=0 +AnAm +�u +κ +� +cos (αmn) ∂2(v+u−κ) +∂z2(v+u−κ) +mn +� +sin(k +� +β2 + z2mn) +k +� +β2 + z2mn +� +Ωmn = Ω(pn, pm, αn, αm, θ0, φ0) = k [xnm sin θ0 cos φ0 + ynm sin θ0 sin φ0 + znm cos θ0] + αnm +pn = xnˆi + ynˆj + znˆk, +β = +� +x2nm + y2nm, +xnm = (xn − xm), +ynm = (yn − ym), +zmn = (zn − zm) +ξmn = +� +1 +m ̸= n +1 +2 cos [Ωmn] +m = n +, +Re{v} > −1 +2, +Re{u} > −1, +αmn = (αn − αm). +(9) +where pn is the position vector of n-th antenna element; αn is the phase of n-th antenna element; +An is the amplitude of n-th antenna element; θ0 is the desired elevation angle, where directivity is +evaluated; φ0 represents the desired azimuth angle, where directivity is evaluated; and k is the wave +number of the transmission. +3. Directivity Optimization Problem in Omnidirectional Scenarios +Given a desired angle of departure (θ0, φ0) over an antenna array composed of N elements, we are +looking for the parameter values of an array configuration that result in the directivity maximization +in such an AoD. +We start the analysis assuming non-zero-phase and the non-unitary amplitude in each antenna; +hence, we will solve the proposed problem to obtain a geometric configuration for each desired angle. +For typical applications, the desired direction changes regularly, which becomes a problem, given +that changes in the positions of antennas are physically impossible. However, in this formulation, +the phase element was not considered, leaving space to adopt virtualization techniques, which highly +depend on the phase elements and will be the subject of our future research. +Based on the directivity expression provided in eq. +(8), we state the following optimization +problem: +maximize +P +D(θ0, φ0) = f1 +f2 +subject to +pn ⪯ pmax +for +n = 1, 2, 3, . . . , N +(10) +where the negative semi-definite pn ⪯ pmax defines the 3-D parallelepiped volume bounded by +[xmax, ymax, zmax]. +Besides, the optimization problem in (8) can be rewritten as a minimization +problem: +minimize +P +f2(P , θ0, φ0) +f1(P , θ0, φ0) +subject to +pn ⪯ pmax +for +n = 1, 2, 3, . . . , N +(11) +Therefore, the objective of this problem can be reinterpreted as minimizing f2(P ). At the same time, +simultaneously maximize f1(P ) subject to the bounds in the space configuration, given, for instance, +by the 2-norm. +5 + +The omnidirectional scenario has been selected due to the concise expressions, analysis, and +manageable complexity while the applicability remains in many exciting fields. The omnidirectional +radiated power is also scattered symmetrically but not equally in all directions, creating a 3-D torus +radiation pattern, which can be formulated as the expression Υ(θ) = cos θ for all azimuth angles +(φ). Hence, considering the element pattern given in eq. (2), for the omnidirectional scenario, we set +u = 0 and v = 1. Such condition simplifies the auxiliary functions as: +f o +1 (θ0) = cos2 (θ0) +N +� +m,n=1 +AnAmξmn cos [Ωmn] +(12) +and +f o +2 = +N +� +n=1 +A2 +n +6 − 2 +N +� +m,n=1 +m̸=n +n>m +AnAm +∂2 +∂z2mn +� +sin(k +� +β2 + z2mn) +k +� +β2 + z2mn +� +(13) +where +dmn = k +� +β2 + z2mn +(14) +Generically, dmn represents the Euclidean distance between the m-th and n-th antenna element in +the 3-D space, given by dmn = k +� +x2mn + y2mn + z2mn. +In Eq. (10), we are interested in maximizing f o +1 while minimize f o +2 . For this purpose, we can +verify in Eq. (12) that the only way to ensure these conditions consists in maximizing cos [Ωmn], i.e. +Ωmn = c1π, +for c1 +2 ∈ Z +c1 even +(15) +Using the definitions in (9), and satisfying the condition above, for Ωmn we have: +xnm sin θ0 cos φ0 + ynm sin θ0 sin φ0 + znm cos θ0 = c1π − (αn − αm) +k +(16) +Eq. (16) can be interpreted as the plane equation containing all the vectors of position difference +(pmn = pn − pm), to fulfill this condition entirely it is necessary all vectors be restrained in the +same plane of the differences position vectors since the constant c1 is not unique. For each value, +we have a different plane, the solution of this condition is given by infinite parallel planes. However, +the combination of position vectors in these parallel planes does not fulfill the condition, given that +the difference vector for points in different parallel planes will result in a vector difference in another +plane. +Moreover, minimizing f o +2 results in a more complicated task, given that the directivity problem +in (7) requires minimization of the summation of all terms in the right-hand side of (13), instead of +minimizing each term. Hence, the f o +2 minimization problem can be recast in an alternative version +as: +maximize +P +N +� +m,n=1 +m̸=n +n>m +AnAm +∂2 +∂z2mn +� +sin(k +� +β2 + z2mn) +k +� +β2 + z2mn +� +(17) +Besides, the second derivative in (17) can be re-written as: +∂2 +∂z2mn +� +sin( +� +β2 + z2mn) +� +β2 + z2mn +� += (β2 − 2z2 +mn) cos dmn +d4mn +− +� +(β2 − 2)z2 +mn + β2 + z4 +mn +� +sin dmn +d5mn +(18) +6 + +which is function of the xmn, ymn and zmn, and related to β and dmn through the constraints in +eq. (8) and (14), respectively. Notice that in this formulation, the omnidirectional configuration has +influence only in (18). For another scenarios, i.e., different values of u, v ̸= 0, the only change will +be the objective function (OF) in (17), where the order and number of derivatives dependent on the +values of u, v ∈ Z+. +3.1. Recasting the Simplified Omnidirectional Problem +One of the most important constraints resulting by the omnidirectional scenario is given in eq. +(16), and where the values of c1 can be chosen arbitrarily; hence, aiming to simplify the analysis we +set c1 = 0. Furthermore, considering all null phases, for the n-th and m-th antennas, we obtain: +sin θ0 cos φ0xn + sin θ0 sin φ0yn + cos θ0zn = 0 +sin θ0 cos φ0xm + sin θ0 sin φ0ym + cos θ0zm = 0 +(19) +By subtracting both previous expressions, one can obtain: +znm = tan θ0(cos φ0xnm + sin φ0ynm) +(20) +The restriction (20) can be incorporated into (18), finally resulting in a simplified optimization prob- +lem: +minimize +x,y +G(x, y) = − +N +� +m,n=1 +m̸=n +n>m +F(xmn, ymn) +subject to +|xmn| ≤ xmax, +|ymn| ≤ ymax +for +n = 1, 2, 3, . . . , N +(21) +with OF defined by (18), and substituting β2 = d2 +mn +k +− z2 +mn, results: +F(xmn, ymn) = +� +(d2 +mn − 3k)z2 +mn + d2 +mn +� +sin dmn +kd5mn +− (d2 +mn − 3kz2 +mn) cos dmn +kd4mn +(22) +The function G can be rewritten using the definition in (13), implying the following relation: +f o +2 = +N +� +n=1 +A2 +n +6 + 2G +(23) +Since f o +2 represents the denominator integral in (7), for the omnidirectional scenario this values must +be positive; therefore, the function G has a lower bound for each value of N, given by: +GN +bound = − +N +� +n=1 +A2 +n +12 +(24) +The bound on zmn is implicit, having dependency on xmax, ymax, and on the desired angle of departure +(θ0, φ0). The solution of (21) give us the optimal coordinates xn and yn, ∀n which can be used to +find the optimal zn, ∀n = 1, . . . , N coordinate from (20). +3.2. Geometric Planar Arrays Comparison +In this section, different geometric planar array structures are considered for a directivity com- +parison purpose. It is shown the importance of setting the adequate dmin and the associated area +occupied by the array, aiming to exploit the potential of the directivity improvement. All the selected +arrays are planar, aiming to preserve the constraint in (20). Adopted parameter values and antenna +structure include: +7 + +Parameter +Adopted Values +# antennas +N = 16 +Normalized wavelength +λ = 1 [m] +Desired direction +θ0 = φ0 = π/4 +uniform planar array +UPA +uniform circular array +UCA +uniform hexagonal planar array +UHPA +The selected uniform planar array geometries with N = 16 elements are depicted in Fig. +2 +with the minimum distance between antenna elements dmin represented by red arrows. Fig. 2(d) +exhibits the directivity dependence regarding the increasing dmin values for each selected uniform +array geometries. The planar and the hexagon geometry attained superior directivity than the circular +dmin +-5 +0 +-15 +15 +Y Axis +5 +-10 +10 +Z Axis +-5 +X Axis +10 +5 +0 +0 +15 +-5 +(a) Uniform Planar Array (UPA) +X Axis +dmin +Y Axis +-5 +-4 +-3 +5 +-2 +-1 +0 +Z Axis +0 +1 +2 +3 +4 +0 +5 +-5 +(b) Uniform Circular Array (UCA) +Y Axis +X Axis +dmin +-6 +-4 +-5 +-2 +5 +0 +Z Axis +2 +0 +4 +0 +6 +8 +-5 +5 +(c) Uniform Hexagonal Planar Array (UHPA) +0 +1 +2 +3 +4 +5 +dmin (m) +2 +4 +6 +8 +10 +12 +14 +16 +18 +Directivity (dBi) +UHPA +UCA +UPA +(d) Directivity versus the minimum distance for the +three different geometry arrays: UPA, UCA, and UHPA +Figure 2: Different planar array geometries and respective attained directivity with N = 16 elements on the desired +plane defined by θ0 = φ0 = π/4. +array; the optimal value of dmin for both structures are close to each other and significantly smaller +than the circular array dmin value. For this application, the UCA has demonstrated do not be suitable +and will be suppressed hereafter in further analyses. +8 + +These illustrative results express the importance of setting the adequate dmin to exploit the di- +rectivity enhancement’s potential. However, the area occupied by the array is also essential, since +compact antenna arrays are more valuable. Although the value of dmin is smaller for the UPA, the +area is not necessarily smaller than the UHPA. To investigate this aspect, Fig. 3 depicts directivity +and the normalized area given by the convex hull of the arrangement of points for both promising +array structures; in (a) the value of directivity and in (b) the value of area occupied by the convex +hull of the array, selecting the optimal value of dmin. It is apparent in both graphics the remarkable +directivity performance proximity for both structures and respective normalized areas with the in- +crease in the number of antennas. The area occupied by the convex hull for small values of antennas +are very similar; however, for the maximum directivity configuration and under a significant number +of antenna elements, the UPA demonstrated a better performance covering a smaller region with the +increasing of the antennas; besides, the points of the UHPA where the performance are significantly +better is given when the hexagon formed by the array is perfect. Therefore, on can conjecture that +both configurations have a very similar directivity performance. At the same time, for a significant +number of antennas, the UPA surpasses the UHPA directivity vs. area performance tradeoff unless +the hexagon formed by the UHPA is perfect. +0 +50 +100 +150 +200 +250 +Number of Antennas (N) +14 +16 +18 +20 +22 +24 +26 +28 +30 +32 +Directivity (dBi) +UPA +UHPA +(a) Directivity vs. the number of antennas increasing (N) +0 +50 +100 +150 +200 +250 +Number of Antennas (N) +0 +50 +100 +150 +200 +250 +Area (m2) +(b) Array area vs. the number of antennas (N) increasing. +Figure 3: UPA against UHPA in terms of directivity and area, both with optimal setting of dmin. +Given the attained near directivity between UPA and UHPA, both geometries could be selected +without losses in terms of directivity gain and occupied area. Therefore, the structure selected for +the optimization method in the next section is the UPA, due to the compactness in describing the +antenna elements positioning over a rectangular coordinates system. +4. Optimizing the Position of UPA Elements +This section describes the proposed method to achieve high directivity using the regular planar +array structure. This method provides reasonable solutions for the proposed optimization problem +of Eq. (21) with low computation effort considering the high-complexity operation of searching the +geometric locus. +The proposed method consists in assuming a given number of waves (k), a desired direction +(θ0,φ0), and a number of antennas (N), finding the optimal UPA on the desired plane defined by +(20) aiming to attain a remarkable improvement in the directivity in a desired direction. The most +critical parameter to be defined to achieve this goal is dmin. The following describes a procedure for +finding this parameter for a generic UPA. Firstly, we can consider the positioning of the antennas on +the xy plane, considering a UPA with N1 × N2 antenna elements regularly distributed, as in Fig. 4. +9 + +-0.5 +0 +0.5 +1 +1.5 +2 +X axis (dmin ) +-0.5 +0 +0.5 +1 +1.5 +2 +Y axis (dmin ) +... +... +... +... +... +... +... +... +... +1 +2 +3 +2N1 +3 + N1 +2 + N1 +1 + N1 +1 + 2N1 +3 + 2N1 +3N1 +2 + 2N1 +N2N1 +N1 +2 + (N2-1)N1 +1 + (N2-1)N1 +3 + (N2-1)N1 +Figure 4: UPA elements disposition and labeling for N1 × N2 antennas. +From this antenna-elements distribution, one can define the distance between two generic elements, +i.e, dmn for m ̸= n, which is an important parameter on the use of (22). The distance dmn can be +formulated as: +dmn = + + + + + +|m−n|dmin +N1 +if ψ1 = 0 +|m − n|dmin +if ψ2 = 0 +and +|m − n| < N1 +dmin +� +ψ2 +1 + ψ2 +2 +c.c +(25) +where: +ψ1 = mod(m − 1, N1) − mod(n − 1, N1); +ψ2 = +�m − 1 +N1 +� +− +�n − 1 +N1 +� +(26) +To allocate the array on the desired plane without altering the distances, it is possible to use a +rotation matrix considering the desired direction, defined by: +Rv = + + +sin2 φ0µγ + cos γ +− cos φ0 sin φ0µγ +− cos φ0 sin γ +− sin φ0 cos φ0µγ +cos2 φ0µγ + cos γ +− sin φ0 sin γ +cos φ0 sin γ +sin φ0 sin γ +cos γ + + +(27) +where γ = acos | cos θ0| and µγ = (1 − cos γ); notice that for θ0 = 0, the restriction (20) becomes the +xy-plane and this rotation matrix becomes an identity matrix. The position matrix of the array on +the desired plane can be expressed as: +�P = P Rv +(28) +Besides, the variable zmn, which consists in a combination of subtractions on the last column of �P +can be described as: +zmn = −(ψ1 cos φ0 sin γ + ψ2 sin φ0 sin γ) +(29) +Now, with the parameters dmn and zmn available, the OF analytical expression in (21) can be +evaluated; however, this is a non-convex function of one variable. +At this point, the directivity +10 + +optimization problem can not be solved using convex optimization tools. On the other hand, notice +that finding the global optimum of this function, either by exhaustive search or by quasi-optimal +evolutionary heuristic methods, the optimal UPA to be placed on the desired plane can be found, +given the advantage of allocating the antenna-elements on the same plane, while finding the optimal +general UPA in terms of directivity in that plane. +Remark 1: The general methodology is focused on finding the optimal position of each antenna +element that maximizes the antenna array directivity. In this section, we have developed a solution +confined into a specific plane, where the antenna-elements position on this plane resembles a sequence +of equilateral triangles; hence, the idea of finding the optimal UPA antenna-elements placement in +the such plane is described by the position matrix, Eq. (28), and solved applying the simplified +optimization problem, Eq. (21). +4.1. Successive Evaluation and Validation (SEV) +With the dmn and zmn values, the OF of the directivity optimization problem in (21) for the +UPA can be expressed and therefore emerges the necessity of finding the optimal value for dmin that +enhances the array directivity. Based on numerical results, we concluded that the first local minimum +of the OF in (21) is also the global optimum. Fig. 5 depicts the value of G for different values of +N1 and N2. As suspected, for all analyzed N1 × N2 configurations, the first minimum local is also +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +dmin (m) +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +1 + 2 +2 + 2 +5 + 2 +4 + 4 +9 + 2 +Global min +Figure 5: Values of G against dmin values for different configurations of N1 × N2 +the minimum global. Taking advantage of this aspect, we propose in Algorithm 1 the successive +evaluation and validation (SEV) method, which basically is a line search procedure, evaluating the +function at the point (d0 = c), with an increment closely to the origin (0+); then, an increment is +computed again for (d1 = d0 + c). Hence, if the increment brought a decreasing of the OF, i.e., +Gn+1 < Gn, then the process is repeated; if not the point dn is declared the global optimum. +A straightforward idea is behind the SEV algorithm: for a minimal input parameter c value, the +number of iterations increases; however, the precision of the solution improves; on the other hand, +for large c values, the number of iterations and precision can be reduced remarkably, and depending +on how significant is such value, the first local minimum could be inadvertently skipped, losing the +11 + +Algorithm 1 SEV – Successive Evaluation and Validation +1: Input: c +2: n = 0 +3: dn = (n + 1)c +4: Gn = − +N +� +m,n=1 +m̸=n +n>m +F(dn) : +use (21), (25), (29) with dn = dmin +5: while Gn+1 − Gn ≤ 0 do +6: +n = n + 1 +7: end while +8: dmin = dn +9: Output: d⋆ +min +optimum solution. Hence, the step parameter c must be carefully selected to the SEV achieves a +good precision-complexity tradeoff. +4.2. Proposed Optimal Uniform Planar Array (OUPA) +The technique that allocates omnidirectional element-antennas arranged as uniform planar array +(UPA) on a specific znm plane, given by (20), and using the SEV method to select the d⋆ +min is +denominated hereafter optimal uniform planar array (OUPA). We propose this straightforward, low- +computational cost, quasi-optimal method solution that significantly enhances the directivity in UPA +antennas; besides, OUPA design requires few parameters: the carrier frequency/wave number (f/λ), +the desired directivity angles (θ0, φ0) and the geometric UPA structure configuration, i.e., N1 × N2. +A pseudocode for the OUPA technique is shown in Algorithm 2. +Algorithm 2 OUPA – Optimal Uniform Planar Array +1: Input: f/λ, θ0, φ0, N1, N2 +2: Calculate all possible dmn values using (25) +3: Calculate all possible zmn values using (29) +4: Evaluate G in (21) using the values of dmn and zmn +5: Determine d⋆ +min via SEV method (Algorithm 1) +6: Determine the UPA matrix of position P using d⋆ +min via (6) +7: Final array position evaluated by �P (θ0, φ0) = P Rv, eq. (28) +8: Output: �P +Remark 2: The OUPA method exploits the OF features, Eq. (22), in the simplified optimization +formulation, Eq. (21), while for generic N1 × N2 antennas configuration, a practical quasi-optimal +solution is developed in the next section (specifically, subsection 5.3) based on the evolutionary +heuristic optimization approach. The array directivity subject to antenna-elements positioning has +been formulated and solved for various elements in the range N ∈ {4; 36}. +5. Numerical Results +Different OUPA performance aspects are numerically evaluated, including a) directivity perfor- +mance, b) occupied area, c) comparison with other directivity optimization methods. The general +parameters for the omnidirectional directivity optimization scenario deployed throughout this section +are summarized in Table 1. +5.1. OUPA Directivity Performance +Numerical simulations have been conducted considering different values of N1 × N2 antenna ar- +rangements. For each configuration, the value of d⋆ +min is obtained using the SEV procedure, and then +the relation directivity-area occupied by the array is examined. The goal is understanding how the +12 + +Table 1: General simulation setup +Parameter +Adopted Values +Angle of Departure +(θ0 ,φ0) = (π +4 , π +4 ) +Element Amplitude +An = 1 for all n. +Element Phase +αn = 0 for all n. +Omni-directional Scenario +u = 0 and v = 1 +occupied area impacts the planar array directivity. The occupied array area is an essential parameter +since compact antenna arrays are far more helpful. Table 2 depicts the selected parameter values +deployed in the simulations. +Table 2: Simulation setup for OUPA directivity-area evaluation +Parameter +Adopted Values +Frequency / Wave Number +5 GHz / k ≈ 104.8 m−1 +SEV parameter (increment) +c = 10−3 +Number of N = N1 × N2 antennas-elements +# Vertical elements +N1 = [2, 3, 4, . . . 50] +# Horizontal elements +N2 = [2, 3, 4, . . . 10] +0 +100 +200 +300 +400 +500 +Number of Antennas (N) +5 +10 +15 +20 +25 +30 +35 +Directivity (dBi) +0 +0.5 +1 +1.5 +Area ( m2) +Directivity +Area +0 +100 +200 +300 +400 +500 +600 +Number of Antennas (N) +40 +42 +44 +46 +48 +50 +52 +54 +56 +dmin (mm) +dmin = 48.28 +dmin = 45.88 +dmin = 43.47 +dmin = 41.06 +dmin = 50.69 +dmin = 53.10 +dmin = 55.51 +Figure 6: a) Directivity and array Area values for different N = N1 × N2 element-antennas configurations; b) corre- +spondent d⋆ +min values obtained via SEV procedure for different configurations of N = N1 × N2. +In Fig. 6, the values of directivity, array area and correspondent d⋆ +min are depicted for a wide range +of N element-antennas values. As expected, increasing N, the directivity and array area are increased +proportionally. Since the array area increment is not desired, the directivity-area tradeoff must be +carefully evaluated. The correspondent d⋆ +min values found by simulations using SEV procedure are +illustrated in Fig. 6.b. The most important feature of this plot is that the optimal value of dmin has +a discrete distribution regarding N1 and N2; besides, the difference between the levels of optimum +values is constant, which is very interesting and can be used to reduce significantly the search space of +the optimal value. Another essential characteristic is that the increase in N not necessarily increases +d⋆ +min value. Finally, the N1 and N2 values impact the optimal dmin value; hence, this effect is analyzed +in more details in the next section. +To gain a deep understanding of the Area and Directivity dependence, Fig. 7 depicts directivity +13 + +values against array area for different configurations of N = N1 × N2. Observing the Pareto frontier +among conflicting directivity vs occupied area by the UPA array is possible. +As expected, it is +necessary to increase the array area to achieve higher directivity values. The Pareto front establishes +the optimal trade-off between both conflicting parameters; possible solutions (points) above this +frontier are unfeasible for such an array structure, and the points below are sub-optimal in the +directivity-area tradeoff. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Area (m 2) +5 +10 +15 +20 +25 +30 +35 +Directivity (dBi) +��Directivity +Pareto Frontier +Figure 7: Pareto front for directivity vs occupied area of UPA arrays. +5.2. Directivity vs Area under Different N1 × N2 Arrangements +Due to constructive aspects and integer combination of N1 and N2, the structure realization +for a given area is constrained, given that the total number of antennas is a multiplication of two +integers, i.e., N1 × N2 = N. Moreover, most of the analyses available in the literature focus on the +UPA structures, with squared N1 = N2 antenna arrangements. In this work, we also have analyzed +rectangular N1 ̸= N2 antenna arrangements. +To investigate the impact on the directivity, four different UPA structures with N = [36; 48; 60; 72] +antennas were compared. +Fig. +8.a) depicts possible achievable directivity values and respective +UPA normalized area (assuming k = 1) for possible N1 and N2 combinations. +The directivity +value and area occupied by N antennas are compared considering different UPA configurations, each +column representing the same number of antennas; the markers on the same column indicate specific +configurations of N1 × N2. As indicated in Fig. +8.b), decreasing the difference between N1 and +N2 results in better values of maximum attainable directivity1 as a function of dmin. However, the +area occupied by the UPA also increases. On the other hand, choosing configurations with a great +disparity among N1 and N2 implies a substantial directivity reduction, being possible to attain a +similar performance, or even be surpassed, when compared with the same UPA structure but with a +smaller number of antennas, for instance, 36 × 2 against 6 × 8, in which, despite the great difference +of 24 antennas, the directivity are quite similar in both array arrangements. +As the focus of this work is the directivity enhancement and aiming at exploiting the UPA +structure, the difference between N1 and N2 must be reduced; therefore, the choice for N1 and N2 +1In Fig. 8.b), the variation of dmin causes an impact on the value of directivity. It is possible to notice that better +directivity values are achieved when N1 = N2. Indeed, these numerical results indicate that the smaller difference +between N1 and N2 better is the array directivity. +14 + +35 +40 +45 +50 +55 +60 +65 +70 +75 +80 +Number of Antennas (N) +18 +19 +20 +21 +22 +23 +24 +25 +Directivity (dBi) +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Area (m2) +Directivity +Area +6×6 +9×4 +12×3 +18×2 +10×6 +12×5 +20×3 +30×2 +24×2 +9×8 +18×4 +24×3 +36×2 +6×8 +12×4 +16×3 +15×4 +12×6 +0 +2 +4 +6 +8 +10 +dmin(m) +8 +10 +12 +14 +16 +18 +20 +22 +Directivity (dBi) +6x6 +9x4 +18x2 +36x1 +(a) +(b) +Figure 8: +a) Directivity and Area vs. +N for different configurations with the same number of antennas N = +[36; 48; 60; 72]. +b) Directivity vs dmin fo four possible N1 × N2 arrangements in an UPA with N = 36 antennas. +should respect the following condition: +|N1 − N2| = +� 0 +squared (UPA) +1 +rectangular (UPA) +(30) +5.3. Genetic Algorithm solving Non-Convex Directivity Problem +Since the original directivity optimization is a non-convex problem, in this subsection, the perfor- +mance of the genetic algorithm (GA) in solving the omnidirectional uniform array directivity problem +is evaluated. The pseudocode for the GA deployed to find the enhanced directivity solutions is pre- +sented in Algorithm 3. Besides, the GA parameter values deployed in this context are summarized +in Table 3. +Algorithm 3 GA-UPA – GA Optimizer for Omnidirectional Uniform Antenna Array +1: Input: nvars, cf, gmax, MGaussian, psize +2: Population ← InitializePopulation(psize,nvars) +3: S = G(Population) +4: sbest ← s which results in the minimum value of G(s) +5: while # number of Generation < gmax do +6: +Parents ← SelectParents(Population,psize) +7: +Children ← ∅ +8: +for [Parent1,Parent2] ∈ Parents do +9: +[Children1,Children2] ← Crossover(Parent1,Parent2,cf) +10: +Children ← MGaussian(Child1) +11: +Children ← MGaussian(Child2) +12: +end for +13: +S = G(Children) +14: +sbest ← s which results in the minimum value of G(s) +15: +Population ← Replace(Population, Children) +16: end while +17: Output: sbest +15 + +Table 3: Simulation Setup used in the GA-UPA directivity +Parameter +Adopted Values +Wave Number / Wavelength +k = 1 [m−1] / λ = 2π [m] +# antenna-elements +N ∈ {4, 5, 6, 7, 8, 9} +Elements placement bounds +[xmax, ymax, zmax] = [5, 5, zmax] +Objective function (OF) +eq. (21) +Number of Variables, (nvars) +2N +Crossover Fraction (cf) +0.7 +Max. # Generations (gmax) +40 +Mutation (MGaussian) +Gaussian +Initial Population +Feasible and Random +Population Size (psize) +200N +Population Type +Double Vector +Search space +[0, xmax]; +[0, ymax] +Remark 3: GA has been selected as a powerful, well-established evolutionary technique to solve +non-convex optimization problems, among other also well-known techniques, such as particle swarm +optimization (PSO), ant colony optimization (ACO), artificial bee colony algorithm (BCA), and other +evolutionary algorithms (EA) optimization. GA or EA applies the natural evolution principles to find +an optimal local solution. In GA, the problem is encoded in a series of bit strings manipulated by the +algorithm; the decision variables and problem functions are deployed directly. The main drawbacks of +an EA are: a) it is much slower2 than alternatives such as the gradient-based and Simplex methods; +b) as problem size scales up, e.g., ten to a hundred or a thousand decision variables, an EA is +often overwhelmed by the dimensionality of the problem, being unable to find a solution close to a +locally optimal solution; c) a solution is acceptable only in comparison to other, previously discovered +solutions; indeed, an EA has no concept of an optimal solution, or any way to test whether a given +solution is optimal, even locally optimal; d) An EA never really knows when to stop, since it does +not know whether a given solution is optimal. Finally, EAs usually finish running manually by the +user, or by a predetermined limit on the number of iterations. +Fig. 9 depicts the solution of problem eq. (21) found by the GA for N = 6, 7 and 8 antennas. +When the solution is visualized in the 3-D plots, it is possible to observe the plane that restrains +the solution and how the solutions found strive to attain a significant number of equal Euclidean +distances. However, when analyzing the projection of the solution in the xy-plane, it is possible to +verify the distortion introduced by the term znm, eq. (20). +The evolution of antenna-element position through the GA-UPA generations is depicted in Fig. +10; the left plots show the placement of the coordinates (x, y) found by the GA solution and the +right plots indicate the distribution of the difference-coordinate points (xmn,ymn), i.e., all possible +differences between the correspondent coordinates (x, y), together with the contour plots of the +objective function F defined in (22). Notice that in the right plots, the number of points on each plot +is given by the combinations of all possible differences (xmn,ymn), which result in N(N − 1) points. +Another interpretation for these coordinates points is that they represent the terms of each summation +in G, eq. (21). The evolution of the antenna-element positions through the GA generations is given +in the xy-plane projection. Moreover, it is possible to observe the evolution of the coordinates points +(xmn, ymn); this evolution leads to an arrangement where all allocated points are concentrated in +sites of small values of F, regarding the range of all the possible values. +The selection of the difference-coordinate points (xmn, ymn) can not be performed freely. Indeed, +these points result in Euclidean distances constraints, which in some cases did not physically feasible; +therefore, the obvious solution consisting in allocating all points to the same minimum of F is not +2often by factors of a hundred times. +16 + +Y Axis +X Axis +-6 +-6 +-4 +-4 +-2 +6 +Z Axis +0 +-2 +4 +2 +2 +0 +4 +0 +2 +-2 +4 +-4 +(a) N = 6 antennas +-6 +-4 +-2 +0 +2 +4 +6 +X Axis +-6 +-4 +-2 +0 +2 +4 +6 +Y Axis +(b) N = 6, xy-plane +-4 +-5 +-2 +0 +Z Axis +5 +2 +X Axis +0 +4 +Y Axis +0 +5 +-5 +(c) N = 8 +-6 +-4 +-2 +0 +2 +4 +6 +X Axis +-6 +-4 +-2 +0 +2 +4 +6 +Y Axis +(d) N = 8, xy-plane +Figure 9: GA solution for N = 6, 8 UPA antennas: (left) final element-antennas position; (right) projection onto the +xy-plane. +17 + +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +X axis +-6 +-4 +-2 +0 +2 +4 +6 +Y Axis +(a) 1th Generation +(b) 1th Generation +-6 +-4 +-2 +0 +2 +4 +6 +X Axis +-6 +-4 +-2 +0 +2 +4 +6 +Y Axis +(c) 35th Generation +(d) 35th Generation +Figure 10: Evolution of GA-UPA solutions for N = 8 antennas: (left) antenna-element position; (right) distribution +of (xmn,ymn) and contour lines for the objective function F of eq. (22). +18 + +10 +8 +6 +44 +6 +8 +102 +mn +0 +-2 +-4 +-6 +-8 +-10 +-10 +-8 +-6 +-4 +-2 +0 +2 +X +mn10 +8 +6 +44 +6 +8 +102 +mn +0 +-2 +-4 +-6 +-8 +-10 +-10 +-8 +-6 +-4 +-2 +0 +2 +X +mnpossible due to the implication that all Euclidean distances must be the same. Indeed, the modifica- +tion of position in one element of the antenna implies in the allocation of N − 1 points; hence, the +allocation of these points must be resolved simultaneously. +Remark 4: The GA solution (Fig. 10) converges to a geometric figure similar to the points on a +regular triangular tiling (RTT); the only reason for the convergence is not exactly an RTT geometry +is found examining plane constrain zmn, eq. +(20); hence, for θ0 = φ0 = 0, the solution will be +given by a set of points allocated on an RTT. This arrangement presents symmetry which imposes +that all possible distances can be reduced remarkably by repetition of points regularity. As a result, +the difference-coordinate points (xmn,ymn) can be distributed exploiting such standard, selecting the +feasible points corresponding to small values of F and using the repetition to allocate more points. +Comparison with the UCA and ULA: The values of directivity found with GA-UPA is compared +with the two classical steering vector beamforming: a) the uniform circular array (UCA), and b) the +uniform linear array (ULA). The directivity using steering vector consists of changing each antenna +element’s phase, aiming to minimize the array factor in (3), depending on the position of each element +in a well-defined geometric structure. For the ULA and UCA the steering vector is given, respectively, +by: +αula +n += −dn cos θ0 ˆk +(31) +αuca +n += −r sin γn sin θ0 cos φ0 ˆi − r cos γn sin θ0 sin φ0 ˆj +where: γn = (n−1)2π +N +, and d is the regular distance between adjacent element. In this formulation, r +is the ratio of the circle that contains all the points of the UCA. This ratio was considered so that +the minimal distance between the antenna’s element was defined as half carrier wavelength, λ +2. +The values of directivity D through the generations of the GA-UPA is compared in Fig. 11.a with +the regular ULA design; it is noticeable the excellent performance increase in the early generations, +being capable of surpassing the conventional ULA design after the first generation. The convergence of +the proposed algorithm is observable with the stationary directivity performance over the generations, +which occurs after g = 21, 20 and 35 generations for N = 6, 7 and 8 antenna elements, respectively. +The sum of all the difference-coordinates points (xmn,ymn) corresponding to the OF in eq. (21) +typically evolves through generations, as depicted in Fig. +11.b for N = 6, 7 and 8; also, lower +bound values can be calculated using eq. +(24) and considering An = 1, ∀ n. +The bound values +G6 +bound = − 1 +2, G7 +bound = − 7 +12 and G8 +bound = − 2 +3 are included for comparison purpose. Hence, after +algorithm convergence, the GA-UPA performance gap can be calculated as: +∆GN = GN +bound − GN +(32) +where GN is the OF value attained with the GA for N antenna-elements after convergence. Applying +(32) on the values found in Fig. 11.b, the gaps can be established: +∆G6 = 0.0418; +∆G7 = 0.0318; +∆G8 = 0.0423 +The gap indicates the proximity with the best solution possible; hence, for all three simulation +scenarios, the GA-UPA method achieves almost the best solution in terms of the performance gap. +Moreover, by increasing N = 6 to 8, it is possible to verify a decrease in G values for the same number +of generations. Indeed, a decreasing in G is responsible for the gain in the directivity, Fig. 11.a. +5.4. Improved GA-aided UPA Directivity Methods +The GA-aided optimization directivity approach of the previous section demonstrated certain +difficulty in finding the optimal solution when the number of antennas increases. Indeed, considering +exhaustive combinations in the numerical simulations, the GA-ULA could not outperform the pro- +posed OUPA directivity method, in a manageable time, for N > 12, with the constrain of searching +the solution only on the desired plane, which significantly reduces the GA-ULA complexity. +19 + +0 +10 +20 +30 +40 +Generations (g) +2 +4 +6 +8 +10 +12 +14 +Directivity (dBi) + GA | N = 6 + GA | N = 7 + GA | N = 8 +ULA | N = 6 +ULA | N = 7 +ULA | N = 8 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Generation ( g) +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +G +N = 6 +N = 7 +N = 8 +Gbound +6 +Gbound +7 +Gbound +8 +10 +20 +30 +40 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +(a) Directivity +(b) Objective function G values +Figure 11: Directivity and OF values G through the GA-UPA generations for N = 6, 7 and 8. ULA directivity is +included in (a); a lower bound for GN following (24) is defined by dashed lines in (b). +Aiming at using the GA heuristic optimization approach in scenarios with more antennas, we +propose changing the GA initialization parameters. Hence, to solve the UPA element-antenna position +problem in a manageable time, the first approach named GA-marginal includes an initial population +based on a near-local solution. This approach’s main changes are summarized in Table 4. +Table 4: GA-marginal input parameters +Parameter +Alteration +Initial Population +Near-local solution +Mutation (MUniform) +Uniform Mutation +Population Size (psize) +8N 2 +Elements placement bounds +[xmax, ymax] = [2pmax, 2pmax] +Max. # Generations (gmax) +Unlimited +Stopping Criterion +Outperform OUPA +Mutation rate +Determined via simulation +Crossover Fraction (cf) +Determined via simulation +The near-local solution strategy consists in adding the OUPA solution to the initial population, +but with a small perturbation, preventing the GA search from being restricted to the local solutions. +Besides, the mutation function was changed to be uniform, increasing the diversity to escape local +optima. The population size was considerably increased to expand the chances of finding the better +solution at the cost of increasing algorithm complexity. Moreover, the bounds of element placement +were altered considering the OUPA solution, the new values are dependent on pmax, which is the +max value of the coordinate x or y of the OUPA solution. The stopping criterion was modified in a +way that the gmax became unlimited, and the new criterion consisted in incrementing the generations +until GA-marginal outperforms the OUPA method. Furthermore, the mutation rate and the crossover +fraction is now determined by a simulation wherein a combination of both values are evaluated jointly +20 + +for every 100 generations, and then, the combination that achieves the best value of directivity is +selected; the range of both values are set equal and multiple percentages of 10%. +Remark 5: the idea of introducing modifications on the initial population generation in the GA-aided +directivity method aims to facilitate the guided search to attain incremental improvements over the +OUPA solution, in a manageable time, given the non-convexity nature of the problem. Indeed, the +computational difficulty of outperforming our proposed OUPA method in such burden computational +conditions is a clear indication of the expedited and advantageous solution given by the proposed +OUPA method. +However, limiting the GA technique to surpass the proposed OUPA marginally will not give us a +promising perspective on how much gain one can attain in the UPA directivity context using heuristic +evolutionary techniques, and how much the cost to achieve such improvement. Hence, to address this +issue, another GA modification is proposed herein, using almost the same set of parameters described +for the GA-marginal, but changing only the stopping criterion that now consists of achieving hundred +consecutive stalled generations, without directivity gain. This method henceforth will be called GA- +stall. +5.5. Array Directivity Methods: Numerical Comparison +Considering a small number of element-antennas yet, Table 5 exposes values of array directivity +found by the OUPA method, by GA-aided UPA improved solutions, and the two classical steering +vector beamforming, the UCA, and the conventional ULA, as well as an improved directivity method +for the omnidirectional UCA proposed in [15]. +Table 5: Directivity Methods Comparison for small N’s +N +6 +8 +9 +OUPA +11.70 dBi +12.91 dBi +14.12 dBi +GA-marginal +11.81 dBi +13.19 dBi +14.12 dBi +GA-stall +12.35 dBi +13.49 dBi +14.5 dBi +UCA +7.96 dBi +8.73 dBi +9.17 dBi +ULA +9.17 dBi +10.38 dBi +10.88 dBi +UCA [15] +n.a +12.00 dBi +n.a. +The last row in Table 5 depicts the directivity found in [15], which consists in a technique based +on the subspace changes for the omnidirectional UCA directivity. For instance, considering N = 8, +the maximum directivity value found is close to D = 12 dBi. Such value remains remarkably reduced +compared with the optimization methods proposed herein, i.e., ∼ 1.5 dBi less compared with the +GA-stall approach and ∼ 0.9 dBi less when compared with our OUPA approach. +For computational complexity analysis, Fig. +12 exhibits the simulation time vs. # antennas +N ∈ [4; 36], considering the OUPA improved GA-marginal and GA-stall methods proposed, the +GA-marginal and GA-stall. Aiming to maximize the directivity, the quasi-squared UPA condition in +eq. (30) is applied; hence, possible combinations of N1 and N2 was bounded by max(N1, N2) = 6, +implying in the following possible antenna-elements arrangements +N = N1 × N2 → +2 × 2 +2 × 3 +3 × 3 +3 × 4 +4 × 4 +4 × 5 +5 × 5 +5 × 6 +6 × 6 +, +(33) +which are identified by the respective markers in Fig. 12. +The OUPA technique proposed herein, even with far less complexity, can follow the performance +of both GA’s techniques very closely. As expected, the GA-marginal was able to surpass the OUPA +method marginally, proving its sub-optimality. However, the time spent by both GA heuristic meth- +ods was far superior, configuring a far inferior performance-complexity tradeoffs. The GA-stall aided +21 + +UPA directivity optimization method was able to achieve a considerable and consistent improvement +when compared with the OUPA, the amount of time consumed to find such a solution is extensively +large. +5 +10 +15 +20 +25 +30 +35 +Number of Antennas ( N) +10 +12 +14 +16 +18 +20 +22 +Directivity (dBi) +OUPA +GA-marginal +GA-stall +5 +10 +15 +20 +25 +30 +35 +Number of Antennas ( N) +10-2 +10-1 +100 +101 +102 +103 +104 +105 +106 +Time (s) +OUPA +GA-marginal +GA-stall +(a) Directivity +(b) Computational Time +Figure 12: Directivity and run-time comparison between OUPA, GA-marginal and GA with the increasing in the +number of antennas N = N1 × N2. +Remark 6: One can conjecture that among the three proposed directivity methods, the improvement +beyond the OUPA solution provided by the GA-stall aided UPA algorithm comes with a much high +computational cost. Moreover, both GA-stall and GA-marginal optimization schemes have resulted +in less effectiveness than the proposed OUPA method in terms of performance-complexity tradeoff. +Finally, Table 6 shows the crossover fraction (cf) and the mutation rate (mr) values as a function +of dimensionality problem (N), required in the GA-aided methods that maximize the directivity in +the simulation of Fig. 12. For most configurations, the crossover fraction percentage has high values, +Table 6: Input parameters configuration for both GA-marginal and GA-stall algorithms +N +4 +6 +9 +12 +16 +20 +25 +30 +36 +cf (%) +80 +40 +10 +90 +40 +80 +80 +60 +80 +mr (%) +100 +20 +10 +10 +10 +70 +10 +10 +20 +while the mutation rate presents a low percentage. Such a combination lead to a population that has +several changes to escape local optima between the generations, nonetheless, the value of mr remaining +low can be interpreted as a mean of guaranteeing an improvement around the solution. Notice that +in the case N = 20, both GA-stall directivity optimization algorithms selected a high mr percentual +22 + +value; as a result, the simulation time resulted significantly smaller; however, the performance was +incremental better when compared to the OUPA and GA-marginal. The discrepant values found for +N = 4 antennas can be explained by the simplicity of the geometric configuration, therefore, given a +population, a volatile evolution did not prevent the method from finding a remarkable solution. +6. Conclusions and Final Remarks +This work proposes a new approach to maximize the directivity of an omnidirectional volumetric +antenna array. This technique assumes a uniform planar array (UPA) confined on a specific plane +with a minimum distance between the antenna elements optimized by the successive evaluation and +validation (SEV) procedure, namely optimal uniform planar array (OUPA). Moreover, evolutionary +heuristic GA optimization is employed to validate the proposed methodology; hence, for a large +number of antennas, two modifications on the initial parameters are suggested, denoted GA-marginal +and GA-stall, both made the GA a promising optimization tool to solve UPA directivity problem in +such large scale antenna scenario. +An exciting finding is that a plane space constrains the element positioning solutions as a function +of the desired elevation (θ0) and azimuth (φ0) angles. A new method based on the UPA structure is in- +troduced to address this constraint to solve the directivity optimization problem with low-complexity +effort. Then an evolutionary heuristic technique is selected to implement the directivity optimization +analysis devised herein. Indeed, the genetic algorithm (GA) was selected as an expedited optimization +tool. +The proposed methodology is focused on finding the optimal position of each antenna element +that maximizes the antenna array directivity. The solution was confined to a specific plane, where +the antenna elements position on this plane resembling a sequence of equilateral triangles; hence, the +idea of finding the optimal UPA in such a plane is introduced. The OUPA method exploits the OF +features, while in the evolutionary heuristic optimization approach, the directivity based on antenna- +elements position has been formulated and solved for a different number of antenna-elements in the +range N ∈ {4; 36}. Indeed, numerical results deploying the proposed GA-marginal and GA-stall +heuristic evolutionary techniques against the also proposed OUPA method for a different number +of antennas has demonstrated higher directivity gains when compared with the well-known regular +UCA and ULA arrangements, as well as when compared with recent literature design for the specific +case of N = 8 antennas. +The computational complexity reveals a superior performance of the OUPA method, implying +in an excellent performance-complexity tradeoff. Moreover, the OUPA design method has achieved +impressive performance in massive MIMO scenarios (N ≥ 50), achieving a directivity of 30 dBi in an +extensive quasi-squared planar configuration, i.e., N = 15 × 16 antennas. +Acknowledgement +This work was partly supported by The National Council for Scientific and Technological Devel- +opment (CNPq) of Brazil under Grants 310681/2019-7, partly by the CAPES- Brazil - Finance Code +001, and the Londrina State University - Parana State Government (UEL). +References +[1] P. Angeletti and G. Toso, “Array Antennas With Jointly Optimized Elements Positions and +Dimensions Part I: Linear Arrays,” IEEE Transactions on Antennas and Propagation, vol. 62, +pp. 1619–1626, April 2014. +[2] P. Angeletti, G. Toso and G. Ruggerini, “Array Antennas With Jointly Optimized Elements +Positions and Dimensions Part II: Planar Circular Arrays,” IEEE Transactions on Antennas and +Propagation, vol. 62, pp. 1627–1639, April 2014. +23 + +[3] H. R. E. Bouchekara, A. Orlandi, M. Al-Qdah and F. de Paulis, “Most Valuable Player Algo- +rithm for Circular Antenna Arrays Optimization to Maximum Sidelobe Levels Reduction,” IEEE +Transactions on Electromagnetic Compatibility, vol. 60, pp. 1655–1661, December 2018. +[4] J. Diao, J. W. Kunzler and K. F. Warnick, “Sidelobe Level and Aperture Efficiency Optimization +for Tiled Aperiodic Array Antennas,” IEEE Transactions on Antennas and Propagation, vol. 65, +pp. 7083–7090, December 2017. +[5] S. K. Goudos, K. A. Gotsis, K. Siakavara, E. E. Vafiadis and J. N. Sahalos, “A Multi-Objective +Approach to Subarrayed Linear Antenna Arrays Design Based on Memetic Differential Evolu- +tion,” IEEE Transactions on Antennas and Propagation, vol. 61, pp. 3042–3052, June 2013. +[6] S. Jayaprakasam, S. K. Abdul Rahim, C. Y. Leow and M. F. Mohd Yusof, “Beampatten op- +timization in distributed beamforming using multiobjective and metaheuristic method,” 2014 +IEEE Symposium on Wireless Technology and Applications (ISWTA), pp. 86–91, September +2014. +[7] S. Jayaprakasam, S. K. Abdul Rahim, C. Y. Leow, T. O. Ting and A. A. Eteng, “Multiobjective +Beampattern Optimization in Collaborative Beamforming via NSGA-II With Selective Distance,” +IEEE Transactions on Antennas and Propagation, vol. 65, pp. 2348–2357, May 2017. +[8] Tripathy, Malay Ranjan, and Ranjan, Priya, “Multi-objective optimization for planar antenna +array design,” 2019 URSI Asia-Pacific Radio Science Conference (AP-RASC), pp. 1–3, 2019. +DOI: 10.23919/URSIAP-RASC.2019.8738225 +[9] K. Y. Reddy, R. B. Kumar, M. Jijenth, V. S. Gangwar, K. K. Suman and R. K. Gangwar, “An +expeditious synthesis of thinned planar antenna array by exploitation of multi-objective opti- +mization technique,” 2018 3rd International Conference on Microwave and Photonics (ICMAP), +pp. 1–2, February 2018. +[10] A. Rezagholi and F. Mohajeri, “Directivity optimization of fractal antenna arrays using PSO +algorithm,” 2016 24th Iranian Conference on Electrical Engineering (ICEE), pp. 1224–1228, +May 2016. +[11] S. S. Saleem, M. M. Ahmed, U. Rafique and U. F. Ahmed, “Optimization of Linear Antenna Array +for Low SLL and High Directivity,” 2016 19th International Multi-Topic Conference (INMIC), +pp. 1–6, December 2016. +[12] P. Swain, S. K. Mohanty and B. B. Mangaraj, “Linear Dipole Antenna Array design and opti- +mization using Gravitational Search Algorithm,” 2016 2nd International Conference on Advances +in Electrical, Electronics, Information, Communication and Bio-Informatics (AEEICB), pp. 514– +518, February 2016. +[13] S. Yang and J. Kiang, “Optimization of Sparse Linear Arrays Using Harmony Search Algorithms,” +IEEE Transactions on Antennas and Propagation, vol. 63, pp. 4732–4738, November 2015. +[14] B. Pavan Kumar, C. Kumar, V. Senthil Kumar and V. V. Srinivasan, “Optimal Radiation Pat- +tern of the Element for a Spherical Phased Array With Hemispherical Scan Capability,” IEEE +Antennas and Wireless Propagation Letters, vol. 16, pp. 2780–2782, 2017. +[15] G. Huang, J. Benestry and J. Chen, “Subspace Superdirective Beamforming with Uniform Cir- +cular Microphone Arrays” 2016 IEEE International Workshop on Acoustic Signal Enhancement +(IWAENC), Xi’an, 2016, pp. 1-5. +[16] A. Trucco and M. Crocco, “Design of an Optimum Superdirective Beamformer Through General- +ized Directivity Maximization,” IEEE Transactions on Signal Processing, vol. 62, pp. 6118–6129, +December 2014. +24 + +[17] W. Tan and S. Ma, “Antenna Array Topologies for mmWave Massive MIMO Systems: Spec- +tral Efficiency Analysis,” IEEE Transactions on Vehicular Technology, vol., no., pp. 1-15, DOI: +10.1109/TVT.2022.3197600. +[18] B. F. Costa and T. Abrão, “Closed-Form Directivity Expression for Arbitrary Volumetric An- +tenna Arrays,” IEEE Transactions on Antennas and Propagation, 2018, vol. 66, issue: 12 pp.7443 +- 7448, DOI:10.1109/TAP.2018.2869243 +[19] Szymanski, John E., “Basic Mathematics for Electronic Engineers:Models and Applications,” +Taylor Francis. p. 154. ISBN 0278000681. +25 + diff --git a/UtE1T4oBgHgl3EQfIgOA/content/tmp_files/load_file.txt b/UtE1T4oBgHgl3EQfIgOA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b184a971ce86f702cb29583b06e8900fa95d4f29 --- /dev/null +++ b/UtE1T4oBgHgl3EQfIgOA/content/tmp_files/load_file.txt @@ -0,0 +1,844 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf,len=843 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='02940v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='SY] 7 Jan 2023 GA-Aided Directivity in Volumetric and Planar Massive-Antenna Array Design Bruno Felipe Costa, Taufik Abrão Londrina State University (UEL), Department of Electrical Engineering, Londrina, 86057-970, Parana, Brazil Abstract The problem of directivity enhancement, leading to the increase in the directivity gain over a certain desired angle of arrival/departure (AoA/AoD), is considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' A new formulation of the volumetric array directivity problem is proposed using the rectangular coordinates to describe each antenna element, and the desired azimuth and elevation angles with a general element pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Such a directivity problem is formulated to find the optimal minimum distance between the antenna elements dmin aiming to achieve as high directivity gains as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' An expedited implementation method is developed to place the antenna elements in a distinctive plane dependent on (θ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' A novel concept on optimizing directivity for the uniform planar array (OUPA) is introduced to find a quasi-optimal solution for the non-convex optimization problem with low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This solution is reached by deploying the proposed successive evaluation and validation (SEV) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, the genetic algorithm (GA) method was deployed to find the directivity optimization solution expeditiously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For a small number of antenna elements , typically N ∈ [4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' , 9], the achievable directivity by GA optimization demonstrates gains of ∼ 3 dBi compared with the traditional beamforming technique, using steering vector for uniform linear arrays (ULA) and uniform circular arrays (UCA), while gains of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 dBi is attained when compared with an improved UCA directivity method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For a larger number of antenna elements , two improved GA procedures, namely GA-marginal and GA-stall, were proposed and compared with the OUPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' OUPA also indicates promising directivity gains surpassing 30 dBi for massive MIMO scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Keywords: Directivity, Antenna Array, 5G, Optimization, Omnidirectional, Genetic Algorithm (GA), Optimal Uniform Planar Array (OUPA) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Introduction Array antennas can provide many advantages in flexibility and scalability concerning conven- tional antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This is related to the capacity of array pattern reconfiguration and adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Such adaptability enables various applications, including radar communications, satellite communi- cations, wireless communications, radio-astronomy, remote sensing, and direction of arrival (DoA) estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Optimally designed arrays are used to obtain high directivity patterns, improving the array performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Uniform array configurations, such as uniform linear (ULA), planar (UPA), rectangular (URA), circular or cylindrical (UCA) antenna arrays, have been deployed to improve the performance of massive antenna-based 5G and B5G communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Since planar antenna array can deploy a higher number of antenna elements at a limited physical space and capture the beamspace in both the horizontal and vertical directions in three-dimensional (3D) propagation, the development of different planar array configurations, such as URA and UCA, have become a natural choice among the different possible array geometries, representing a great interest in the mmWave massive MIMO system designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Array directivity optimization remains an open issue, especially in cases where the antenna- element position is considered as an optimization variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Indeed, the choice of antenna-elements Preprint submitted to Arxiv (https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='org/) January 10, 2023 position as a parameter to be optimized can be justified by the concept of array virtualization, in which antenna-elements cooperate among each other, forming a virtual antenna array (VAA) with better directivity performance that the original physical antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Recent works on array optimization have designed linear sparse array antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For instance, the algorithm proposed in [1] jointly and deterministically maximizes the aperture efficiency and directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Besides, [2] proposes a planar circular sparse array antenna design, where the positions and dimensions of the radiators are jointly optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In [3] an optimal circular antenna array for maximum sidelobe levels (SLLs) reduction was investigated using a metaheuristic approach called player algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' An optimization of sidelobe level and aperture efficiency for aperiodic array antennas was proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Multiobjective optimization (MOO) is deployed in [5] to efficiently and effectively design subarrays in linear antenna array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the optimization was made using a memetic differential evolution (mDE) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Following the same research direction, the authors in [6, 7] used MOO and metaheuristic methods to optimize the beampattern in collaborative array beamforming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In [8], the MOO technique is deployed to discuss the planar antenna array design under different optimization criteria, such as the side lobe, beam steering quality, minimum power beam width, and maximum directivity, as a function of element lengths and distances between these elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' A fundamental question is exploring the Pareto front modeling optimal trade-offs among multiple conflicting objectives, including the structure size, its performance, and the minimization of cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The well-known heuristic algorithm nondominated sorting genetic algorithm II (NSGA-II) has been used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Many other works are deploying MOO and/or metaheuristics solutions, such as [9], [10], [11], [12] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In [14], the optimal radiation pattern is generated using a spherical phased array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In contrast, in [15], a technique based on changes in the subspace is deployed to improve the directivity of the omnidirectional UCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, the optimum directive beamformer has been proposed in [16], by deploying the concept of generalized directivity maximization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the concept of directivity is based on the mean beam power pattern, which is crucial in assessing and optimizing the performance of directivity arrays subjected to array imperfections and sensor mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Recently, by adopting a 3D wireless channel model, and taking into account both the azimuth and elevation angles of departure (AoD), the authors of [17] derive the probability density function for the distances between randomly distributed users and the BS antenna array, considering three URA, ULA and UCA uniform array topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Also, they derived closed-form expressions for the squared inner product of different channel vectors, facilitating the interuser interference analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Against this background, the current work proposes a new methodology to maximize the directiv- ity of omnidirectional volumetric antenna arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' By assuming a uniform planar array (UPA) confined on a specific plane with a minimum distance between the antenna elements optimized, the proposed method takes in hand the successive evaluation and validation (SEV) procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' To demonstrate effectiveness, the proposed method is implemented by deploying evolutionary heuristic optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, to implement the directivity optimization methodology under a large number of antennas in planar arrays, we suggested two variants of the genetic algorithm: GA-marginal and GA-stall, both made the GA a promising optimization tool to solve UPA directivity problem in such large scale antenna scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' An exciting finding of the proposed methodology is that a plane space constraints the antenna-element positioning solutions as a function of the desired elevation and azimuth angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The contribution of this work is fourfold: a) formulation of optimization problem for directivity maximization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the novelty of this design is brought by the optimization of each antenna-element position bounded by a volumetric constraint on the search space, which is given by a parallelepiped with a central point at the origin and all the points bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' b) a geometric interpretation for the solution in case of omnidirectional scenarios, more specifically, this interpretation results in restraining the position of each antenna element to a plane equation dependent on the desired angles (θ0, φ0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 2 c) using a novel concept based on the UPA geometry, an expedited implementation method is developed consisting in placing the antenna elements in a distinctive plane dependent on θ0 and φ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, using the geometry properties of the UPA, a directivity optimization problem was formulated to find the optimal minimum distance between the antenna-elements dmin aiming to achieve as high directivity gains as possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' d) to corroborate the effectiveness of the proposed method, a genetic algorithm (GA) is deployed to implement the directivity enhancement method with low complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' when compared with a conventional geometric design, the proposed method has demonstrated a significant improve- ment in directivity gains on the desired angle of departure (DoD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Section 2 describes the concept of the ra- diation pattern, directivity optimization via position antenna-element placement, and the adopted system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Section 3 formulates the directivity optimization problem, with particular attention on the optimization problem for the omnidirectional case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' besides, geometric interpretation is evoked, and then the problem is recast in a simplified way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For comparison purposes in terms of directivity, different geometric array structures are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' A novel directivity optimization method applied to the UPA geometries is devised in Section 4 as an effective low-complexity optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Numerical results are developed in Section 5 by deploying heuristic GA as a practical implementa- tion tool supporting our finding on the UPA-basis antenna array directivity optimization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Concluding remarks are pointed out in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Radiation Pattern and Directivity via Element Positioning For non-isotropic antennas, the directivity is the level of radiated signal power in a specific di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This is especially important for applications such as beamforming in massive MIMO 5G wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 1 depicts the general concept of directivity enhancement via an array of antenna elements with low directivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the constructive and destructive interference can be guided, by changing the position and/or phase of each element, aiming to achieve a high directivity antenna pattern [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In this work, the omnidirectional antenna-element configuration has been chosen to implement the array directivity enhancement, which is a more realistic scenario compared to the isotropic sources, while remaining a suitable configuration to be analytically analyzed due to its sim- ple element factor, expressed by Υ(θ) = cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Furthermore, the applicability of this configuration is interesting in many fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Such configuration dramatically simplifies the analytical directivity expression, while the non-convex directivity optimization problem remains a special issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Radiation Pattern of an Antenna Array An antenna array is a grouping of N antennas whose geometry can assume many different forms, such as uniform linear, circular, rectangular, non-uniform grouping, and so forth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' moreover, all ele- ments in the array work jointly to increase the directivity of the arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The entire field of an antenna array is obtained by multiplying the field of a single element Υe(θ, φ) and the array factor Υa(θ, φ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' this is called the radiation pattern of an antenna array and can be written as: Υ(θ, φ) = Υe(θ, φ)Υa(θ, φ) (1) The element factor (EF) is the radiation pattern of a single-element antenna, and the model must resemble the pattern of a real antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Using spherical coordinates, the EF is periodic in θ and φ angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Thus, the radiated power of an arbitrary antenna can be described using Fourier analysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' it can be written using a linear combination of powers of cosine and sine functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Hence, the EF can be written: Υe(θ) = sinu (θ) cosv (θ) (2) The array factor (AF) describes a combination of radiating elements in an array without con- sidering the element radiation pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Thus, the array factor can be interpreted as the radiation 3 Figure 1: Directivity representation in a) isotropic source, b) omnidirectional source, and c) via constructive and destructive interferences of low-directivity sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' pattern by replacing the actual elements by the isotropic (point) sources, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', Υe(θ, φ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The AF has a dependency in terms of position, relative phase and relative amplitude of each antenna element and is given by: Υa(θ, φ) = N � n=1 Anej(αn+kpn·ap) (3) where N is the number of antenna elements, An is the relative amplitude of n-th antenna element, αn is the relative phase of n-th antenna element, pn is the position vector of n-th antenna element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' k is the wave number, and ap is the unit vector of observation point in spherical coordinates, given by: ap = sin(θ) cos(φ)ˆi + sin(θ) sin(φ)ˆj + cos(θ)ˆk (4) The position vector can be written as a combination of the elements in the 3-D cartesian axis: pn = xnˆi + ynˆj + znˆk (5) In this work, we have considered the position of each antenna element of the array as the variable to be optimized, which the matrix can mathematically defined: P = � p1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' pN �T (6) where the position vector pn defines the localization of the n-th antenna element in the 3-D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Directivity for Arbitrary Volumetric Antenna Arrays Directivity can be defined as the level of irradiation signal power in a specific direction to the detriment of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This is especially important for some applications, such as beamforming in non-isotropic massive MIMO antennas in 5G wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Antenna directivity can be written as the ratio between the radiation intensity in a desired angle and the sum of the radiation intensity in all the other directions: D(θ0, φ0) = |Υ(θ0, φ0)|2 1 4π 2π � 0 π� 0 |Υ(θ, φ)|2 sin (θ)dθdφ (7) 4 High Directivity Patternenna ray Constructive and Destructive InterferencesAnt Al n-th Antenna Yncos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0 nal SourceMe= 1 Isotropic Source Omnidirecti Element Factor with low Directivitywhere |Υ(θ, φ)|2 is the radiation intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Besides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' for an arbitrary array with the general element pattern (cosv θ sinu θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' ∀ u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the directivity expression can be defined as [18]: D(θ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' φ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' v) = f1 f2 = sin2u (θ0) cos2v (θ0) N � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='n=1 AnAmξmn cos [Ωmn] In(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' v) + Im(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' v) (8) where In(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' v) = N � n=1 A2 n �1 8((−1)2v + 1)B(u + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' v + 1 2) � Im(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' v) = 2 (−1)(v+2u) N � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='m=1 m̸=n n>m u � κ=0 AnAm �u κ � cos (αmn) ∂2(v+u−κ) ∂z2(v+u−κ) mn � sin(k � β2 + z2mn) k � β2 + z2mn � Ωmn = Ω(pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' pm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' αn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' αm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' θ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' φ0) = k [xnm sin θ0 cos φ0 + ynm sin θ0 sin φ0 + znm cos θ0] + αnm pn = xnˆi + ynˆj + znˆk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' β = � x2nm + y2nm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' xnm = (xn − xm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' ynm = (yn − ym),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' zmn = (zn − zm) ξmn = � 1 m ̸= n 1 2 cos [Ωmn] m = n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Re{v} > −1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Re{u} > −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' αmn = (αn − αm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (9) where pn is the position vector of n-th antenna element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' αn is the phase of n-th antenna element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' An is the amplitude of n-th antenna element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' θ0 is the desired elevation angle, where directivity is evaluated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' φ0 represents the desired azimuth angle, where directivity is evaluated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' and k is the wave number of the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Directivity Optimization Problem in Omnidirectional Scenarios Given a desired angle of departure (θ0, φ0) over an antenna array composed of N elements, we are looking for the parameter values of an array configuration that result in the directivity maximization in such an AoD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' We start the analysis assuming non-zero-phase and the non-unitary amplitude in each antenna;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, we will solve the proposed problem to obtain a geometric configuration for each desired angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For typical applications, the desired direction changes regularly, which becomes a problem, given that changes in the positions of antennas are physically impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' However, in this formulation, the phase element was not considered, leaving space to adopt virtualization techniques, which highly depend on the phase elements and will be the subject of our future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Based on the directivity expression provided in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (8), we state the following optimization problem: maximize P D(θ0, φ0) = f1 f2 subject to pn ⪯ pmax for n = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' , N (10) where the negative semi-definite pn ⪯ pmax defines the 3-D parallelepiped volume bounded by [xmax, ymax, zmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Besides, the optimization problem in (8) can be rewritten as a minimization problem: minimize P f2(P , θ0, φ0) f1(P , θ0, φ0) subject to pn ⪯ pmax for n = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' , N (11) Therefore, the objective of this problem can be reinterpreted as minimizing f2(P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' At the same time, simultaneously maximize f1(P ) subject to the bounds in the space configuration, given, for instance, by the 2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5 The omnidirectional scenario has been selected due to the concise expressions, analysis, and manageable complexity while the applicability remains in many exciting fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The omnidirectional radiated power is also scattered symmetrically but not equally in all directions, creating a 3-D torus radiation pattern, which can be formulated as the expression Υ(θ) = cos θ for all azimuth angles (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Hence, considering the element pattern given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (2), for the omnidirectional scenario, we set u = 0 and v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Such condition simplifies the auxiliary functions as: f o 1 (θ0) = cos2 (θ0) N � m,n=1 AnAmξmn cos [Ωmn] (12) and f o 2 = N � n=1 A2 n 6 − 2 N � m,n=1 m̸=n n>m AnAm ∂2 ∂z2mn � sin(k � β2 + z2mn) k � β2 + z2mn � (13) where dmn = k � β2 + z2mn (14) Generically, dmn represents the Euclidean distance between the m-th and n-th antenna element in the 3-D space, given by dmn = k � x2mn + y2mn + z2mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (10), we are interested in maximizing f o 1 while minimize f o 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For this purpose, we can verify in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (12) that the only way to ensure these conditions consists in maximizing cos [Ωmn], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Ωmn = c1π, for c1 2 ∈ Z c1 even (15) Using the definitions in (9), and satisfying the condition above, for Ωmn we have: xnm sin θ0 cos φ0 + ynm sin θ0 sin φ0 + znm cos θ0 = c1π − (αn − αm) k (16) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (16) can be interpreted as the plane equation containing all the vectors of position difference (pmn = pn − pm), to fulfill this condition entirely it is necessary all vectors be restrained in the same plane of the differences position vectors since the constant c1 is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For each value, we have a different plane, the solution of this condition is given by infinite parallel planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' However, the combination of position vectors in these parallel planes does not fulfill the condition, given that the difference vector for points in different parallel planes will result in a vector difference in another plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, minimizing f o 2 results in a more complicated task, given that the directivity problem in (7) requires minimization of the summation of all terms in the right-hand side of (13), instead of minimizing each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Hence, the f o 2 minimization problem can be recast in an alternative version as: maximize P N � m,n=1 m̸=n n>m AnAm ∂2 ∂z2mn � sin(k � β2 + z2mn) k � β2 + z2mn � (17) Besides, the second derivative in (17) can be re-written as: ∂2 ∂z2mn � sin( � β2 + z2mn) � β2 + z2mn � = (β2 − 2z2 mn) cos dmn d4mn − � (β2 − 2)z2 mn + β2 + z4 mn � sin dmn d5mn (18) 6 which is function of the xmn, ymn and zmn, and related to β and dmn through the constraints in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (8) and (14), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Notice that in this formulation, the omnidirectional configuration has influence only in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For another scenarios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', different values of u, v ̸= 0, the only change will be the objective function (OF) in (17), where the order and number of derivatives dependent on the values of u, v ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Recasting the Simplified Omnidirectional Problem One of the most important constraints resulting by the omnidirectional scenario is given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (16), and where the values of c1 can be chosen arbitrarily;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, aiming to simplify the analysis we set c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' considering all null phases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' for the n-th and m-th antennas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' we obtain: sin θ0 cos φ0xn + sin θ0 sin φ0yn + cos θ0zn = 0 sin θ0 cos φ0xm + sin θ0 sin φ0ym + cos θ0zm = 0 (19) By subtracting both previous expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' one can obtain: znm = tan θ0(cos φ0xnm + sin φ0ynm) (20) The restriction (20) can be incorporated into (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' finally resulting in a simplified optimization prob- lem: minimize x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='y G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' y) = − N � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='n=1 m̸=n n>m F(xmn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' ymn) subject to |xmn| ≤ xmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' |ymn| ≤ ymax for n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' , N (21) with OF defined by (18), and substituting β2 = d2 mn k − z2 mn, results: F(xmn, ymn) = � (d2 mn − 3k)z2 mn + d2 mn � sin dmn kd5mn − (d2 mn − 3kz2 mn) cos dmn kd4mn (22) The function G can be rewritten using the definition in (13), implying the following relation: f o 2 = N � n=1 A2 n 6 + 2G (23) Since f o 2 represents the denominator integral in (7), for the omnidirectional scenario this values must be positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' therefore, the function G has a lower bound for each value of N, given by: GN bound = − N � n=1 A2 n 12 (24) The bound on zmn is implicit, having dependency on xmax, ymax, and on the desired angle of departure (θ0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The solution of (21) give us the optimal coordinates xn and yn, ∀n which can be used to find the optimal zn, ∀n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' , N coordinate from (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Geometric Planar Arrays Comparison In this section, different geometric planar array structures are considered for a directivity com- parison purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' It is shown the importance of setting the adequate dmin and the associated area occupied by the array, aiming to exploit the potential of the directivity improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' All the selected arrays are planar, aiming to preserve the constraint in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Adopted parameter values and antenna structure include: 7 Parameter Adopted Values # antennas N = 16 Normalized wavelength λ = 1 [m] Desired direction θ0 = φ0 = π/4 uniform planar array UPA uniform circular array UCA uniform hexagonal planar array UHPA The selected uniform planar array geometries with N = 16 elements are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 2 with the minimum distance between antenna elements dmin represented by red arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 2(d) exhibits the directivity dependence regarding the increasing dmin values for each selected uniform array geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The planar and the hexagon geometry attained superior directivity than the circular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='dmin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='Y Axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='Z Axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='X Axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='(a) Uniform Planar Array (UPA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='X Axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='dmin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='Y Axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='(b) Uniform Circular Array (UCA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='Y Axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='X Axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='dmin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='Z Axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='(c) Uniform Hexagonal Planar Array (UHPA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='dmin (m) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='Directivity (dBi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='UHPA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='UCA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='UPA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='(d) Directivity versus the minimum distance for the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='three different geometry arrays: UPA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' UCA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' and UHPA Figure 2: Different planar array geometries and respective attained directivity with N = 16 elements on the desired plane defined by θ0 = φ0 = π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the optimal value of dmin for both structures are close to each other and significantly smaller than the circular array dmin value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For this application, the UCA has demonstrated do not be suitable and will be suppressed hereafter in further analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 8 These illustrative results express the importance of setting the adequate dmin to exploit the di- rectivity enhancement’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' However, the area occupied by the array is also essential, since compact antenna arrays are more valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Although the value of dmin is smaller for the UPA, the area is not necessarily smaller than the UHPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' To investigate this aspect, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 3 depicts directivity and the normalized area given by the convex hull of the arrangement of points for both promising array structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' in (a) the value of directivity and in (b) the value of area occupied by the convex hull of the array, selecting the optimal value of dmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' It is apparent in both graphics the remarkable directivity performance proximity for both structures and respective normalized areas with the in- crease in the number of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The area occupied by the convex hull for small values of antennas are very similar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' however, for the maximum directivity configuration and under a significant number of antenna elements, the UPA demonstrated a better performance covering a smaller region with the increasing of the antennas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' besides, the points of the UHPA where the performance are significantly better is given when the hexagon formed by the array is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Therefore, on can conjecture that both configurations have a very similar directivity performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' At the same time, for a significant number of antennas, the UPA surpasses the UHPA directivity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' area performance tradeoff unless the hexagon formed by the UHPA is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 0 50 100 150 200 250 Number of Antennas (N) 14 16 18 20 22 24 26 28 30 32 Directivity (dBi) UPA UHPA (a) Directivity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the number of antennas increasing (N) 0 50 100 150 200 250 Number of Antennas (N) 0 50 100 150 200 250 Area (m2) (b) Array area vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the number of antennas (N) increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Figure 3: UPA against UHPA in terms of directivity and area, both with optimal setting of dmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Given the attained near directivity between UPA and UHPA, both geometries could be selected without losses in terms of directivity gain and occupied area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Therefore, the structure selected for the optimization method in the next section is the UPA, due to the compactness in describing the antenna elements positioning over a rectangular coordinates system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Optimizing the Position of UPA Elements This section describes the proposed method to achieve high directivity using the regular planar array structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This method provides reasonable solutions for the proposed optimization problem of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (21) with low computation effort considering the high-complexity operation of searching the geometric locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The proposed method consists in assuming a given number of waves (k), a desired direction (θ0,φ0), and a number of antennas (N), finding the optimal UPA on the desired plane defined by (20) aiming to attain a remarkable improvement in the directivity in a desired direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The most critical parameter to be defined to achieve this goal is dmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The following describes a procedure for finding this parameter for a generic UPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Firstly, we can consider the positioning of the antennas on the xy plane, considering a UPA with N1 × N2 antenna elements regularly distributed, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 2 X axis (dmin ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 2 Y axis (dmin ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 1 2 3 2N1 3 + N1 2 + N1 1 + N1 1 + 2N1 3 + 2N1 3N1 2 + 2N1 N2N1 N1 2 + (N2-1)N1 1 + (N2-1)N1 3 + (N2-1)N1 Figure 4: UPA elements disposition and labeling for N1 × N2 antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' From this antenna-elements distribution, one can define the distance between two generic elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e, dmn for m ̸= n, which is an important parameter on the use of (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The distance dmn can be formulated as: dmn = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 |m−n|dmin N1 if ψ1 = 0 |m − n|dmin if ψ2 = 0 and |m − n| < N1 dmin � ψ2 1 + ψ2 2 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='c (25) where: ψ1 = mod(m − 1, N1) − mod(n − 1, N1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' ψ2 = �m − 1 N1 � − �n − 1 N1 � (26) To allocate the array on the desired plane without altering the distances, it is possible to use a rotation matrix considering the desired direction, defined by: Rv = \uf8ee \uf8f0 sin2 φ0µγ + cos γ − cos φ0 sin φ0µγ − cos φ0 sin γ − sin φ0 cos φ0µγ cos2 φ0µγ + cos γ − sin φ0 sin γ cos φ0 sin γ sin φ0 sin γ cos γ \uf8f9 \uf8fb (27) where γ = acos | cos θ0| and µγ = (1 − cos γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' notice that for θ0 = 0, the restriction (20) becomes the xy-plane and this rotation matrix becomes an identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The position matrix of the array on the desired plane can be expressed as: �P = P Rv (28) Besides, the variable zmn, which consists in a combination of subtractions on the last column of �P can be described as: zmn = −(ψ1 cos φ0 sin γ + ψ2 sin φ0 sin γ) (29) Now, with the parameters dmn and zmn available, the OF analytical expression in (21) can be evaluated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' however, this is a non-convex function of one variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' At this point, the directivity 10 optimization problem can not be solved using convex optimization tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' On the other hand, notice that finding the global optimum of this function, either by exhaustive search or by quasi-optimal evolutionary heuristic methods, the optimal UPA to be placed on the desired plane can be found, given the advantage of allocating the antenna-elements on the same plane, while finding the optimal general UPA in terms of directivity in that plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Remark 1: The general methodology is focused on finding the optimal position of each antenna element that maximizes the antenna array directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In this section, we have developed a solution confined into a specific plane, where the antenna-elements position on this plane resembles a sequence of equilateral triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, the idea of finding the optimal UPA antenna-elements placement in the such plane is described by the position matrix, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (28), and solved applying the simplified optimization problem, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Successive Evaluation and Validation (SEV) With the dmn and zmn values, the OF of the directivity optimization problem in (21) for the UPA can be expressed and therefore emerges the necessity of finding the optimal value for dmin that enhances the array directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Based on numerical results, we concluded that the first local minimum of the OF in (21) is also the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5 depicts the value of G for different values of N1 and N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' As suspected, for all analyzed N1 × N2 configurations, the first minimum local is also 1 2 3 4 5 6 7 8 9 10 dmin (m) 3 2 1 0 1 2 3 4 5 1 2 2 2 5 2 4 4 9 2 Global min Figure 5: Values of G against dmin values for different configurations of N1 × N2 the minimum global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Taking advantage of this aspect, we propose in Algorithm 1 the successive evaluation and validation (SEV) method, which basically is a line search procedure, evaluating the function at the point (d0 = c), with an increment closely to the origin (0+);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' then, an increment is computed again for (d1 = d0 + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Hence, if the increment brought a decreasing of the OF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', Gn+1 < Gn, then the process is repeated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' if not the point dn is declared the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' A straightforward idea is behind the SEV algorithm: for a minimal input parameter c value, the number of iterations increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' however, the precision of the solution improves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' on the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' for large c values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the number of iterations and precision can be reduced remarkably,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' and depending on how significant is such value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the first local minimum could be inadvertently skipped,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' losing the 11 Algorithm 1 SEV – Successive Evaluation and Validation 1: Input: c 2: n = 0 3: dn = (n + 1)c 4: Gn = − N � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='n=1 m̸=n n>m F(dn) : use (21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (29) with dn = dmin 5: while Gn+1 − Gn ≤ 0 do 6: n = n + 1 7: end while 8: dmin = dn 9: Output: d⋆ min optimum solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Hence, the step parameter c must be carefully selected to the SEV achieves a good precision-complexity tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Proposed Optimal Uniform Planar Array (OUPA) The technique that allocates omnidirectional element-antennas arranged as uniform planar array (UPA) on a specific znm plane, given by (20), and using the SEV method to select the d⋆ min is denominated hereafter optimal uniform planar array (OUPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' We propose this straightforward, low- computational cost, quasi-optimal method solution that significantly enhances the directivity in UPA antennas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' besides, OUPA design requires few parameters: the carrier frequency/wave number (f/λ), the desired directivity angles (θ0, φ0) and the geometric UPA structure configuration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', N1 × N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' A pseudocode for the OUPA technique is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Algorithm 2 OUPA – Optimal Uniform Planar Array 1: Input: f/λ, θ0, φ0, N1, N2 2: Calculate all possible dmn values using (25) 3: Calculate all possible zmn values using (29) 4: Evaluate G in (21) using the values of dmn and zmn 5: Determine d⋆ min via SEV method (Algorithm 1) 6: Determine the UPA matrix of position P using d⋆ min via (6) 7: Final array position evaluated by �P (θ0, φ0) = P Rv, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (28) 8: Output: �P Remark 2: The OUPA method exploits the OF features, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (22), in the simplified optimization formulation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (21), while for generic N1 × N2 antennas configuration, a practical quasi-optimal solution is developed in the next section (specifically, subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='3) based on the evolutionary heuristic optimization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The array directivity subject to antenna-elements positioning has been formulated and solved for various elements in the range N ∈ {4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 36}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Numerical Results Different OUPA performance aspects are numerically evaluated, including a) directivity perfor- mance, b) occupied area, c) comparison with other directivity optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The general parameters for the omnidirectional directivity optimization scenario deployed throughout this section are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' OUPA Directivity Performance Numerical simulations have been conducted considering different values of N1 × N2 antenna ar- rangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For each configuration, the value of d⋆ min is obtained using the SEV procedure, and then the relation directivity-area occupied by the array is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The goal is understanding how the 12 Table 1: General simulation setup Parameter Adopted Values Angle of Departure (θ0 ,φ0) = (π 4 , π 4 ) Element Amplitude An = 1 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Element Phase αn = 0 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Omni-directional Scenario u = 0 and v = 1 occupied area impacts the planar array directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The occupied array area is an essential parameter since compact antenna arrays are far more helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Table 2 depicts the selected parameter values deployed in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Table 2: Simulation setup for OUPA directivity-area evaluation Parameter Adopted Values Frequency / Wave Number 5 GHz / k ≈ 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='8 m−1 SEV parameter (increment) c = 10−3 Number of N = N1 × N2 antennas-elements # Vertical elements N1 = [2, 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 50] # Horizontal elements N2 = [2, 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 10] 0 100 200 300 400 500 Number of Antennas (N) 5 10 15 20 25 30 35 Directivity (dBi) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 Area ( m2) Directivity Area 0 100 200 300 400 500 600 Number of Antennas (N) 40 42 44 46 48 50 52 54 56 dmin (mm) dmin = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='28 dmin = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='88 dmin = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='47 dmin = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='06 dmin = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='69 dmin = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='10 dmin = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='51 Figure 6: a) Directivity and array Area values for different N = N1 × N2 element-antennas configurations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' b) corre- spondent d⋆ min values obtained via SEV procedure for different configurations of N = N1 × N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 6, the values of directivity, array area and correspondent d⋆ min are depicted for a wide range of N element-antennas values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' As expected, increasing N, the directivity and array area are increased proportionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Since the array area increment is not desired, the directivity-area tradeoff must be carefully evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The correspondent d⋆ min values found by simulations using SEV procedure are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The most important feature of this plot is that the optimal value of dmin has a discrete distribution regarding N1 and N2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' besides, the difference between the levels of optimum values is constant, which is very interesting and can be used to reduce significantly the search space of the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Another essential characteristic is that the increase in N not necessarily increases d⋆ min value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Finally, the N1 and N2 values impact the optimal dmin value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, this effect is analyzed in more details in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' To gain a deep understanding of the Area and Directivity dependence, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 7 depicts directivity 13 values against array area for different configurations of N = N1 × N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Observing the Pareto frontier among conflicting directivity vs occupied area by the UPA array is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' As expected, it is necessary to increase the array area to achieve higher directivity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The Pareto front establishes the optimal trade-off between both conflicting parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' possible solutions (points) above this frontier are unfeasible for such an array structure, and the points below are sub-optimal in the directivity-area tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4 Area (m 2) 5 10 15 20 25 30 35 Directivity (dBi) ��Directivity Pareto Frontier Figure 7: Pareto front for directivity vs occupied area of UPA arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Directivity vs Area under Different N1 × N2 Arrangements Due to constructive aspects and integer combination of N1 and N2, the structure realization for a given area is constrained, given that the total number of antennas is a multiplication of two integers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', N1 × N2 = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, most of the analyses available in the literature focus on the UPA structures, with squared N1 = N2 antenna arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In this work, we also have analyzed rectangular N1 ̸= N2 antenna arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' To investigate the impact on the directivity, four different UPA structures with N = [36;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 48;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 72] antennas were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='a) depicts possible achievable directivity values and respective UPA normalized area (assuming k = 1) for possible N1 and N2 combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The directivity value and area occupied by N antennas are compared considering different UPA configurations, each column representing the same number of antennas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the markers on the same column indicate specific configurations of N1 × N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' As indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='b), decreasing the difference between N1 and N2 results in better values of maximum attainable directivity1 as a function of dmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' However, the area occupied by the UPA also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' On the other hand, choosing configurations with a great disparity among N1 and N2 implies a substantial directivity reduction, being possible to attain a similar performance, or even be surpassed, when compared with the same UPA structure but with a smaller number of antennas, for instance, 36 × 2 against 6 × 8, in which, despite the great difference of 24 antennas, the directivity are quite similar in both array arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' As the focus of this work is the directivity enhancement and aiming at exploiting the UPA structure, the difference between N1 and N2 must be reduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' therefore, the choice for N1 and N2 1In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='b), the variation of dmin causes an impact on the value of directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' It is possible to notice that better directivity values are achieved when N1 = N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Indeed, these numerical results indicate that the smaller difference between N1 and N2 better is the array directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 14 35 40 45 50 55 60 65 70 75 80 Number of Antennas (N) 18 19 20 21 22 23 24 25 Directivity (dBi) 10 20 30 40 50 60 70 80 90 100 Area (m2) Directivity Area 6×6 9×4 12×3 18×2 10×6 12×5 20×3 30×2 24×2 9×8 18×4 24×3 36×2 6×8 12×4 16×3 15×4 12×6 0 2 4 6 8 10 dmin(m) 8 10 12 14 16 18 20 22 Directivity (dBi) 6x6 9x4 18x2 36x1 (a) (b) Figure 8: a) Directivity and Area vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' N for different configurations with the same number of antennas N = [36;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 48;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' b) Directivity vs dmin fo four possible N1 × N2 arrangements in an UPA with N = 36 antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' should respect the following condition: |N1 − N2| = � 0 squared (UPA) 1 rectangular (UPA) (30) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Genetic Algorithm solving Non-Convex Directivity Problem Since the original directivity optimization is a non-convex problem, in this subsection, the perfor- mance of the genetic algorithm (GA) in solving the omnidirectional uniform array directivity problem is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The pseudocode for the GA deployed to find the enhanced directivity solutions is pre- sented in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Besides, the GA parameter values deployed in this context are summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Algorithm 3 GA-UPA – GA Optimizer for Omnidirectional Uniform Antenna Array 1: Input: nvars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' cf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' gmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' MGaussian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' psize 2: Population ← InitializePopulation(psize,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='nvars) 3: S = G(Population) 4: sbest ← s which results in the minimum value of G(s) 5: while # number of Generation < gmax do 6: Parents ← SelectParents(Population,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='psize) 7: Children ← ∅ 8: for [Parent1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='Parent2] ∈ Parents do 9: [Children1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='Children2] ← Crossover(Parent1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='Parent2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='cf) 10: Children ← MGaussian(Child1) 11: Children ← MGaussian(Child2) 12: end for 13: S = G(Children) 14: sbest ← s which results in the minimum value of G(s) 15: Population ← Replace(Population,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Children) 16: end while 17: Output: sbest 15 Table 3: Simulation Setup used in the GA-UPA directivity Parameter Adopted Values Wave Number / Wavelength k = 1 [m−1] / λ = 2π [m] # antenna-elements N ∈ {4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 9} Elements placement bounds [xmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' ymax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' zmax] = [5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' zmax] Objective function (OF) eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (21) Number of Variables, (nvars) 2N Crossover Fraction (cf) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='7 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' # Generations (gmax) 40 Mutation (MGaussian) Gaussian Initial Population Feasible and Random Population Size (psize) 200N Population Type Double Vector Search space [0, xmax];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' [0, ymax] Remark 3: GA has been selected as a powerful, well-established evolutionary technique to solve non-convex optimization problems, among other also well-known techniques, such as particle swarm optimization (PSO), ant colony optimization (ACO), artificial bee colony algorithm (BCA), and other evolutionary algorithms (EA) optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' GA or EA applies the natural evolution principles to find an optimal local solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In GA, the problem is encoded in a series of bit strings manipulated by the algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the decision variables and problem functions are deployed directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The main drawbacks of an EA are: a) it is much slower2 than alternatives such as the gradient-based and Simplex methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' b) as problem size scales up, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', ten to a hundred or a thousand decision variables, an EA is often overwhelmed by the dimensionality of the problem, being unable to find a solution close to a locally optimal solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' c) a solution is acceptable only in comparison to other, previously discovered solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' indeed, an EA has no concept of an optimal solution, or any way to test whether a given solution is optimal, even locally optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' d) An EA never really knows when to stop, since it does not know whether a given solution is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Finally, EAs usually finish running manually by the user, or by a predetermined limit on the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 9 depicts the solution of problem eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (21) found by the GA for N = 6, 7 and 8 antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' When the solution is visualized in the 3-D plots, it is possible to observe the plane that restrains the solution and how the solutions found strive to attain a significant number of equal Euclidean distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' However, when analyzing the projection of the solution in the xy-plane, it is possible to verify the distortion introduced by the term znm, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The evolution of antenna-element position through the GA-UPA generations is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the left plots show the placement of the coordinates (x, y) found by the GA solution and the right plots indicate the distribution of the difference-coordinate points (xmn,ymn), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', all possible differences between the correspondent coordinates (x, y), together with the contour plots of the objective function F defined in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Notice that in the right plots, the number of points on each plot is given by the combinations of all possible differences (xmn,ymn), which result in N(N − 1) points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Another interpretation for these coordinates points is that they represent the terms of each summation in G, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The evolution of the antenna-element positions through the GA generations is given in the xy-plane projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, it is possible to observe the evolution of the coordinates points (xmn, ymn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' this evolution leads to an arrangement where all allocated points are concentrated in sites of small values of F, regarding the range of all the possible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The selection of the difference-coordinate points (xmn, ymn) can not be performed freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Indeed, these points result in Euclidean distances constraints, which in some cases did not physically feasible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' therefore, the obvious solution consisting in allocating all points to the same minimum of F is not 2often by factors of a hundred times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 16 Y Axis X Axis 6 6 4 4 2 6 Z Axis 0 2 4 2 2 0 4 0 2 2 4 4 (a) N = 6 antennas 6 4 2 0 2 4 6 X Axis 6 4 2 0 2 4 6 Y Axis (b) N = 6, xy-plane 4 5 2 0 Z Axis 5 2 X Axis 0 4 Y Axis 0 5 5 (c) N = 8 6 4 2 0 2 4 6 X Axis 6 4 2 0 2 4 6 Y Axis (d) N = 8, xy-plane Figure 9: GA solution for N = 6, 8 UPA antennas: (left) final element-antennas position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (right) projection onto the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 17 5 4 3 2 1 0 1 2 3 4 5 X axis 6 4 2 0 2 4 6 Y Axis (a) 1th Generation (b) 1th Generation 6 4 2 0 2 4 6 X Axis 6 4 2 0 2 4 6 Y Axis (c) 35th Generation (d) 35th Generation Figure 10: Evolution of GA-UPA solutions for N = 8 antennas: (left) antenna-element position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (right) distribution of (xmn,ymn) and contour lines for the objective function F of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 18 10 8 6 44 6 8 102 mn 0 2 4 6 8 10 10 8 6 4 2 0 2 X mn10 8 6 44 6 8 102 mn 0 2 4 6 8 10 10 8 6 4 2 0 2 X mnpossible due to the implication that all Euclidean distances must be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Indeed, the modifica- tion of position in one element of the antenna implies in the allocation of N − 1 points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, the allocation of these points must be resolved simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Remark 4: The GA solution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 10) converges to a geometric figure similar to the points on a regular triangular tiling (RTT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the only reason for the convergence is not exactly an RTT geometry is found examining plane constrain zmn, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, for θ0 = φ0 = 0, the solution will be given by a set of points allocated on an RTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This arrangement presents symmetry which imposes that all possible distances can be reduced remarkably by repetition of points regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' As a result, the difference-coordinate points (xmn,ymn) can be distributed exploiting such standard, selecting the feasible points corresponding to small values of F and using the repetition to allocate more points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Comparison with the UCA and ULA: The values of directivity found with GA-UPA is compared with the two classical steering vector beamforming: a) the uniform circular array (UCA), and b) the uniform linear array (ULA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The directivity using steering vector consists of changing each antenna element’s phase, aiming to minimize the array factor in (3), depending on the position of each element in a well-defined geometric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For the ULA and UCA the steering vector is given, respectively, by: αula n = −dn cos θ0 ˆk (31) αuca n = −r sin γn sin θ0 cos φ0 ˆi − r cos γn sin θ0 sin φ0 ˆj where: γn = (n−1)2π N , and d is the regular distance between adjacent element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' In this formulation, r is the ratio of the circle that contains all the points of the UCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This ratio was considered so that the minimal distance between the antenna’s element was defined as half carrier wavelength, λ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The values of directivity D through the generations of the GA-UPA is compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='a with the regular ULA design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' it is noticeable the excellent performance increase in the early generations, being capable of surpassing the conventional ULA design after the first generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The convergence of the proposed algorithm is observable with the stationary directivity performance over the generations, which occurs after g = 21, 20 and 35 generations for N = 6, 7 and 8 antenna elements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The sum of all the difference-coordinates points (xmn,ymn) corresponding to the OF in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (21) typically evolves through generations, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='b for N = 6, 7 and 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' also, lower bound values can be calculated using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (24) and considering An = 1, ∀ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The bound values G6 bound = − 1 2, G7 bound = − 7 12 and G8 bound = − 2 3 are included for comparison purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Hence, after algorithm convergence, the GA-UPA performance gap can be calculated as: ∆GN = GN bound − GN (32) where GN is the OF value attained with the GA for N antenna-elements after convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Applying (32) on the values found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='b, the gaps can be established: ∆G6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0418;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' ∆G7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0318;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' ∆G8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='0423 The gap indicates the proximity with the best solution possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, for all three simulation scenarios, the GA-UPA method achieves almost the best solution in terms of the performance gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, by increasing N = 6 to 8, it is possible to verify a decrease in G values for the same number of generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Indeed, a decreasing in G is responsible for the gain in the directivity, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Improved GA-aided UPA Directivity Methods The GA-aided optimization directivity approach of the previous section demonstrated certain difficulty in finding the optimal solution when the number of antennas increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Indeed, considering exhaustive combinations in the numerical simulations, the GA-ULA could not outperform the pro- posed OUPA directivity method, in a manageable time, for N > 12, with the constrain of searching the solution only on the desired plane, which significantly reduces the GA-ULA complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 19 0 10 20 30 40 Generations (g) 2 4 6 8 10 12 14 Directivity (dBi) GA | N = 6 GA | N = 7 GA | N = 8 ULA | N = 6 ULA | N = 7 ULA | N = 8 0 5 10 15 20 25 30 35 40 Generation ( g) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 2 G N = 6 N = 7 N = 8 Gbound 6 Gbound 7 Gbound 8 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='3 (a) Directivity (b) Objective function G values Figure 11: Directivity and OF values G through the GA-UPA generations for N = 6, 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' ULA directivity is included in (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' a lower bound for GN following (24) is defined by dashed lines in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Aiming at using the GA heuristic optimization approach in scenarios with more antennas, we propose changing the GA initialization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Hence, to solve the UPA element-antenna position problem in a manageable time, the first approach named GA-marginal includes an initial population based on a near-local solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This approach’s main changes are summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Table 4: GA-marginal input parameters Parameter Alteration Initial Population Near-local solution Mutation (MUniform) Uniform Mutation Population Size (psize) 8N 2 Elements placement bounds [xmax, ymax] = [2pmax, 2pmax] Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' # Generations (gmax) Unlimited Stopping Criterion Outperform OUPA Mutation rate Determined via simulation Crossover Fraction (cf) Determined via simulation The near-local solution strategy consists in adding the OUPA solution to the initial population, but with a small perturbation, preventing the GA search from being restricted to the local solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Besides, the mutation function was changed to be uniform, increasing the diversity to escape local optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The population size was considerably increased to expand the chances of finding the better solution at the cost of increasing algorithm complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, the bounds of element placement were altered considering the OUPA solution, the new values are dependent on pmax, which is the max value of the coordinate x or y of the OUPA solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The stopping criterion was modified in a way that the gmax became unlimited, and the new criterion consisted in incrementing the generations until GA-marginal outperforms the OUPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Furthermore, the mutation rate and the crossover fraction is now determined by a simulation wherein a combination of both values are evaluated jointly 20 for every 100 generations, and then, the combination that achieves the best value of directivity is selected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' the range of both values are set equal and multiple percentages of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Remark 5: the idea of introducing modifications on the initial population generation in the GA-aided directivity method aims to facilitate the guided search to attain incremental improvements over the OUPA solution, in a manageable time, given the non-convexity nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Indeed, the computational difficulty of outperforming our proposed OUPA method in such burden computational conditions is a clear indication of the expedited and advantageous solution given by the proposed OUPA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' However, limiting the GA technique to surpass the proposed OUPA marginally will not give us a promising perspective on how much gain one can attain in the UPA directivity context using heuristic evolutionary techniques, and how much the cost to achieve such improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Hence, to address this issue, another GA modification is proposed herein, using almost the same set of parameters described for the GA-marginal, but changing only the stopping criterion that now consists of achieving hundred consecutive stalled generations, without directivity gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This method henceforth will be called GA- stall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Array Directivity Methods: Numerical Comparison Considering a small number of element-antennas yet, Table 5 exposes values of array directivity found by the OUPA method, by GA-aided UPA improved solutions, and the two classical steering vector beamforming, the UCA, and the conventional ULA, as well as an improved directivity method for the omnidirectional UCA proposed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Table 5: Directivity Methods Comparison for small N’s N 6 8 9 OUPA 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='70 dBi 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='91 dBi 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='12 dBi GA-marginal 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='81 dBi 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='19 dBi 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='12 dBi GA-stall 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='35 dBi 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='49 dBi 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 dBi UCA 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='96 dBi 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='73 dBi 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='17 dBi ULA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='17 dBi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='38 dBi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='88 dBi UCA [15] n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='a 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='00 dBi n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The last row in Table 5 depicts the directivity found in [15], which consists in a technique based on the subspace changes for the omnidirectional UCA directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For instance, considering N = 8, the maximum directivity value found is close to D = 12 dBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Such value remains remarkably reduced compared with the optimization methods proposed herein, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='5 dBi less compared with the GA-stall approach and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='9 dBi less when compared with our OUPA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For computational complexity analysis, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 12 exhibits the simulation time vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' # antennas N ∈ [4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 36], considering the OUPA improved GA-marginal and GA-stall methods proposed, the GA-marginal and GA-stall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Aiming to maximize the directivity, the quasi-squared UPA condition in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' (30) is applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, possible combinations of N1 and N2 was bounded by max(N1, N2) = 6, implying in the following possible antenna-elements arrangements N = N1 × N2 → 2 × 2 2 × 3 3 × 3 3 × 4 4 × 4 4 × 5 5 × 5 5 × 6 6 × 6 , (33) which are identified by the respective markers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The OUPA technique proposed herein, even with far less complexity, can follow the performance of both GA’s techniques very closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' As expected, the GA-marginal was able to surpass the OUPA method marginally, proving its sub-optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' However, the time spent by both GA heuristic meth- ods was far superior, configuring a far inferior performance-complexity tradeoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The GA-stall aided 21 UPA directivity optimization method was able to achieve a considerable and consistent improvement when compared with the OUPA, the amount of time consumed to find such a solution is extensively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 5 10 15 20 25 30 35 Number of Antennas ( N) 10 12 14 16 18 20 22 Directivity (dBi) OUPA GA-marginal GA-stall 5 10 15 20 25 30 35 Number of Antennas ( N) 10-2 10-1 100 101 102 103 104 105 106 Time (s) OUPA GA-marginal GA-stall (a) Directivity (b) Computational Time Figure 12: Directivity and run-time comparison between OUPA, GA-marginal and GA with the increasing in the number of antennas N = N1 × N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Remark 6: One can conjecture that among the three proposed directivity methods, the improvement beyond the OUPA solution provided by the GA-stall aided UPA algorithm comes with a much high computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, both GA-stall and GA-marginal optimization schemes have resulted in less effectiveness than the proposed OUPA method in terms of performance-complexity tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Finally, Table 6 shows the crossover fraction (cf) and the mutation rate (mr) values as a function of dimensionality problem (N), required in the GA-aided methods that maximize the directivity in the simulation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' For most configurations, the crossover fraction percentage has high values, Table 6: Input parameters configuration for both GA-marginal and GA-stall algorithms N 4 6 9 12 16 20 25 30 36 cf (%) 80 40 10 90 40 80 80 60 80 mr (%) 100 20 10 10 10 70 10 10 20 while the mutation rate presents a low percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Such a combination lead to a population that has several changes to escape local optima between the generations, nonetheless, the value of mr remaining low can be interpreted as a mean of guaranteeing an improvement around the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Notice that in the case N = 20, both GA-stall directivity optimization algorithms selected a high mr percentual 22 value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' as a result, the simulation time resulted significantly smaller;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' however, the performance was incremental better when compared to the OUPA and GA-marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The discrepant values found for N = 4 antennas can be explained by the simplicity of the geometric configuration, therefore, given a population, a volatile evolution did not prevent the method from finding a remarkable solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Conclusions and Final Remarks This work proposes a new approach to maximize the directivity of an omnidirectional volumetric antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' This technique assumes a uniform planar array (UPA) confined on a specific plane with a minimum distance between the antenna elements optimized by the successive evaluation and validation (SEV) procedure, namely optimal uniform planar array (OUPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, evolutionary heuristic GA optimization is employed to validate the proposed methodology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, for a large number of antennas, two modifications on the initial parameters are suggested, denoted GA-marginal and GA-stall, both made the GA a promising optimization tool to solve UPA directivity problem in such large scale antenna scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' An exciting finding is that a plane space constrains the element positioning solutions as a function of the desired elevation (θ0) and azimuth (φ0) angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' A new method based on the UPA structure is in- troduced to address this constraint to solve the directivity optimization problem with low-complexity effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Then an evolutionary heuristic technique is selected to implement the directivity optimization analysis devised herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Indeed, the genetic algorithm (GA) was selected as an expedited optimization tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The proposed methodology is focused on finding the optimal position of each antenna element that maximizes the antenna array directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The solution was confined to a specific plane, where the antenna elements position on this plane resembling a sequence of equilateral triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' hence, the idea of finding the optimal UPA in such a plane is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The OUPA method exploits the OF features, while in the evolutionary heuristic optimization approach, the directivity based on antenna- elements position has been formulated and solved for a different number of antenna-elements in the range N ∈ {4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 36}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Indeed, numerical results deploying the proposed GA-marginal and GA-stall heuristic evolutionary techniques against the also proposed OUPA method for a different number of antennas has demonstrated higher directivity gains when compared with the well-known regular UCA and ULA arrangements, as well as when compared with recent literature design for the specific case of N = 8 antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' The computational complexity reveals a superior performance of the OUPA method, implying in an excellent performance-complexity tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Moreover, the OUPA design method has achieved impressive performance in massive MIMO scenarios (N ≥ 50), achieving a directivity of 30 dBi in an extensive quasi-squared planar configuration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', N = 15 × 16 antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' Acknowledgement This work was partly supported by The National Council for Scientific and Technological Devel- opment (CNPq) of Brazil under Grants 310681/2019-7, partly by the CAPES- Brazil - Finance Code 001, and the Londrina State University - Parana State Government (UEL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' References 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 66, issue: 12 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='7443 7448, DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content='2869243 [19] Szymanski, John E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=', “Basic Mathematics for Electronic Engineers:Models and Applications,” Taylor Francis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' ISBN 0278000681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfIgOA/content/2301.02940v1.pdf'} diff --git a/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf b/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e9f76970a2acb0990cc1e87c7e47d379b48d2478 --- /dev/null +++ b/V9E4T4oBgHgl3EQfng2k/content/2301.05177v1.pdf @@ -0,0 +1,3 @@ +version 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a/VdE0T4oBgHgl3EQflwF_/content/tmp_files/2301.02490v1.pdf.txt b/VdE0T4oBgHgl3EQflwF_/content/tmp_files/2301.02490v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a32778700ae482deaf21ca3a229fedca734f8577 --- /dev/null +++ b/VdE0T4oBgHgl3EQflwF_/content/tmp_files/2301.02490v1.pdf.txt @@ -0,0 +1,1032 @@ +Fuzzers for stateful systems: Survey and Research Directions +CRISTIAN DANIELE, SEYED BEHNAM ANDARZIAN, and ERIK POLL, Radboud University, The Nether- +lands +Fuzzing is a security testing methodology effective in finding bugs. In a nutshell, a fuzzer sends multiple slightly malformed messages +to the software under test, hoping for crashes or weird system behaviour. The methodology is relatively simple, although applications +that keep internal states are challenging to fuzz. The research community has responded to this challenge by developing fuzzers +tailored to stateful systems, but a clear understanding of the variety of strategies is still missing. In this paper, we present the first +taxonomy of fuzzers for stateful systems and provide a systematic comparison and classification of these fuzzers. +CCS Concepts: • Security and privacy → Software security engineering. +Additional Key Words and Phrases: stateful fuzzing, state model, active learning +1 +INTRODUCTION +With fuzzing (or fuzz testing) a system is fed a large collection of automatically generated inputs to find security +vulnerabilities, in particular memory corruption bugs. Fuzzing is a great way to improve software security: it can find +lots of bugs with relatively little effort. The idea of fuzzing goes back to the late 1980s [32] but interest in fuzzing +exploded in the 2000s. Major game changers here have been the advent of white-box fuzzing using symbolic (or more +precisely, concolic) execution in the SAGE fuzzer [19] and the advent of grey-box fuzzing, also known as evolutionary +fuzzing, pioneered in the fuzzer AFL [48], where code execution paths are monitored to guide the generation of inputs. +Both approaches avoid the need of having the user provide an explicit grammar describing the input format. +Traditionally, most fuzzers target stateless systems, where the system under test takes a single input, say a JPEG +image; the fuzzer then tries many possible inputs, including many malformed ones. This survey focuses on the fuzzing +of stateful systems. By a stateful system, we mean a system that takes a sequence of messages as input, producing outputs +along the way, and where each input may result in an internal state change. Most protocols, including most network +protocols, are stateful. So when people talk about ‘network fuzzing’ [21] or ‘network protocol fuzzing’ [20, 23], they +are usually talking about the fuzzing of stateful systems. Obviously, fuzzing stateful systems is harder than fuzzing +stateless systems, as the internal state changes increase the state space that we try to explore. Moreover, it may be hard +for a fuzzer to reach ‘deeper’ states. Indeed, fuzzing stateful systems is listed as one of the challenges in fuzzing by +Boehme et al. [11]. +There are some good survey papers about fuzzing, e.g. [31, 50], but these do not investigate the issues of fuzzing +stateful systems in any depth, even though these surveys do include some fuzzers for stateful systems. There is a +growing number of fuzzers specifically for stateful systems, which raises questions about the approaches these fuzzers +take, their commonalities and differences, and their pros and cons, which this paper tries to answer. +The outline of this paper is as follows. Section 2 defines some basic concepts and terminology for discussing stateful +systems. Section 3 presents the traditional classification for fuzzers used in the literature and discusses some of the +differences between fuzzing stateless and stateful systems. Section 4 presents our taxonomy for fuzzers for stateful +systems, classifies existing fuzzers and compares the various approaches. +2023. Manuscript submitted to ACM +1 +arXiv:2301.02490v1 [cs.CR] 6 Jan 2023 + +, , +Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll +2 +CONCEPTS AND TERMINOLOGY +By a stateful system we mean a system that takes a sequence of messages as input, producing outputs along the way, +and where each input may result in an internal state change. To avoid confusion, we reserve the term message or input +message for the individual input that the System Under Test (SUT) consumes at each step and the term trace for a +sequence of such messages that make up the entire input. +We use the term response for the output that the SUT produces along the way. In the case of a synchronous protocol, +there is usually one response after each input message. In this case, the state machine describing the input-output +behaviour will be a Mealy machine. +The input language of a stateful system consists of two levels1: 1) the language of the individual messages, which we +will refer to as the message format, and 2) the language of traces, built on top of that. A description or specification of +such an input language will usually come in two parts, one for each of the levels: for example, a context-free grammar +for the message format and a finite state machine describing sequences of these messages. We will call the latter the +state model or, if it is described as a state machine, the protocol state machine. +The state model usually involves a notion of message type where messages of one type trigger a different transition +than messages of another type. So then the set of messages types is the input alphabet of any protocol state machine. +This alphabet will typically abstract away from payloads inside messages. For some protocols, messages simply include +an instruction byte in a header that determines the message type. For output messages, it is common to distinguish +between error responses and non-error responses, and possibly a finer-grained distinction of error responses based on +the error code. Fuzzers that try to infer the protocol state machine (using active or passive learning, as will be discussed +later) may require the user to specify the abstraction function that maps concrete messages to message types or even to +provide an implementation of this function. There can be two abstraction functions, one for input messages and one for +responses. +In some protocols the format of input messages and responses is very similar. For fuzzing it is then a good strategy +to also include responses as inputs as this may trigger unexpected behaviour: client and server are likely to share a part +of the codebase and one may then accidentally will process messages only intended for the other2 +We use the term protocol state to refer to the abstract state of an SUT that determines how future input messages are +handled. The SUT will have a concrete program state, which is related to this protocol state, but which usually carries +much more detail. +The term ‘state’ can quickly become overloaded: not only the SUT has state, even if it implements a stateless protocol, +but the fuzzer itself also has a state. We use the term stateful fuzzing to refer to the fuzzing of stateful systems, but we +avoid the term ‘stateful fuzzer’ as even a fuzzer for stateless systems will have internal state. +There are basically two ways for an SUT to record the protocol state: 1) it can keep track of the protocol state using +program variables that record state information or 2) the state can be more implicitly recorded by the program point +where the execution is at (and possibly the call stack). Of course, these two ways can be combined. For fuzzers that use +a grey- or white-box approach (discussed in more detail later), the difference can be important: white-box approaches +that observe the values of program variables will work better for 1) than for 2), whereas grey-box approaches that +observe execution paths will work better for 2) than for 1). +1For text-based protocols, as opposed to binary protocols, there may even be a third level, namely the character set or character encoding used, but none +of the fuzzers studied use that. +2CVE-2018-10933 is an interesting example of a bug of this kind in Libssh: if the message that the server sends to a client to confirm that the client has +successfully authenticated is sent to the server, a malicious client could skip the whole authentication phase. +2 + +Fuzzers for stateful systems +, , +Stateless vs stateful systems. There is not always a clear border between stateless systems and stateful systems. For +example, if a system has a memory leak then it is (unintentionally) stateful, even though its behaviour will appear to be +stateless for a very long time, namely until it runs out of memory and crashes. +More generally, we can think of any stateful system that takes a sequence of messages as input as a stateless system +which takes that whole trace as single input. Conversely, we can view a stateless system that takes a single complex +value as input as a stateful system that processes a sequence of smaller inputs, say bits or bytes. For instance, a stateless +program that processes JPEGs, which will always process the same JPG in the same way, can be viewed as a stateful +program that takes a sequence of bytes as input, and which will process the same byte in different ways depending +on where in the JPEG image it occurs. But a fundamental difference between a stateful system and a stateless system +viewed as stateful one in this way is that the former will typically provide some observable output after processing each +input. Another difference is knowing that inputs are made up of smaller messages can help in making useful mutations, +by swapping the order of messages or repeating messages. +Some stateless systems can process sequences of inputs like stateful systems do, but then the idea is that previous +inputs do not have any effect on how subsequent inputs are handled. Some fuzzers use this possibility to avoid the +overhead of having to restart the SUT between inputs. This is called persistent fuzzing. This does then involve a sequence +of inputs, but it is the polar opposite of stateful fuzzing: the goal is not to explore the statefulness of the SUT, but rather +it presupposes that there is no statefulness. +3 +BACKGROUND +This section discusses existing classifications of fuzzers used in the literature, as this provides a starting point for our +classification of stateful fuzzing, and makes some initial observations about how and why fuzzing stateful systems is +different. +There are some good surveys papers about fuzzing, but none pay much attention to issues specific to stateful fuzzing. +The survey by Zhu et al. [50] classifies close to 40 fuzzers. Only two of these, namely AFLNet [36] and de Ruiter et al. +[15] specifically target stateful systems. The more extensive survey by Manes et al. [31] categorises over 60 fuzzers. +Thirteen of these are fuzzers for ‘network protocols’ so presumably these are fuzzers geared to fuzzing stateful systems; +all of these 13 fuzzers are black-box fuzzers. One section in this survey (Section 5.1.2) discusses the statefulness of the +SUT: here it discusses the inference of state machine models. +The only attempt at classifying fuzzers for stateful systems that we are aware of is given in the paper by Yu et al. +about SGPFuzzer [47]: Table 1 in this paper lists twelve other fuzzers for stateful systems (or "protocol fuzzers" in the +terminology used in the paper), namely AutoFuzz [20], AspFuzz, SECFuzz [45], Sulley 3, BooFuzz [35], Peach 4, SNOOZE +[7], PULSAR [18], TLS-fuzzer, DTLS-fuzzer, ICS-fuzzer, and NLP-fuzzer. The authors identify four challenges based on +the shortcomings of these 12 fuzzers and then design SGPFuzzer to address these. Unfortunately, the definitions of +the features used for the comparison are left implicit and the comparison fails to point out some very fundamental +differences between tools, for instance that the man-in-the-middle nature of AutoFuzz and SecFuzz comes with an +important inherent limitation, namely that the order of messages cannot be fuzzed (as we discuss in Section 4.6). +3https://github.com/OpenRCE/sulley +4https://wiki.mozilla.org/Security/Fuzzing/Peach +3 + +, , +Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll +3.1 +Existing classifications of fuzzers +The standard classification of fuzzers in the literature (e.g. [11, 31, 50]) distinguishes black-box, grey-box and white-box +fuzzers, where the black-box fuzzers are sub-divided into grammar-based fuzzers and mutation-based fuzzers. Even +though this classification is fairly standard, the terminology varies between papers, and there are combinations of +approaches that do not neatly fit into one of these categories. We discuss this classification in more detail below. +Black-box fuzzers. As the name suggest, black-box fuzzers only observe the SUT from the outside. To stand any +change of producing interesting inputs, black-box fuzzers require some knowledge of the input format. One approach +here, taken by generation-based aka grammar-based fuzzers, is that the fuzzer is given knowledge of the input format in +the form of grammar or model. A downside of such fuzzers is the effort required to produce such a model or grammar. +Another approach, taken by mutation-based fuzzers, is that the fuzzer is supplied with a set of sample inputs which are +then mutated in the hope of discovering bugs. +Most grammar-based fuzzers allow users to supply grammar for an arbitrary input format (or protocol) they want +to fuzz, but there are also grammar-based fuzzers which have a specific input format hard-coded in them, such as +the KiF fuzzer [1] for the SIP protocol. There are also commercial fuzzers for fuzzing specific protocols, for example, +Codenomicon’s DEFENSIS fuzzer 5 (since acquired by Synopsys), which grew out of the PROTOS [25] project at the +University of Oulu in Finland started in 1999. +Some fuzzers combine the generation-based and mutation-based approach. A grammar-based fuzzer should not only +produce grammatically correct inputs, but also malformed ones; this either has to involve some form of mutation or the +grammar has to be too ‘loose’ to begin with. Conversely, a mutation-based fuzzer can be given some knowledge about +the input format, for instance by providing a list of keywords or byte values with a special meaning, which the fuzzer +can use in mutations in the hope of generating interesting mutations. +White-box fuzzers. White-box fuzzers require access to the (source or binary) program code and analyse that code +in order to provide interesting inputs. With access to the code, it is possible to see which branches there are to then +construct inputs that trigger particular branches. Typically white-box fuzzers use symbolic or concolic execution to +construct interesting test cases. Microsoft’s SAGE fuzzer [19] is the best-known example of this class. +Grey-box fuzzers. Grey-box fuzzers occupy the middle ground and can observe some aspects of the SUT as it executes +and use this feedback to steer the fuzzer. This is also called evolutionary fuzzing as the inputs will gradually evolve into +more interesting mutations. Grey-box fuzzers can be considered as a special kind of mutational fuzzers because the +evolution always involves mutation. Grey-box fuzzers are sometimes called smart mutational fuzzers; the black-box +mutational fuzzers that lack a feedback mechanism to guide the evolution are then called dumb mutational fuzzers. +Grey-box fuzzers often require some instrumentation of the code or running the code in some emulator. The approach +has been popularised by the fuzzer AFL, which observes the code execution path – or, more precisely, the branches are +taken in the execution – to see if some input mutation results in new execution paths. Grey-box fuzzers that observe +the execution path in this way are also called coverage-guided greybox fuzzers (CGF). This approach has proved to be +very successful, providing much better coverage than ‘dumb’ mutational fuzzers but without the work of having to +provide a grammar. +5www.codenomicon.com/defensics/ +4 + +Fuzzers for stateful systems +, , +All fuzzers that involve mutation – dumb mutational fuzzers, evolutionary fuzzers, but also grammar-based fuzzers +that use mutation – can be parameterised by mutation primitives, for instance random bit flips, repeating sub-sequences +of the input, or inserting specific bytes, characters, or keywords. +Alternative classifications. Instead of classifying fuzzers into white-, grey- and black-box, an orthogonal classification +is to consider the kind of applications targeted and the kind of input this involves [31]: e.g. some fuzzers are geared +towards fuzzing file formats, others to network traffic, and others still to web applications or OS kernels. Fuzzers for +web applications are often called ‘scanners’. There is a relation between this classification and statefulness: applications +that take a file as input are usually not stateful, whereas applications that implement a network protocol usually are. +3.2 +White-, grey-, and black-box fuzzing for stateful systems +For stateful systems some basic observations about the classification into white-, grey-, and black-box can be made: +• The terms ‘grey-box fuzzing’ and ‘evolutionary fuzzing’ are often used as synonyms, but for stateful systems, +they are not: for a stateful system the evolution of inputs can also be steered by the outputs of the SUT, which is +then evolutionary but black-box. This is a key difference between a stateful and a stateless system: the response +that the SUT produces is an observation that the fuzzer can make without any instrumentation of the code. +• For grammar-based fuzzers it does not really matter if the SUT is stateful or not: the grammar describing the +system can describe both the message format and the protocol state machine. +• For dumb mutational black-box fuzzers it also does not matter that much if the SUT is stateful or not. Of +course, it helps if fuzzer is aware of the fact that inputs are traces of messages, so that it can try swapping +the order of messages, removing messages or repeating messages as interesting mutations. The same goes for +any grammar-based fuzzer, which should also try re-ordering, repeating or dropping messages as interesting +corruptions of the grammar. +• The techniques used in grey-box and white-box fuzzing to observe program execution may shed some light on +the state that the SUT is in. But as discussed in Section 2, there are different ways in which the SUT can record +protocol state: the state can be recorded in some program variables, it can be recorded in the program point that +the SUT is in (and possibly the call stack), or a combination of these. The way in which the SUT does this can +make a difference in how well some grey- or white-box technique can observe the protocol state. +The statefulness of the SUT may complicate observation for a grey- or whitebox fuzzer. For white-box fuzzers +that rely on symbolic or concolic execution input, the statefulness of the SUT is obviously a serious complication: +a symbolic execution of a program handling a single input can already be tricky, and the execution for sequence +of symbolic inputs will be an order of magnitude harder. For example, if the SUT implements some loop to handle +incoming messages then that loop would have to be unwound. +3.3 +Bug detection for stateful systems +In addition to a mechanism to generate inputs, a fuzzer also requires some mechanism to observe the SUT to detect +if an input triggered a bug. Typically fuzzers look for memory corruption bugs that crash the SUT using sanitisers +such as ASan (AddressSanitizer) [40], MSan (MemorySanitizer) [43], or older less efficient sanitisers such as Valgrind. +When fuzzing programs written in memory-safe languages, e.g. when fuzzing Java programs with Kelinci [26], instead +of looking for memory corruption bugs we can look for uncaught runtime exceptions; even if these bugs cannot be +exploited in the way memory corruption can, they can still lead to Denial of Service problems. +5 + +, , +Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll +Section +Category +Input required +Generate +state model +Human +interaction +4.1 +Grammar-based +Grammar +No +No +4.2 +Grammar-learner +Sample traces +Yes +Yes / No +4.3 +Evolutionary +Sample traces +No +Yes / No +4.4 +Evolutionary grammar-based +Grammar +No +No +4.5 +Evolutionary grammar-learner +Sample traces +Yes +No +4.6 +Man-in-the-Middle fuzzers +Live traffic +No +Yes / No +4.7 +Machine learning fuzzers +Many sample traces +No +No +Table 1. The seven categories of fuzzers with their main characteristics. Human interaction refers to manual code or grammar +annotation. +Fig. 1. The five categories of fuzzers that involve a grammar, an evolutionary feedback mechanism, or both. +A type of bug that is specific to stateful systems are deviations from the correct state behaviour: if a system is +expected to behave in a specific way, for instance by only responding to certain inputs after authentication, then a +deviation from this behaviour may be a security issue. For security protocols such as TLS any deviations from the +expected state behaviour are highly suspicious: security protocols are very fragile and even small deviations may break +the security guarantees the protocol aims to provide. Unlike the detection of memory corruption bugs, this cannot be +totally automated: it either requires a specification of expected behaviour (or, conversely, of unwanted behaviour), for +instance with a state machine or in temporal logic, or it requires some post-hoc manual analysis of the state behaviour +inferred by the fuzzer. +4 +FUZZERS FOR STATEFUL SYSTEMS +We have identified seven categories of fuzzers for stateful fuzzing. Table 1 summarises the main characteristics of each +category. Some categories can be regarded as a combination or sub-category of other categories, as illustrated in Fig. 1. +6 + +GRAMMAR +LEARNER +EVOLUTIONARY +GRAMMAR +LEARNER +EVOLUTIONARY +EVOLUTIONARY +GRAMMAR BASED +GRAMMAR BASEDFuzzers for stateful systems +, , +Before we discuss each category in more detail in the sections below, we first discuss common ingredients involved in +some of them: +• Some fuzzers require sample traces as inputs, either a few traces to act as seeds to further mutation, or many +traces so that grammar can be inferred or machine model can be trained. +• Many fuzzers involve some form of grammar. This can be a grammar describing just for the message format, a +grammar describing just the protocol state machine, or both. Some fuzzers require such grammars as inputs, but +others can provide grammars that are inferred during the fuzzing as output. +• Many fuzzers use some form of learning to infer information about the message format, the protocol state +machine, or both. Evolution can be regarded as a form of learning because it produces and uses new knowledge +about the input format, even though this knowledge is (usually) not expressed in the form of a regular expression, +state machine, or context-fee grammar. +Evolution is a form of active learning because it involves interaction with the SUT, where the next input we try +can depend on the outcome of previous tests. Some fuzzers use forms of passive learning instead of (or in addition +to) such active learning. By this we mean approaches where information about the input format is inferred after +a set of traces has been collected, so without interactively trying new experiments. +There is a long line of research into algorithms for inferring formal language descriptions, either actively or +passively, which includes research into regular inference and grammatical inference that focus specifically on +inference of regular expressions and context-free grammar, respectively. Research in this field is presented at +the bi-annual International Conference on Grammatical Inference (ICGI) and there are entire textbooks on the +subject (e.g. [14]). For active learning of a protocol state machine, an algorithm that can be used is L* [2] or one +of its improvements, e.g. the TTT algorithm used in LearnLib [24]. For the passive learning of protocol state +machines, some fuzzers use ad-hoc solutions. For instance, the fuzzer by Hsu et al. [22] uses an algorithm called +partial finite state automaton reduction. An important limitation of some learning algorithms, notably L* and its +improvements, is that they cannot deal with the non-deterministic behaviour of the SUT, as it will cause the +algorithm to diverge. +A very different form of (passive) learning used by some fuzzers is machine learning. This does not produce +knowledge in a nice concrete format like a regular expression, finite state machine, or context-free grammar. +Also, it typically requires more samples. Still, possible advantages are that there are many existing machine +learning approaches that can be used and that these may cope more easily with non-deterministic behaviour of +the SUT. +Below we first give a general description of the seven categories of fuzzers. In the subsequent sections, we discuss each +category in more detail: +(1) Grammar-based fuzzers Any grammar-based fuzzer can be used to fuzz stateful systems without any special +adaptions. The grammar that is supplied to the fuzzer will have to describe the two levels of the input language, +with some rules of the grammar describing the message format and some rules describing the protocol state +machine. Apart from that, no change in the fuzzer itself is needed, except that course, swapping, dropping and +repeating messages are useful – if not essential – mutation strategies for the fuzzer to include. But for a stateless +SUT where the format of the inputs is quite complex it can also be useful to include swapping, dropping and +repeating sub-components of inputs as mutation strategies. +7 + +, , +Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll +(2) Grammar-learner fuzzers Whereas the grammar-based fuzzers require the users to provide a grammar, these +fuzzers are able to extract a grammar from a set of sample traces. They can be considered as the sequential +composition of two tools: a grammar extractor that infers the grammar from a set of sample traces (using so-called +passive grammatical inference) and a grammar-based fuzzer that then does the actual fuzzing using this inferred +grammar. +As for the grammar-based fuzzers, for the grammar-learner fuzzers the statefulness of the SUT does not make +any fundamental difference: it only means that the grammar will have two levels. So grammar-learner fuzzers +can be applied to stateless as well as stateful SUTs. +(3) Evolutionary fuzzers These fuzzers basically take the same approach as stateless evolutionary fuzzers such +as AFL: they take some sample traces as initial input and mutate these using a feedback system to steer the +mutation process. Of course, evolutionary fuzzers for stateful systems should be aware that an input trace is a +sequence of messages and should include swapping, omitting or repeating these messages as mutation strategies. +A difference between stateful and stateless systems when it comes to evolutionary approaches of fuzzing is that +the responses that a stateful SUT provides after individual messages can be used in the feedback to guide the +evolution, as mentioned before in Section 3.1. +(4) Evolutionary grammar-based fuzzers These fuzzers use both a grammar provided to the user to generate +(correct, protocol-compliant) traces and an evolution mechanism to mutate these traces. We can think of them +as evolutionary fuzzers that use a grammar instead of a set of sample input traces to provide the initial traces +that will be mutated. We can also think of them as grammar-based fuzzers that include a feedback mechanism +to steer the evolution of mutations. So in Fig. 1 they are the intersection of the evolutionary fuzzers and the +grammar-based fuzzers. +(5) Evolutionary grammar-learner fuzzers This is the most complex category of fuzzers. These tools all use +some form of grammar to describe the protocol state machine; one also uses a grammar to describe the message +format. They involve two feedback mechanisms to steer two forms of evolution: (i) one for the mutation of +individual messages, in the style of conventional evolutionary fuzzers like AFL, and (ii) another for the mutation +of sequences, which then infers a protocol state machine. The second form of evolution is based on the response +that the SUT provides as feedback, so it is black-box. +The final two categories of fuzzers are very different from the five above: +(6) Man-in-the-Middle fuzzers: These fuzzers sit in the middle between the SUT and a program interacting with +it and modify messages going to the SUT, as illustrated in Fig. 7. Responses coming back from the SUT are left +untouched. +These fuzzers can take a dumb mutational approach to modify the messages, but they may leverage a protocol +specification (automatically inferred or given as input) to modify messages. +(7) Machine learning fuzzers These fuzzers use a Machine Learning (ML) model trained on a large set of input +traces. The model outputs slightly different — hopefully malicious — mutated traces. Machine learning methods +used by these fuzzers include Seq2seq and Seq-GAN. +These fuzzers are similar to the grammar learner fuzzers in that they require a set of sample traces as input that +is then used to infer a model of the input format which is then the basis for the fuzzing. The key difference is +that for the grammar learner fuzzers this model is a grammar, whereas for these machine learning fuzzers the +model is an ML model. +8 + +Fuzzers for stateful systems +, , +Fig. 2. Evolutionary fuzzers +Fig. 3. Grammar-based fuzzers +Fig. 4. Grammar learner fuzzers +Fig. 5. Evolutionary grammar-based fuzzers +Fig. 6. Evolutionary grammar learner fuzzers +9 + +CRAFTER +SYSTEMUNDERTEST +FEEDBACKSYSTEM一》 +CRAFTER +SYSTEMUNDERTEST +GRAMMAR (PROVIDED +BY THE USER)SET OF TRACES +GRAMMAR +SYSTEM UNDER TEST- +GRAMMAR(PROVIDED +CRAFTER +SYSTEMUNDERTEST +BY THE USER) +FEEDBACKSYSTEMCRAFTER. +SYSTEM UNDER TEST +GRAMMAR EXTRACTOR +SET OF TRACES +GRAMMAR +FEEDBACK SYSTEM (U) +Optional +SYSTEM UNDER TEST +RESPONSE (II), , +Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll +Fig. 7. Man-in-the-middle fuzzers +Fig. 8. Machine learning fuzzers +Fuzzer +Based on +Mutates +Peach +- +- Message +SNOOZE [7] +- +- Message +PROTOS[25] +- +- Message +Sulley +- +- Message +BooFuzz [35] +Sulley +- Message +Fuzzowski 6 +BooFuzz [35] +- Message +- Trace +AspFuzz [27] +- +- Message +- Trace +Table 2. Grammar-based fuzzers +4.1 +Grammar-based fuzzers +Table 2 lists the grammar-based fuzzers. As shown in Fig. 3, these fuzzers use a grammar provided by the user. In the +case of a stateful SUT, this grammar should describe the syntax of the messages and the protocol state machine. For +some fuzzers, e.g. Peach7 and SNOOZE [7], this grammar is supplied in some XML format. +The obvious downside of these fuzzers is that they require an accurate grammar. Producing one can be a lot of work +and it can be challenging and error-prone. Not all errors will matter – or matter equally: if the grammar is a bit too +‘loose’ this is not much of a problem, but if the grammar omits interesting parts of the language it may be, as this +would mean that the fuzzer will not explore that part of the language. Ideally the documentation of the SUT, or the +specification of the protocol it implements, simply provides a formal grammar that can be used. However, this will +7Here we mean the community edition, available at https://gitlab.com/gitlab-org/security-products, which lacks some features of Peach Fuzzer Professional. +10 + +CLIENT (OR SERVER) +CRAFTER +SYSTEM UNDERTEST +GRAMMAR EXTRACTOR +GRAMMAR +SET OFTRACES +OptionalSETOFTRACES +CRAFTER(MLMODEL) +SYSTEMUNDERTESTFuzzers for stateful systems +, , +Fuzzer +Learns +Based on +Input needed +PULSAR [18] +- State model +- Message fields +- Passive learning +(using PRISMA [18]) +- Traces +GLADE+ [8] +- Message fields +- Active learning +(using GLADE [8] ) +- Traces +Hsu et al.[22] +- State model +- Passive learning +(using partial finite state +automaton reduction [22]) +- Message field +specification +- Traces +Table 3. Grammar learner fuzzers +often not be the case: documentation or specifications may be unclear or incomplete. That the SUT is stateful does not +make a difference here, Still, in earlier research [37] we found that documentation is more likely to include a clear (but +informal) specification for the message format than for the protocol state machine. The protocol state machine are +often – very poorly – specified in prose scattered throughout specification documents. +Some grammar-based fuzzers, e.g. SNOOZE [7] and Sulley, come with grammars for some standard protocols, so that +for these the hard work to produce a grammar has already been done for the user, but for other protocols the user still +has to do it themselves. +4.2 +Grammar learner fuzzers +Table 3 presents the grammar learner fuzzers. These fuzzers operate in two phases: first, they infer a grammar from a +set of collected traces; then they do the actually fuzzing using that inferred grammar just like a grammar-based fuzzer +would do. So each of these fuzzers is effectively the composition of two tools: +(1) a grammar learner: a special component with the goal to build a grammar as much as possible similar to the real +one +(2) an actual fuzzer: in principle any of the grammar-based fuzzers discussed in the previous section. +All these fuzzers will require a comprehensive and complete set of traces, as e.g. the makers of PULSAR explicitly point +out [18], to give good fuzzing performance. +For the first phase, the fuzzers in Table 3 not only use different inference techniques, but also try to infer different +aspects of the input format: +• PULSAR [18] infers both the message format and a protocol state machine, passively, from observed traffic. The +learning techniques it uses are the ones developed earlier for the PRISMA fuzzer [28]. These can also infer rules +for dependencies between messages, such as increasing sequence numbers. +As the authors note, the approach relies on the completeness of the set of observed network traces and will be +unable to model protocol paths not included in this traffic. +• GLADE [8] uses a new active learning algorithm for inferring context-free grammars which can infer both the +message format and the protocol state machine. +Strictly speaking, GLADE is not a fuzzer, but just a tool for inferring a context-free grammar. This inference +uses active learning, so it does involve some fuzzing of the SUT. But GLADE has been extended to be used as a +front-end for a grammar-based fuzzer. In Table 3 we refer to this extension as GLADE+ to avoid confusion. +The algorithm used by GLADE+ is shown to have better precision and recall that the active learning algorithms +L* [2] and RNPI [34] for the case studies tried out by the makers [8]. The results of GLADE+ are also compared +11 + +, , +Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll +Fuzzer +Feedback +system +Based on +Input needed +nyx-net [39] +- Coverage +- AFL +- Target binary +- Protocol specification +- Seed inputs (optional) +FitM fuzzer [30] +- Coverage +- AFL +- Client binary +- Server binary +- Seed inputs +SNPSfuzzer [29] +- Coverage +- AFL +- Target binary +- Seed inputs +Chen et al. [13] +- Coverage +- Branches +- AFL +- Manual code +annotation +- Target binary +- Seed inputs +SGFuzz [6] +- Coverage +- Variables +- AFL +- Automatic code +annotation +- Target binary +- Seed inputs +IJON [4] +- Coverage +- Variables +- AFL +- Manual code +annotation +- Target source code +Table 4. Evolutionary fuzzers +with AFL for some of these case studies. However, the case studies are not typical stateful protocols but include +interpreters for Python, Ruby and JavaScript. As AFL is best at fuzzing binary formats, it is maybe not that +surprising that GLADE+ beats AFL here. +• Unlike PULSAR and GLADE, the fuzzer by Hsu et al. [22] cannot infer the message format: it only infers a +protocol state machine. The tool requires that the message format is known; in fact, it needs an (un)parser for +the message format to be supplied. The protocol state machine is then inferred from observed traffic – i.e. using +passive learning – using a new algorithm they introduce. Once this state machine is inferred, the SUT can be +fuzzed. Here a collection of mutation primitives is used, including mutations to mutate individual messages and +mutations to reorder messages in the input trace. +The first phase of this fuzzer, i.e. inferring a protocol state machine given a known message format, is very +similar to what tools like LearnLib [38] do. But it uses passive learning, whereas LearnLib uses active learning +with a variant of L*. Hsu et al. report that they also tried active learning for this initial phase, using a variant of +L*, as they also did in earlier work [41], but abandoned that approach because of 1) the difficulty in constructing +concrete messages that active learning requires and 2) it being inefficient and not learning an accurate model. +4.3 +Evolutionary fuzzers +Table 4 presents the evolutionary fuzzers. As shown in Fig. 2, these use feedback to guide the mutation of inputs. This +feedback can use different types of observation, namely the five options listed below or a combination: +F1 Response. Some fuzzers use the response of the SUT. This is the only type of observation that can be done +black-box. +12 + +Fuzzers for stateful systems +, , +F2 Coverage. Some fuzzers observe branch coverage in the style of AFL, i.e. using a bitmap to observe branches +taken during execution. This is a greybox approach that either requires re-compilation to instrument the code or +running code in some emulator, just like AFL does. +F3 Branches: Some fuzzers observe branch coverage not by observing all branches like AFL does, but by observing +specific branches that are manually marked as interesting to the user. This is a white-box approach and requires +manual annotation of code by the user. +F4 Variables: Some fuzzers observe the value of specific program variables. This is a white-box approach and requires +manual annotation of code by the user. The idea is that the program variables observed record information about +the protocol state. +F5 Memory: Instead of observing specific individual program variables, one fuzzer observes memory segments: it +takes snapshots of memory areas to see if inputs affect these. The idea is that changes in the memory signal +change the protocol state. The only fuzzer using this, StateAFL, is not an evolutionary fuzzer but one of the more +complicated evolutionary grammar learner, so it is discussed in Section 4.5. +All the evolutionary fuzzers in Table 4 are based on AFL, so all of them at least observe branch coverage in the style of +ALF (i.e. F2, Coverage) to steer the evolution, but some tools use an additional feedback mechanism on top of this. +Regarding F3: the fuzzer by Chen et al. allows the user to mark some specific branches in the code. The idea is that +taking these marked branches is an indication of the SUT moving to a different protocol state. Given that the AFL +instrumentation already observes branch coverage, it is somewhat surprising that additional observation of selected +branches improves the performance of the fuzzer. The fuzzer not just observes if these branches are taken in execution, +but when this happens it effectively starts a new AFL session for this specific state (i.e. using a new bitmap for recording +branches and creating a new queue of messages to mutate). So whereas AFL and all the other AFL-like evolutionary +fuzzers in Table 4 maintain a single bitmap to record which branches have been taken, the fuzzer by Chen et al. has one +such bitmap for each of the marked branches. This allows it to learn different strategies for generating test cases for +different protocol states. Intuitively this makes sense: messages in different stages of a protocol may have different +formats, so learning different mutation strategies, each tailored to a specific protocol state, can improve the fuzzing. +Regarding F4: IJON [4] observes specific program variables during the fuzzing. The user has to mark these in the +source code. The idea is that the user marks variables that record information about the protocol state. SGFuzz is an +improvement of this: instead of the user having to annotate code to specify which program variables record interesting +state information, the fuzzer automatically infers which program variables have an enumeration type, and it assumes +that all these program variables record state information. +As discussed in Section 2, there are different ways in which the SUT can record its protocol state. If the protocol +state is recorded in program variables, approach F4 of IJON and SGFuzz can be expected to work well. If the program +point is used the protocol state, approach F3 as used by Chen et al. might work better. +All evolutionary fuzzers require initial seeds as input traces to start fuzzing. The choice of these initial seeds can +influence the performance. Some fuzzers provide some automation to create initial seeds: for instance, Nyx-net [39] +provides functionality (in the form of a Python library) to generate seeds messages from PCAP network dumps. The +creators of IJON [4] note that in some cases IJON’s feedback mechanism works so good that manually picking good +seeds is no longer necessary to obtain good coverage; in some experiments, they could simply use a single uninformative +seed containing only the character ‘a’ [4] . +13 + +, , +Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll +Fuzzer +Feedback +system +Based on +Inputs needed +RESTler [5] +- Response +- +- State model specification +- Target source file +SPFuzz [42] +- Coverage +- AFL +- Protocol specification +- Target source code +- Initial seeds +EPF [21] +- Coverage +- AFL +- Fuzzowski +- Protocol specification +- Target source code +- PCAP files (as initial seeds) +Table 5. Evolutionary grammar-based fuzzers +4.4 +Evolutionary grammar-based fuzzers +Table 5 presents the evolutionary grammar-based fuzzers. These fuzzers combine the grammar-based and evolutionary +approaches, as shown in Fig. 5: they require a grammar as the starting point to generate messages but they also include +some feedback to observe the effects of inputs in an effort to fuzz more intelligently. +RESTler [5] is an open-source fuzzer by Microsoft for fuzzing REST APIs. It uses a grammar in the form of an +OpenAPI8 specification (as can be produced by Swagger tools) to generate messages but then observes responses +combinations of messages that always lead to the same error. The fact that RESTful API typically come with grammar +in the form of an OpenAPI spec is a big win: it means we can use a grammar-based approach but avoid the downside of +having to produce a grammar. +SPFuzz [42] and EPF [21] observe branch coverage in the style of AFL to get information about coverage (i.e. F2)), +whereas RESTler only uses the response of the SUT (i.e. F1). . +RESTler does not require any initial seeds provided by the user, as you would expect of a fuzzer that has a grammar +that can be used to generate inputs. However, SPFuzz and EPF do require the user to provide initial seeds. For EPF these +are provided in the form of a PCAP file. +SPFuzz does not use some standard specification format like OpenAPI, but it has its own format to describe the +protocol grammar and dependencies. +These dependencies, like the ones between requests and responses [5], or the ones between the length field, the +content of the message or the data types [21, 42], significantly influence the quality of the inputs generated by the +fuzzer. +4.5 +Evolutionary grammar learner fuzzers +Table 6 presents the evolutionary grammar-learner fuzzers. This is the most complex category of fuzzers. These fuzzers +involve a grammar, which only describes (an approximation of) the protocol state machine. They use two forms of +evolution, illustrated by the two feedback loops in Figure 6: +(i) Message evolution: like for the evolutionary fuzzers discussed in Section 4.3, feedback from the system is used to +mutate traces, using one or several of the five types of observation discussed there. +(ii) State machine evolution: here feedback from the system is used to improve an approximation of the protocol state +machine. This comes down to a form of active state machine learning. +8https://www.openapis.org +14 + +Fuzzers for stateful systems +, , +Fuzzer +Learns +Feedback (i) +Feedback (ii) +Inputs needed +Based on +AFLNet [36] +- State model +- Coverage +- Response +- Target binary +- Sample traces +- AFL +FFUZZ [10] +- State model +- Coverage +- Response +- Target binary +- Sample traces +- AFLNet +StateAFL [33] +- State model +- Coverage +- Memory +- Target binary +- Sample traces +- AFLNet +SGPFuzzer [47] +- State model +- Message fields +- Coverage +- Response +- Target binary +- PCAP file +- AFL +LearnLib [38] +- State model +N/A +- Response +- Set of messages +- TTT [24] +Doupé et al. [16] +- State model +N/A +- Response +No input required +- Web crawling +Table 6. Evolutionary grammar learner fuzzers +Fuzzer +Limitations +Uses +Input needed +AutoFuzz [20] +- Cannot fuzz the message order +Passive +learning [9] [22] +Live traffic +Black-Box Live +Protocol Fuzzing [44] +- Cannot fuzz the message order +- User needs to specify +the fields to fuzz +N/A +Live traffic +SECFuzz [45] +- Limited fuzzing of message order +N/A +Live traffic +Table 7. Man-in-the-middle fuzzers +For all tools except StateAFL the feedback used here is the response from the SUT (i.e. F1) or some information +extracted from that response; for example, for AFLNet it is the response code in the response, for EPF it is just +information about whether the connection was dropped. StateAFL observes whether the content of long-lived +memory areas has changed (i.e. F5). +LearnLib and the fuzzer by Doupé et al. are odd ones out in Table 6 in that they are very limited in the kind of fuzzing +they do. They do not mutate individual messages but only try combinations of a fixed set of input messages to infer the +state machine. Here LearnLib uses the TTT algorithm [24], an improvement of L*. The fuzzer of Doupé et al. uses an +ad-hoc algorithm developed for the tool: it is a fuzzer for web applications, so the response of the SUT is a web page, +and the tool analyses these web pages for similarity in an attempt to crawl the entire website. +4.6 +Man-in-the-middle fuzzers +Table 7 presents the man-in-the-middle-fuzzers. As shown in Fig. 7, these fuzzers sit between the SUT and another +application that interacts with the SUT to intercept the communication and modify the communication going to the +SUT. If the SUT is a server then this other application will be a client. +An fundamental limitation of these fuzzers is that they are only able to modify the order of the messages in a limited +way. In fact, AutoFuzz [20] and Black-Box Live Protocol Fuzzing [44] do not modify the order of messages at all, so the +exploration of the protocol state machine will be very limited. SECFuzz [45] does fuzz the order of the messages, but +only a little bit, namely by inserting well-formed messages at random positions in the input trace (i.e. in the sequence +of messages sent by the other application). +Even though the overall set-up is the same, the fuzzers use different techniques: +15 + +, , +Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll +Fuzzer +Based on +Input needed +GANFuzz [23] +seq-gan model +Traces +Machine Learning for +Black-box Fuzzing of +Network Protocols [17] +seq2seq model +Traces +SeqFuzzer [49] +seq2seq model +Traces +Table 8. Machine learning fuzzers +• Like the grammar learner fuzzers, AutoFuzz operates in two phases. Prior to the actual fuzzing it starts with +a passive learning phase to infer an approximation of the protocol state machine. For this AutoFuzz uses the +same algorithm as Hsu et al. [22], i.e. partial finite state automaton reduction. So, as for Hsu et al. the user has to +supply implementations of abstraction functions that map concrete messages to some abstract alphabet. During +the fuzzing AutoFuzz then its knowledge of the protocol state machine to guide the input selection. +• Black-Box Live Protocol Fuzzing uses a function to generate the message field specification from a PCAP file, but +the user is required to choose the fields of the messages to fuzz. +• SECFuzz is able to deal with the cryptography of the protocol. To do that, the client has to share with the fuzzer +(through a log file) all the information necessary for the decryption. +4.7 +Machine learning fuzzers +Table 8 presents the machine learning fuzzers. As shown in Fig. 8, like the grammar learner and the evolutionary +grammar learner fuzzers, these fuzzers require a set of traces that are used as dataset to train the machine learning +model. Once trained, the machine learning model is able to output traces that slightly differ from legit, correct traces. +Likewise the man-in-the-middle fuzzers, the machine learning fuzzers observe protocol executions that follow the +happy flow. This cause an unbalanced dataset in favour of the correct traces and the model’s inability to outcomes +traces with messages in the wrong order. +Although these fuzzers use a machine learning model — trained on real protocol execution — to output traces to +forward to the SUT, they employ different strategies. GANFuzz [23] uses a generative adversarial network (GAN) and +an RNN (recursive neural network), while the fuzzer by Fan et al. [17] and SeqFuzzer [49] use seq2seq. We refer to the +review by Wang et al. [46] for a more exhaustive explanation of fuzzing using machine learning. +5 +GENERIC FUZZERS IMPROVEMENTS +Irrespective of the category of fuzzer, there are some generic improvements that several fuzzers include. +Pre-processing of raw network traffic. Many fuzzers take raw network traffic in the formal of a PCAP file as input +and provide some automated pre-processing of that input. Each tool implements it in their own way but it includes +some common ingredients, such as chopping up the traces to extract the individual messages to then clustering similar +messages or recognizing specific fields in the messages. +Using snapshots. One factor that makes fuzzing of stateful systems slow is that a fuzzer often needs to repeat a +sequence of inputs to get the SUT in a particular state, to then start the actual fuzzing in that program state. To avoid +the overhead, some fuzzers [29, 30, 39] use snapshots (aka checkpoints) to capture the program state of the SUT at a +particular point in time, to then be able to quickly re-start fuzzing from that point on. (The same idea is behind the +16 + +Fuzzers for stateful systems +, , +use of forking by AFL, where even for stateless SUTs it has been shown to improve performance off.) This can speed +up fuzzing, as the initial trace to reach some specific state does not have to be repeated, but taking and re-starting +snapshots also introduces overhead, so in the end it may not be faster. Depending on the execution platform there are +different snapshotting techniques that can be used. For instance, FitM and SNPSFuzzer use CRIU’s userspace snapshots +and nyx-net uses hypervisor-based snapshot. For SNSP different snapshotting technologies have been compared [29]: +CRIU 9, DMTCP [3], BLCR 10, and OpenCZ 11. +Mutation primitives and heuristics. Any fuzzer that uses some form of mutation (of individual messages or of traces) +can use a variety of strategies and primitives to do this. For individual messages this may include random bit-flipping, +deleting some parts of a message or inserting some data. For traces as opposed to individual messages) interesting +mutation primitives are of course removal, insertion, or repetition of messages. +The fuzzers we discussed come with variety of primitives for all this. Some offer possibilities for the user to provide +their own custom mutators. We have not gone into the details of this, as the focus was on understanding the overall +approach. Some fuzzers, notably SNOOZE, PROTOS, SPFuzz, SGPFuzzer, and the fuzzer by Hsu et al. [22], provide more +advanced heuristics and tricks for mutations than some of the others. For example, SNOOZE can provide mutations to +try out SQL or command injection or use specific numbers to test boundary conditions. SPFuzz distinguishes different +types of data inside messages (e.g. headers vs payloads) to then use different mutation strategies for specific types +of data. In practice it may of course make a big difference for a particular case study which mutation primitives or +heuristics are used. +6 +CONCLUSIONS +It took us quite some effort to disentangle the ways that various fuzzers for stateful systems work and arrive at the +classification we presented. It seems like every fuzzer picks another set of building blocks, combines them in its own +way, and then adds some ad-hoc heuristics and possibly performance optimisations. New fuzzers are typically evaluated +on some case studies and then compared with some other fuzzers, but it is hard to draw broader conclusions that then +go beyond a particular case study or a particular pair of fuzzers. This underlines the importance of initiatives such +as ProFuzzBench [33] for bench-marking stateful fuzzing approaches. Benchmarking has also been pointed out as a +challenge for fuzzers in general, not just for stateful fuzzing [11]. +We have noted some apparent contradictory observations – though this may simply be because researchers looked +at different case studies. For instance, Shu et al. [41] abandoned the use active learning of protocol state machine using +L* (or its variants) because they found it too slow and inaccurate, while in other research this has proved to be very +useful in finding security flaws [15]. +It is not surprising that the performance of fuzzer may depend heavily on the case study. When fuzzing a stateful +system there is a trade-off between a) trying out many variations of individual messages and b) trying out many +different sequences of messages. The complexity of an application (and hence the likely problem spots) application may +more in the message format or more in the protocol state machine; a corresponding strategy when fuzzing, focusing +more on a) or on b), is then most likely to find bugs. Very broadly we can make a rough distinction into three classes of +tools, illustrated in Fig. 9: I) fuzzers that are very good at aggressively exploring the protocol state machine but poor at +trying out variations of messages; III) fuzzers that are good at trying out variations in messages but poor at exploring +9https://github.com/checkpoint-restore/criu +10https://github.com/angelos-se/blcr +11https://openvz.livejournal.com +17 + +, , +Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll +Fig. 9. Cluster of fuzzers +the protocol state machine; and II) fuzzers which try to explore both the protocol state machine and the format of +individual messages. +There is some relation between this classification and the seven categories we have described. For instance, the +man-in-the-middle fuzzers and the machine learning fuzzers are in class III, as they do not explore the protocol state +machine and mainly (or even exclusively) stick to message sequences observed in real communications between +client and server. The grammar-based fuzzers can deal quite well with both dimensions of fuzzing so are in class II. +Evolutionary-based fuzzers that try to infer the protocol state machine (typically using the response of the SUT as +feedback mechanism) are good at exploring the protocol state space, but may lack mutation primitives or observation +mechanisms to aggressively explore the message formats. LearnLib is an extreme instance of class I as it only fuzzes the +message order. +The exact positioning of tools in Figure 9 is not based on experimental data, but more informally based on the general +characteristics of the tools, so should be taken with a pinch of salt. Also, for tools that require grammars as input or +manual code annotation a lot will depend on the quality of these. +It may seem like fuzzers of type II are the best of both worlds, but given the rapid state space explosion when we +fuzz both individual messages and sequences of messages this need not be the case: Using a fuzzer of type I and a +fuzzer of type III to explore different aspects may be more effective than using one fuzzer of type II that tries to do both. +18 + +Sulley +PULSAR +nyx-net +IJON +RESTler +AFLNet +LearnLib +AutoFuzz +GANFuzz +SGFuzz +SECFuzz +FFuzz +Sulley +LearnLib +AFLNet +SGFuZzIJON +FFuzz +RESTler +PULSAR +ABILITYTOEXPLORESTATES +nyx-net +I +II +SECFuzz +GANFuzz +AutoFuzz +ABILITYTOEXPLOREMESSAGES +IIIFuzzers for stateful systems +, , +For fuzzing of non-stateful systems it has already demonstrated that using a combination of tools may be the optimal +approach, especially if these tools can exchange information [12]; we expect that this will be even more so for stateful +systems. +By providing insight into the components used in various fuzzing approaches, our research suggests several interesting +directions for future research. One direction is in trying our new combinations of approaches and components, for +example, using LearnLib as a pre-processing phase may be useful get a good initial approximation of the protocol +state machine, or using the SUT response as feedback in man-in-the-middle fuzzers to build a more accurate protocol +state model. Some of the performance optimisations implemented by specific fuzzers (e.g. the use of snapshots) can be +applied to a broader set of fuzzers. Another direction is in more systematic, empirical comparison: having identified that +some tools use the same overall approach but a different algorithms for some sub-component allows a more systematic +comparison, where we just observe the effect of changing this one sub-component. +REFERENCES +[1] Humberto J. Abdelnur, Radu State, and Olivier Festor. 2007. KiF: A Stateful SIP Fuzzer. In Proceedings of the 1st International Conference on Principles, +Systems and Applications of IP Telecommunications (IPTComm’07). 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ACM Computing Surveys (CSUR) 54, 11s (2022), +1–36. +20 + diff --git a/VdE0T4oBgHgl3EQflwF_/content/tmp_files/load_file.txt b/VdE0T4oBgHgl3EQflwF_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3526473f134bad4082d1af577a2a2218be65726 --- /dev/null +++ b/VdE0T4oBgHgl3EQflwF_/content/tmp_files/load_file.txt @@ -0,0 +1,819 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf,len=818 +page_content='Fuzzers for stateful systems: Survey and Research Directions CRISTIAN DANIELE, SEYED BEHNAM ANDARZIAN, and ERIK POLL, Radboud University, The Nether- lands Fuzzing is a security testing methodology effective in finding bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In a nutshell, a fuzzer sends multiple slightly malformed messages to the software under test, hoping for crashes or weird system behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The methodology is relatively simple, although applications that keep internal states are challenging to fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The research community has responded to this challenge by developing fuzzers tailored to stateful systems, but a clear understanding of the variety of strategies is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In this paper, we present the first taxonomy of fuzzers for stateful systems and provide a systematic comparison and classification of these fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' CCS Concepts: • Security and privacy → Software security engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Additional Key Words and Phrases: stateful fuzzing, state model, active learning 1 INTRODUCTION With fuzzing (or fuzz testing) a system is fed a large collection of automatically generated inputs to find security vulnerabilities, in particular memory corruption bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Fuzzing is a great way to improve software security: it can find lots of bugs with relatively little effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The idea of fuzzing goes back to the late 1980s [32] but interest in fuzzing exploded in the 2000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Major game changers here have been the advent of white-box fuzzing using symbolic (or more precisely, concolic) execution in the SAGE fuzzer [19] and the advent of grey-box fuzzing, also known as evolutionary fuzzing, pioneered in the fuzzer AFL [48], where code execution paths are monitored to guide the generation of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Both approaches avoid the need of having the user provide an explicit grammar describing the input format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Traditionally, most fuzzers target stateless systems, where the system under test takes a single input, say a JPEG image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' the fuzzer then tries many possible inputs, including many malformed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This survey focuses on the fuzzing of stateful systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' By a stateful system, we mean a system that takes a sequence of messages as input, producing outputs along the way, and where each input may result in an internal state change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Most protocols, including most network protocols, are stateful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' So when people talk about ‘network fuzzing’ [21] or ‘network protocol fuzzing’ [20, 23], they are usually talking about the fuzzing of stateful systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Obviously, fuzzing stateful systems is harder than fuzzing stateless systems, as the internal state changes increase the state space that we try to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Moreover, it may be hard for a fuzzer to reach ‘deeper’ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Indeed, fuzzing stateful systems is listed as one of the challenges in fuzzing by Boehme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There are some good survey papers about fuzzing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [31, 50], but these do not investigate the issues of fuzzing stateful systems in any depth, even though these surveys do include some fuzzers for stateful systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There is a growing number of fuzzers specifically for stateful systems, which raises questions about the approaches these fuzzers take, their commonalities and differences, and their pros and cons, which this paper tries to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The outline of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Section 2 defines some basic concepts and terminology for discussing stateful systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Section 3 presents the traditional classification for fuzzers used in the literature and discusses some of the differences between fuzzing stateless and stateful systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Section 4 presents our taxonomy for fuzzers for stateful systems, classifies existing fuzzers and compares the various approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Manuscript submitted to ACM 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='02490v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='CR] 6 Jan 2023 , , Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll 2 CONCEPTS AND TERMINOLOGY By a stateful system we mean a system that takes a sequence of messages as input, producing outputs along the way, and where each input may result in an internal state change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' To avoid confusion, we reserve the term message or input message for the individual input that the System Under Test (SUT) consumes at each step and the term trace for a sequence of such messages that make up the entire input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' We use the term response for the output that the SUT produces along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In the case of a synchronous protocol, there is usually one response after each input message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In this case, the state machine describing the input-output behaviour will be a Mealy machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The input language of a stateful system consists of two levels1: 1) the language of the individual messages, which we will refer to as the message format, and 2) the language of traces, built on top of that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' A description or specification of such an input language will usually come in two parts, one for each of the levels: for example, a context-free grammar for the message format and a finite state machine describing sequences of these messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' We will call the latter the state model or, if it is described as a state machine, the protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The state model usually involves a notion of message type where messages of one type trigger a different transition than messages of another type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' So then the set of messages types is the input alphabet of any protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This alphabet will typically abstract away from payloads inside messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For some protocols, messages simply include an instruction byte in a header that determines the message type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For output messages, it is common to distinguish between error responses and non-error responses, and possibly a finer-grained distinction of error responses based on the error code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Fuzzers that try to infer the protocol state machine (using active or passive learning, as will be discussed later) may require the user to specify the abstraction function that maps concrete messages to message types or even to provide an implementation of this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There can be two abstraction functions, one for input messages and one for responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In some protocols the format of input messages and responses is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For fuzzing it is then a good strategy to also include responses as inputs as this may trigger unexpected behaviour: client and server are likely to share a part of the codebase and one may then accidentally will process messages only intended for the other2 We use the term protocol state to refer to the abstract state of an SUT that determines how future input messages are handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The SUT will have a concrete program state, which is related to this protocol state, but which usually carries much more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The term ‘state’ can quickly become overloaded: not only the SUT has state, even if it implements a stateless protocol, but the fuzzer itself also has a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' We use the term stateful fuzzing to refer to the fuzzing of stateful systems, but we avoid the term ‘stateful fuzzer’ as even a fuzzer for stateless systems will have internal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There are basically two ways for an SUT to record the protocol state: 1) it can keep track of the protocol state using program variables that record state information or 2) the state can be more implicitly recorded by the program point where the execution is at (and possibly the call stack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Of course, these two ways can be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For fuzzers that use a grey- or white-box approach (discussed in more detail later), the difference can be important: white-box approaches that observe the values of program variables will work better for 1) than for 2), whereas grey-box approaches that observe execution paths will work better for 2) than for 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 1For text-based protocols, as opposed to binary protocols, there may even be a third level, namely the character set or character encoding used, but none of the fuzzers studied use that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2CVE-2018-10933 is an interesting example of a bug of this kind in Libssh: if the message that the server sends to a client to confirm that the client has successfully authenticated is sent to the server, a malicious client could skip the whole authentication phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2 Fuzzers for stateful systems , , Stateless vs stateful systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There is not always a clear border between stateless systems and stateful systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For example, if a system has a memory leak then it is (unintentionally) stateful, even though its behaviour will appear to be stateless for a very long time, namely until it runs out of memory and crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' More generally, we can think of any stateful system that takes a sequence of messages as input as a stateless system which takes that whole trace as single input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Conversely, we can view a stateless system that takes a single complex value as input as a stateful system that processes a sequence of smaller inputs, say bits or bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For instance, a stateless program that processes JPEGs, which will always process the same JPG in the same way, can be viewed as a stateful program that takes a sequence of bytes as input, and which will process the same byte in different ways depending on where in the JPEG image it occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' But a fundamental difference between a stateful system and a stateless system viewed as stateful one in this way is that the former will typically provide some observable output after processing each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Another difference is knowing that inputs are made up of smaller messages can help in making useful mutations, by swapping the order of messages or repeating messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some stateless systems can process sequences of inputs like stateful systems do, but then the idea is that previous inputs do not have any effect on how subsequent inputs are handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some fuzzers use this possibility to avoid the overhead of having to restart the SUT between inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This is called persistent fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This does then involve a sequence of inputs, but it is the polar opposite of stateful fuzzing: the goal is not to explore the statefulness of the SUT, but rather it presupposes that there is no statefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 3 BACKGROUND This section discusses existing classifications of fuzzers used in the literature, as this provides a starting point for our classification of stateful fuzzing, and makes some initial observations about how and why fuzzing stateful systems is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There are some good surveys papers about fuzzing, but none pay much attention to issues specific to stateful fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The survey by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [50] classifies close to 40 fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Only two of these, namely AFLNet [36] and de Ruiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [15] specifically target stateful systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The more extensive survey by Manes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [31] categorises over 60 fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Thirteen of these are fuzzers for ‘network protocols’ so presumably these are fuzzers geared to fuzzing stateful systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' all of these 13 fuzzers are black-box fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' One section in this survey (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='2) discusses the statefulness of the SUT: here it discusses the inference of state machine models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The only attempt at classifying fuzzers for stateful systems that we are aware of is given in the paper by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' about SGPFuzzer [47]: Table 1 in this paper lists twelve other fuzzers for stateful systems (or "protocol fuzzers" in the terminology used in the paper), namely AutoFuzz [20], AspFuzz, SECFuzz [45], Sulley 3, BooFuzz [35], Peach 4, SNOOZE [7], PULSAR [18], TLS-fuzzer, DTLS-fuzzer, ICS-fuzzer, and NLP-fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The authors identify four challenges based on the shortcomings of these 12 fuzzers and then design SGPFuzzer to address these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Unfortunately, the definitions of the features used for the comparison are left implicit and the comparison fails to point out some very fundamental differences between tools, for instance that the man-in-the-middle nature of AutoFuzz and SecFuzz comes with an important inherent limitation, namely that the order of messages cannot be fuzzed (as we discuss in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='com/OpenRCE/sulley 4https://wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='mozilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='org/Security/Fuzzing/Peach 3 , , Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='1 Existing classifications of fuzzers The standard classification of fuzzers in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [11, 31, 50]) distinguishes black-box, grey-box and white-box fuzzers, where the black-box fuzzers are sub-divided into grammar-based fuzzers and mutation-based fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Even though this classification is fairly standard, the terminology varies between papers, and there are combinations of approaches that do not neatly fit into one of these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' We discuss this classification in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Black-box fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' As the name suggest, black-box fuzzers only observe the SUT from the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' To stand any change of producing interesting inputs, black-box fuzzers require some knowledge of the input format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' One approach here, taken by generation-based aka grammar-based fuzzers, is that the fuzzer is given knowledge of the input format in the form of grammar or model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' A downside of such fuzzers is the effort required to produce such a model or grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Another approach, taken by mutation-based fuzzers, is that the fuzzer is supplied with a set of sample inputs which are then mutated in the hope of discovering bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Most grammar-based fuzzers allow users to supply grammar for an arbitrary input format (or protocol) they want to fuzz, but there are also grammar-based fuzzers which have a specific input format hard-coded in them, such as the KiF fuzzer [1] for the SIP protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There are also commercial fuzzers for fuzzing specific protocols, for example, Codenomicon’s DEFENSIS fuzzer 5 (since acquired by Synopsys), which grew out of the PROTOS [25] project at the University of Oulu in Finland started in 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some fuzzers combine the generation-based and mutation-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' A grammar-based fuzzer should not only produce grammatically correct inputs, but also malformed ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' this either has to involve some form of mutation or the grammar has to be too ‘loose’ to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Conversely, a mutation-based fuzzer can be given some knowledge about the input format, for instance by providing a list of keywords or byte values with a special meaning, which the fuzzer can use in mutations in the hope of generating interesting mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' White-box fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' White-box fuzzers require access to the (source or binary) program code and analyse that code in order to provide interesting inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' With access to the code, it is possible to see which branches there are to then construct inputs that trigger particular branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Typically white-box fuzzers use symbolic or concolic execution to construct interesting test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Microsoft’s SAGE fuzzer [19] is the best-known example of this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grey-box fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grey-box fuzzers occupy the middle ground and can observe some aspects of the SUT as it executes and use this feedback to steer the fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This is also called evolutionary fuzzing as the inputs will gradually evolve into more interesting mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grey-box fuzzers can be considered as a special kind of mutational fuzzers because the evolution always involves mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grey-box fuzzers are sometimes called smart mutational fuzzers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' the black-box mutational fuzzers that lack a feedback mechanism to guide the evolution are then called dumb mutational fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grey-box fuzzers often require some instrumentation of the code or running the code in some emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The approach has been popularised by the fuzzer AFL, which observes the code execution path – or, more precisely, the branches are taken in the execution – to see if some input mutation results in new execution paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grey-box fuzzers that observe the execution path in this way are also called coverage-guided greybox fuzzers (CGF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This approach has proved to be very successful, providing much better coverage than ‘dumb’ mutational fuzzers but without the work of having to provide a grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 5www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='codenomicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='com/defensics/ 4 Fuzzers for stateful systems , , All fuzzers that involve mutation – dumb mutational fuzzers, evolutionary fuzzers, but also grammar-based fuzzers that use mutation – can be parameterised by mutation primitives, for instance random bit flips, repeating sub-sequences of the input, or inserting specific bytes, characters, or keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Alternative classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Instead of classifying fuzzers into white-, grey- and black-box, an orthogonal classification is to consider the kind of applications targeted and the kind of input this involves [31]: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' some fuzzers are geared towards fuzzing file formats, others to network traffic, and others still to web applications or OS kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Fuzzers for web applications are often called ‘scanners’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There is a relation between this classification and statefulness: applications that take a file as input are usually not stateful, whereas applications that implement a network protocol usually are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='2 White-, grey-, and black-box fuzzing for stateful systems For stateful systems some basic observations about the classification into white-, grey-, and black-box can be made: The terms ‘grey-box fuzzing’ and ‘evolutionary fuzzing’ are often used as synonyms, but for stateful systems, they are not: for a stateful system the evolution of inputs can also be steered by the outputs of the SUT, which is then evolutionary but black-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This is a key difference between a stateful and a stateless system: the response that the SUT produces is an observation that the fuzzer can make without any instrumentation of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For grammar-based fuzzers it does not really matter if the SUT is stateful or not: the grammar describing the system can describe both the message format and the protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For dumb mutational black-box fuzzers it also does not matter that much if the SUT is stateful or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Of course, it helps if fuzzer is aware of the fact that inputs are traces of messages, so that it can try swapping the order of messages, removing messages or repeating messages as interesting mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The same goes for any grammar-based fuzzer, which should also try re-ordering, repeating or dropping messages as interesting corruptions of the grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The techniques used in grey-box and white-box fuzzing to observe program execution may shed some light on the state that the SUT is in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' But as discussed in Section 2, there are different ways in which the SUT can record protocol state: the state can be recorded in some program variables, it can be recorded in the program point that the SUT is in (and possibly the call stack), or a combination of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The way in which the SUT does this can make a difference in how well some grey- or white-box technique can observe the protocol state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The statefulness of the SUT may complicate observation for a grey- or whitebox fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For white-box fuzzers that rely on symbolic or concolic execution input, the statefulness of the SUT is obviously a serious complication: a symbolic execution of a program handling a single input can already be tricky, and the execution for sequence of symbolic inputs will be an order of magnitude harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For example, if the SUT implements some loop to handle incoming messages then that loop would have to be unwound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='3 Bug detection for stateful systems In addition to a mechanism to generate inputs, a fuzzer also requires some mechanism to observe the SUT to detect if an input triggered a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Typically fuzzers look for memory corruption bugs that crash the SUT using sanitisers such as ASan (AddressSanitizer) [40], MSan (MemorySanitizer) [43], or older less efficient sanitisers such as Valgrind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' When fuzzing programs written in memory-safe languages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' when fuzzing Java programs with Kelinci [26], instead of looking for memory corruption bugs we can look for uncaught runtime exceptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' even if these bugs cannot be exploited in the way memory corruption can, they can still lead to Denial of Service problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 5 , , Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll Section Category Input required Generate state model Human interaction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='1 Grammar-based Grammar No No 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='2 Grammar-learner Sample traces Yes Yes / No 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='3 Evolutionary Sample traces No Yes / No 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='4 Evolutionary grammar-based Grammar No No 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='5 Evolutionary grammar-learner Sample traces Yes No 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='6 Man-in-the-Middle fuzzers Live traffic No Yes / No 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='7 Machine learning fuzzers Many sample traces No No Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The seven categories of fuzzers with their main characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Human interaction refers to manual code or grammar annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The five categories of fuzzers that involve a grammar, an evolutionary feedback mechanism, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' A type of bug that is specific to stateful systems are deviations from the correct state behaviour: if a system is expected to behave in a specific way, for instance by only responding to certain inputs after authentication, then a deviation from this behaviour may be a security issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For security protocols such as TLS any deviations from the expected state behaviour are highly suspicious: security protocols are very fragile and even small deviations may break the security guarantees the protocol aims to provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Unlike the detection of memory corruption bugs, this cannot be totally automated: it either requires a specification of expected behaviour (or, conversely, of unwanted behaviour), for instance with a state machine or in temporal logic, or it requires some post-hoc manual analysis of the state behaviour inferred by the fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 4 FUZZERS FOR STATEFUL SYSTEMS We have identified seven categories of fuzzers for stateful fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Table 1 summarises the main characteristics of each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some categories can be regarded as a combination or sub-category of other categories, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 6 GRAMMAR LEARNER EVOLUTIONARY GRAMMAR LEARNER EVOLUTIONARY EVOLUTIONARY GRAMMAR BASED GRAMMAR BASEDFuzzers for stateful systems , , Before we discuss each category in more detail in the sections below, we first discuss common ingredients involved in some of them: Some fuzzers require sample traces as inputs, either a few traces to act as seeds to further mutation, or many traces so that grammar can be inferred or machine model can be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Many fuzzers involve some form of grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This can be a grammar describing just for the message format, a grammar describing just the protocol state machine, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some fuzzers require such grammars as inputs, but others can provide grammars that are inferred during the fuzzing as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Many fuzzers use some form of learning to infer information about the message format, the protocol state machine, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Evolution can be regarded as a form of learning because it produces and uses new knowledge about the input format, even though this knowledge is (usually) not expressed in the form of a regular expression, state machine, or context-fee grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Evolution is a form of active learning because it involves interaction with the SUT, where the next input we try can depend on the outcome of previous tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some fuzzers use forms of passive learning instead of (or in addition to) such active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' By this we mean approaches where information about the input format is inferred after a set of traces has been collected, so without interactively trying new experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There is a long line of research into algorithms for inferring formal language descriptions, either actively or passively, which includes research into regular inference and grammatical inference that focus specifically on inference of regular expressions and context-free grammar, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Research in this field is presented at the bi-annual International Conference on Grammatical Inference (ICGI) and there are entire textbooks on the subject (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For active learning of a protocol state machine, an algorithm that can be used is L* [2] or one of its improvements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' the TTT algorithm used in LearnLib [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For the passive learning of protocol state machines, some fuzzers use ad-hoc solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For instance, the fuzzer by Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [22] uses an algorithm called partial finite state automaton reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' An important limitation of some learning algorithms, notably L* and its improvements, is that they cannot deal with the non-deterministic behaviour of the SUT, as it will cause the algorithm to diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' A very different form of (passive) learning used by some fuzzers is machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This does not produce knowledge in a nice concrete format like a regular expression, finite state machine, or context-free grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Also, it typically requires more samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Still, possible advantages are that there are many existing machine learning approaches that can be used and that these may cope more easily with non-deterministic behaviour of the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Below we first give a general description of the seven categories of fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In the subsequent sections, we discuss each category in more detail: (1) Grammar-based fuzzers Any grammar-based fuzzer can be used to fuzz stateful systems without any special adaptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The grammar that is supplied to the fuzzer will have to describe the two levels of the input language, with some rules of the grammar describing the message format and some rules describing the protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Apart from that, no change in the fuzzer itself is needed, except that course, swapping, dropping and repeating messages are useful – if not essential – mutation strategies for the fuzzer to include.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' But for a stateless SUT where the format of the inputs is quite complex it can also be useful to include swapping, dropping and repeating sub-components of inputs as mutation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 7 , , Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll (2) Grammar-learner fuzzers Whereas the grammar-based fuzzers require the users to provide a grammar, these fuzzers are able to extract a grammar from a set of sample traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' They can be considered as the sequential composition of two tools: a grammar extractor that infers the grammar from a set of sample traces (using so-called passive grammatical inference) and a grammar-based fuzzer that then does the actual fuzzing using this inferred grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' As for the grammar-based fuzzers, for the grammar-learner fuzzers the statefulness of the SUT does not make any fundamental difference: it only means that the grammar will have two levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' So grammar-learner fuzzers can be applied to stateless as well as stateful SUTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' (3) Evolutionary fuzzers These fuzzers basically take the same approach as stateless evolutionary fuzzers such as AFL: they take some sample traces as initial input and mutate these using a feedback system to steer the mutation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Of course, evolutionary fuzzers for stateful systems should be aware that an input trace is a sequence of messages and should include swapping, omitting or repeating these messages as mutation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' A difference between stateful and stateless systems when it comes to evolutionary approaches of fuzzing is that the responses that a stateful SUT provides after individual messages can be used in the feedback to guide the evolution, as mentioned before in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' (4) Evolutionary grammar-based fuzzers These fuzzers use both a grammar provided to the user to generate (correct, protocol-compliant) traces and an evolution mechanism to mutate these traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' We can think of them as evolutionary fuzzers that use a grammar instead of a set of sample input traces to provide the initial traces that will be mutated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' We can also think of them as grammar-based fuzzers that include a feedback mechanism to steer the evolution of mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' So in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 1 they are the intersection of the evolutionary fuzzers and the grammar-based fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' (5) Evolutionary grammar-learner fuzzers This is the most complex category of fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' These tools all use some form of grammar to describe the protocol state machine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' one also uses a grammar to describe the message format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' They involve two feedback mechanisms to steer two forms of evolution: (i) one for the mutation of individual messages, in the style of conventional evolutionary fuzzers like AFL, and (ii) another for the mutation of sequences, which then infers a protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The second form of evolution is based on the response that the SUT provides as feedback, so it is black-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The final two categories of fuzzers are very different from the five above: (6) Man-in-the-Middle fuzzers: These fuzzers sit in the middle between the SUT and a program interacting with it and modify messages going to the SUT, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Responses coming back from the SUT are left untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' These fuzzers can take a dumb mutational approach to modify the messages, but they may leverage a protocol specification (automatically inferred or given as input) to modify messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' (7) Machine learning fuzzers These fuzzers use a Machine Learning (ML) model trained on a large set of input traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The model outputs slightly different — hopefully malicious — mutated traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Machine learning methods used by these fuzzers include Seq2seq and Seq-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' These fuzzers are similar to the grammar learner fuzzers in that they require a set of sample traces as input that is then used to infer a model of the input format which is then the basis for the fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The key difference is that for the grammar learner fuzzers this model is a grammar, whereas for these machine learning fuzzers the model is an ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 8 Fuzzers for stateful systems , , Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Evolutionary fuzzers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grammar-based fuzzers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grammar learner fuzzers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Evolutionary grammar-based fuzzers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Evolutionary grammar learner fuzzers 9 CRAFTER SYSTEMUNDERTEST FEEDBACKSYSTEM一》 CRAFTER SYSTEMUNDERTEST GRAMMAR (PROVIDED BY THE USER)SET OF TRACES GRAMMAR SYSTEM UNDER TEST- GRAMMAR(PROVIDED CRAFTER SYSTEMUNDERTEST BY THE USER) FEEDBACKSYSTEMCRAFTER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' SYSTEM UNDER TEST GRAMMAR EXTRACTOR SET OF TRACES GRAMMAR FEEDBACK SYSTEM (U) Optional SYSTEM UNDER TEST RESPONSE (II), , Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Man-in-the-middle fuzzers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Machine learning fuzzers Fuzzer Based on Mutates Peach Message SNOOZE [7] Message PROTOS[25] Message Sulley Message BooFuzz [35] Sulley Message Fuzzowski 6 BooFuzz [35] Message Trace AspFuzz [27] Message Trace Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grammar-based fuzzers 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='1 Grammar-based fuzzers Table 2 lists the grammar-based fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 3, these fuzzers use a grammar provided by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In the case of a stateful SUT, this grammar should describe the syntax of the messages and the protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For some fuzzers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Peach7 and SNOOZE [7], this grammar is supplied in some XML format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The obvious downside of these fuzzers is that they require an accurate grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Producing one can be a lot of work and it can be challenging and error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Not all errors will matter – or matter equally: if the grammar is a bit too ‘loose’ this is not much of a problem, but if the grammar omits interesting parts of the language it may be, as this would mean that the fuzzer will not explore that part of the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Ideally the documentation of the SUT, or the specification of the protocol it implements, simply provides a formal grammar that can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' However, this will 7Here we mean the community edition, available at https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='com/gitlab-org/security-products, which lacks some features of Peach Fuzzer Professional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 10 CLIENT (OR SERVER) CRAFTER SYSTEM UNDERTEST GRAMMAR EXTRACTOR GRAMMAR SET OFTRACES OptionalSETOFTRACES CRAFTER(MLMODEL) SYSTEMUNDERTESTFuzzers for stateful systems , , Fuzzer Learns Based on Input needed PULSAR [18] State model Message fields Passive learning (using PRISMA [18]) Traces GLADE+ [8] Message fields Active learning (using GLADE [8] ) Traces Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [22] State model Passive learning (using partial finite state automaton reduction [22]) Message field specification Traces Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grammar learner fuzzers often not be the case: documentation or specifications may be unclear or incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' That the SUT is stateful does not make a difference here, Still, in earlier research [37] we found that documentation is more likely to include a clear (but informal) specification for the message format than for the protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The protocol state machine are often – very poorly – specified in prose scattered throughout specification documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some grammar-based fuzzers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' SNOOZE [7] and Sulley, come with grammars for some standard protocols, so that for these the hard work to produce a grammar has already been done for the user, but for other protocols the user still has to do it themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='2 Grammar learner fuzzers Table 3 presents the grammar learner fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' These fuzzers operate in two phases: first, they infer a grammar from a set of collected traces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' then they do the actually fuzzing using that inferred grammar just like a grammar-based fuzzer would do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' So each of these fuzzers is effectively the composition of two tools: (1) a grammar learner: a special component with the goal to build a grammar as much as possible similar to the real one (2) an actual fuzzer: in principle any of the grammar-based fuzzers discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' All these fuzzers will require a comprehensive and complete set of traces, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' the makers of PULSAR explicitly point out [18], to give good fuzzing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For the first phase, the fuzzers in Table 3 not only use different inference techniques, but also try to infer different aspects of the input format: PULSAR [18] infers both the message format and a protocol state machine, passively, from observed traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The learning techniques it uses are the ones developed earlier for the PRISMA fuzzer [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' These can also infer rules for dependencies between messages, such as increasing sequence numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' As the authors note, the approach relies on the completeness of the set of observed network traces and will be unable to model protocol paths not included in this traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' GLADE [8] uses a new active learning algorithm for inferring context-free grammars which can infer both the message format and the protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Strictly speaking, GLADE is not a fuzzer, but just a tool for inferring a context-free grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This inference uses active learning, so it does involve some fuzzing of the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' But GLADE has been extended to be used as a front-end for a grammar-based fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In Table 3 we refer to this extension as GLADE+ to avoid confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The algorithm used by GLADE+ is shown to have better precision and recall that the active learning algorithms L* [2] and RNPI [34] for the case studies tried out by the makers [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The results of GLADE+ are also compared 11 , , Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll Fuzzer Feedback system Based on Input needed nyx-net [39] Coverage AFL Target binary Protocol specification Seed inputs (optional) FitM fuzzer [30] Coverage AFL Client binary Server binary Seed inputs SNPSfuzzer [29] Coverage AFL Target binary Seed inputs Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [13] Coverage Branches AFL Manual code annotation Target binary Seed inputs SGFuzz [6] Coverage Variables AFL Automatic code annotation Target binary Seed inputs IJON [4] Coverage Variables AFL Manual code annotation Target source code Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Evolutionary fuzzers with AFL for some of these case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' However, the case studies are not typical stateful protocols but include interpreters for Python, Ruby and JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' As AFL is best at fuzzing binary formats, it is maybe not that surprising that GLADE+ beats AFL here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Unlike PULSAR and GLADE, the fuzzer by Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [22] cannot infer the message format: it only infers a protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The tool requires that the message format is known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' in fact, it needs an (un)parser for the message format to be supplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The protocol state machine is then inferred from observed traffic – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' using passive learning – using a new algorithm they introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Once this state machine is inferred, the SUT can be fuzzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Here a collection of mutation primitives is used, including mutations to mutate individual messages and mutations to reorder messages in the input trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The first phase of this fuzzer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' inferring a protocol state machine given a known message format, is very similar to what tools like LearnLib [38] do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' But it uses passive learning, whereas LearnLib uses active learning with a variant of L*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' report that they also tried active learning for this initial phase, using a variant of L*, as they also did in earlier work [41], but abandoned that approach because of 1) the difficulty in constructing concrete messages that active learning requires and 2) it being inefficient and not learning an accurate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='3 Evolutionary fuzzers Table 4 presents the evolutionary fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2, these use feedback to guide the mutation of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This feedback can use different types of observation, namely the five options listed below or a combination: F1 Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some fuzzers use the response of the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This is the only type of observation that can be done black-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 12 Fuzzers for stateful systems , , F2 Coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some fuzzers observe branch coverage in the style of AFL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' using a bitmap to observe branches taken during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This is a greybox approach that either requires re-compilation to instrument the code or running code in some emulator, just like AFL does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' F3 Branches: Some fuzzers observe branch coverage not by observing all branches like AFL does, but by observing specific branches that are manually marked as interesting to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This is a white-box approach and requires manual annotation of code by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' F4 Variables: Some fuzzers observe the value of specific program variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This is a white-box approach and requires manual annotation of code by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The idea is that the program variables observed record information about the protocol state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' F5 Memory: Instead of observing specific individual program variables, one fuzzer observes memory segments: it takes snapshots of memory areas to see if inputs affect these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The idea is that changes in the memory signal change the protocol state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The only fuzzer using this, StateAFL, is not an evolutionary fuzzer but one of the more complicated evolutionary grammar learner, so it is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' All the evolutionary fuzzers in Table 4 are based on AFL, so all of them at least observe branch coverage in the style of ALF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' F2, Coverage) to steer the evolution, but some tools use an additional feedback mechanism on top of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Regarding F3: the fuzzer by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' allows the user to mark some specific branches in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The idea is that taking these marked branches is an indication of the SUT moving to a different protocol state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Given that the AFL instrumentation already observes branch coverage, it is somewhat surprising that additional observation of selected branches improves the performance of the fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The fuzzer not just observes if these branches are taken in execution, but when this happens it effectively starts a new AFL session for this specific state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' using a new bitmap for recording branches and creating a new queue of messages to mutate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' So whereas AFL and all the other AFL-like evolutionary fuzzers in Table 4 maintain a single bitmap to record which branches have been taken, the fuzzer by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' has one such bitmap for each of the marked branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This allows it to learn different strategies for generating test cases for different protocol states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Intuitively this makes sense: messages in different stages of a protocol may have different formats, so learning different mutation strategies, each tailored to a specific protocol state, can improve the fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Regarding F4: IJON [4] observes specific program variables during the fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The user has to mark these in the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The idea is that the user marks variables that record information about the protocol state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' SGFuzz is an improvement of this: instead of the user having to annotate code to specify which program variables record interesting state information, the fuzzer automatically infers which program variables have an enumeration type, and it assumes that all these program variables record state information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' As discussed in Section 2, there are different ways in which the SUT can record its protocol state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' If the protocol state is recorded in program variables, approach F4 of IJON and SGFuzz can be expected to work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' If the program point is used the protocol state, approach F3 as used by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' might work better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' All evolutionary fuzzers require initial seeds as input traces to start fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The choice of these initial seeds can influence the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some fuzzers provide some automation to create initial seeds: for instance, Nyx-net [39] provides functionality (in the form of a Python library) to generate seeds messages from PCAP network dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The creators of IJON [4] note that in some cases IJON’s feedback mechanism works so good that manually picking good seeds is no longer necessary to obtain good coverage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' in some experiments, they could simply use a single uninformative seed containing only the character ‘a’ [4] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 13 , , Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll Fuzzer Feedback system Based on Inputs needed RESTler [5] Response State model specification Target source file SPFuzz [42] Coverage AFL Protocol specification Target source code Initial seeds EPF [21] Coverage AFL Fuzzowski Protocol specification Target source code PCAP files (as initial seeds) Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Evolutionary grammar-based fuzzers 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='4 Evolutionary grammar-based fuzzers Table 5 presents the evolutionary grammar-based fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' These fuzzers combine the grammar-based and evolutionary approaches, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 5: they require a grammar as the starting point to generate messages but they also include some feedback to observe the effects of inputs in an effort to fuzz more intelligently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' RESTler [5] is an open-source fuzzer by Microsoft for fuzzing REST APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' It uses a grammar in the form of an OpenAPI8 specification (as can be produced by Swagger tools) to generate messages but then observes responses combinations of messages that always lead to the same error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The fact that RESTful API typically come with grammar in the form of an OpenAPI spec is a big win: it means we can use a grammar-based approach but avoid the downside of having to produce a grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' SPFuzz [42] and EPF [21] observe branch coverage in the style of AFL to get information about coverage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' F2)), whereas RESTler only uses the response of the SUT (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' RESTler does not require any initial seeds provided by the user, as you would expect of a fuzzer that has a grammar that can be used to generate inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' However, SPFuzz and EPF do require the user to provide initial seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For EPF these are provided in the form of a PCAP file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' SPFuzz does not use some standard specification format like OpenAPI, but it has its own format to describe the protocol grammar and dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' These dependencies, like the ones between requests and responses [5], or the ones between the length field, the content of the message or the data types [21, 42], significantly influence the quality of the inputs generated by the fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='5 Evolutionary grammar learner fuzzers Table 6 presents the evolutionary grammar-learner fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This is the most complex category of fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' These fuzzers involve a grammar, which only describes (an approximation of) the protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' They use two forms of evolution, illustrated by the two feedback loops in Figure 6: (i) Message evolution: like for the evolutionary fuzzers discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='3, feedback from the system is used to mutate traces, using one or several of the five types of observation discussed there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' (ii) State machine evolution: here feedback from the system is used to improve an approximation of the protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This comes down to a form of active state machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 8https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='openapis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='org 14 Fuzzers for stateful systems ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Fuzzer Learns Feedback (i) Feedback (ii) Inputs needed Based on AFLNet [36] State model Coverage Response Target binary Sample traces AFL FFUZZ [10] State model Coverage Response Target binary Sample traces AFLNet StateAFL [33] State model Coverage Memory Target binary Sample traces AFLNet SGPFuzzer [47] State model Message fields Coverage Response Target binary PCAP file AFL LearnLib [38] State model N/A Response Set of messages TTT [24] Doupé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [16] State model N/A Response No input required Web crawling Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Evolutionary grammar learner fuzzers Fuzzer Limitations Uses Input needed AutoFuzz [20] Cannot fuzz the message order Passive learning [9] [22] Live traffic Black-Box Live Protocol Fuzzing [44] Cannot fuzz the message order User needs to specify the fields to fuzz N/A Live traffic SECFuzz [45] Limited fuzzing of message order N/A Live traffic Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Man-in-the-middle fuzzers For all tools except StateAFL the feedback used here is the response from the SUT (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' F1) or some information extracted from that response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' for example, for AFLNet it is the response code in the response, for EPF it is just information about whether the connection was dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' StateAFL observes whether the content of long-lived memory areas has changed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' F5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' LearnLib and the fuzzer by Doupé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' are odd ones out in Table 6 in that they are very limited in the kind of fuzzing they do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' They do not mutate individual messages but only try combinations of a fixed set of input messages to infer the state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Here LearnLib uses the TTT algorithm [24], an improvement of L*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The fuzzer of Doupé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' uses an ad-hoc algorithm developed for the tool: it is a fuzzer for web applications, so the response of the SUT is a web page, and the tool analyses these web pages for similarity in an attempt to crawl the entire website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='6 Man-in-the-middle fuzzers Table 7 presents the man-in-the-middle-fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 7, these fuzzers sit between the SUT and another application that interacts with the SUT to intercept the communication and modify the communication going to the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' If the SUT is a server then this other application will be a client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' An fundamental limitation of these fuzzers is that they are only able to modify the order of the messages in a limited way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In fact, AutoFuzz [20] and Black-Box Live Protocol Fuzzing [44] do not modify the order of messages at all, so the exploration of the protocol state machine will be very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' SECFuzz [45] does fuzz the order of the messages, but only a little bit, namely by inserting well-formed messages at random positions in the input trace (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' in the sequence of messages sent by the other application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Even though the overall set-up is the same, the fuzzers use different techniques: 15 , , Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll Fuzzer Based on Input needed GANFuzz [23] seq-gan model Traces Machine Learning for Black-box Fuzzing of Network Protocols [17] seq2seq model Traces SeqFuzzer [49] seq2seq model Traces Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Machine learning fuzzers Like the grammar learner fuzzers, AutoFuzz operates in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Prior to the actual fuzzing it starts with a passive learning phase to infer an approximation of the protocol state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For this AutoFuzz uses the same algorithm as Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [22], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' partial finite state automaton reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' So, as for Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' the user has to supply implementations of abstraction functions that map concrete messages to some abstract alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' During the fuzzing AutoFuzz then its knowledge of the protocol state machine to guide the input selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Black-Box Live Protocol Fuzzing uses a function to generate the message field specification from a PCAP file, but the user is required to choose the fields of the messages to fuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' SECFuzz is able to deal with the cryptography of the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' To do that, the client has to share with the fuzzer (through a log file) all the information necessary for the decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='7 Machine learning fuzzers Table 8 presents the machine learning fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 8, like the grammar learner and the evolutionary grammar learner fuzzers, these fuzzers require a set of traces that are used as dataset to train the machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Once trained, the machine learning model is able to output traces that slightly differ from legit, correct traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Likewise the man-in-the-middle fuzzers, the machine learning fuzzers observe protocol executions that follow the happy flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This cause an unbalanced dataset in favour of the correct traces and the model’s inability to outcomes traces with messages in the wrong order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Although these fuzzers use a machine learning model — trained on real protocol execution — to output traces to forward to the SUT, they employ different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' GANFuzz [23] uses a generative adversarial network (GAN) and an RNN (recursive neural network), while the fuzzer by Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [17] and SeqFuzzer [49] use seq2seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' We refer to the review by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [46] for a more exhaustive explanation of fuzzing using machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 5 GENERIC FUZZERS IMPROVEMENTS Irrespective of the category of fuzzer, there are some generic improvements that several fuzzers include.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Pre-processing of raw network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Many fuzzers take raw network traffic in the formal of a PCAP file as input and provide some automated pre-processing of that input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Each tool implements it in their own way but it includes some common ingredients, such as chopping up the traces to extract the individual messages to then clustering similar messages or recognizing specific fields in the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Using snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' One factor that makes fuzzing of stateful systems slow is that a fuzzer often needs to repeat a sequence of inputs to get the SUT in a particular state, to then start the actual fuzzing in that program state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' To avoid the overhead, some fuzzers [29, 30, 39] use snapshots (aka checkpoints) to capture the program state of the SUT at a particular point in time, to then be able to quickly re-start fuzzing from that point on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' (The same idea is behind the 16 Fuzzers for stateful systems , , use of forking by AFL, where even for stateless SUTs it has been shown to improve performance off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=') This can speed up fuzzing, as the initial trace to reach some specific state does not have to be repeated, but taking and re-starting snapshots also introduces overhead, so in the end it may not be faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Depending on the execution platform there are different snapshotting techniques that can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For instance, FitM and SNPSFuzzer use CRIU’s userspace snapshots and nyx-net uses hypervisor-based snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For SNSP different snapshotting technologies have been compared [29]: CRIU 9, DMTCP [3], BLCR 10, and OpenCZ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Mutation primitives and heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Any fuzzer that uses some form of mutation (of individual messages or of traces) can use a variety of strategies and primitives to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For individual messages this may include random bit-flipping, deleting some parts of a message or inserting some data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For traces as opposed to individual messages) interesting mutation primitives are of course removal, insertion, or repetition of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The fuzzers we discussed come with variety of primitives for all this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some offer possibilities for the user to provide their own custom mutators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' We have not gone into the details of this, as the focus was on understanding the overall approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some fuzzers, notably SNOOZE, PROTOS, SPFuzz, SGPFuzzer, and the fuzzer by Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [22], provide more advanced heuristics and tricks for mutations than some of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For example, SNOOZE can provide mutations to try out SQL or command injection or use specific numbers to test boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' SPFuzz distinguishes different types of data inside messages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' headers vs payloads) to then use different mutation strategies for specific types of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In practice it may of course make a big difference for a particular case study which mutation primitives or heuristics are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 6 CONCLUSIONS It took us quite some effort to disentangle the ways that various fuzzers for stateful systems work and arrive at the classification we presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' It seems like every fuzzer picks another set of building blocks, combines them in its own way, and then adds some ad-hoc heuristics and possibly performance optimisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' New fuzzers are typically evaluated on some case studies and then compared with some other fuzzers, but it is hard to draw broader conclusions that then go beyond a particular case study or a particular pair of fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' This underlines the importance of initiatives such as ProFuzzBench [33] for bench-marking stateful fuzzing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Benchmarking has also been pointed out as a challenge for fuzzers in general, not just for stateful fuzzing [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' We have noted some apparent contradictory observations – though this may simply be because researchers looked at different case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For instance, Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [41] abandoned the use active learning of protocol state machine using L* (or its variants) because they found it too slow and inaccurate, while in other research this has proved to be very useful in finding security flaws [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' It is not surprising that the performance of fuzzer may depend heavily on the case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' When fuzzing a stateful system there is a trade-off between a) trying out many variations of individual messages and b) trying out many different sequences of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The complexity of an application (and hence the likely problem spots) application may more in the message format or more in the protocol state machine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' a corresponding strategy when fuzzing, focusing more on a) or on b), is then most likely to find bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Very broadly we can make a rough distinction into three classes of tools, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 9: I) fuzzers that are very good at aggressively exploring the protocol state machine but poor at trying out variations of messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' III) fuzzers that are good at trying out variations in messages but poor at exploring 9https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='com/checkpoint-restore/criu 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='com/angelos-se/blcr 11https://openvz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='livejournal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='com 17 , , Cristian Daniele, Seyed Behnam Andarzian, and Erik Poll Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Cluster of fuzzers the protocol state machine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' and II) fuzzers which try to explore both the protocol state machine and the format of individual messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' There is some relation between this classification and the seven categories we have described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' For instance, the man-in-the-middle fuzzers and the machine learning fuzzers are in class III, as they do not explore the protocol state machine and mainly (or even exclusively) stick to message sequences observed in real communications between client and server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The grammar-based fuzzers can deal quite well with both dimensions of fuzzing so are in class II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Evolutionary-based fuzzers that try to infer the protocol state machine (typically using the response of the SUT as feedback mechanism) are good at exploring the protocol state space, but may lack mutation primitives or observation mechanisms to aggressively explore the message formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' LearnLib is an extreme instance of class I as it only fuzzes the message order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' The exact positioning of tools in Figure 9 is not based on experimental data, but more informally based on the general characteristics of the tools, so should be taken with a pinch of salt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Also, for tools that require grammars as input or manual code annotation a lot will depend on the quality of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' It may seem like fuzzers of type II are the best of both worlds, but given the rapid state space explosion when we fuzz both individual messages and sequences of messages this need not be the case: Using a fuzzer of type I and a fuzzer of type III to explore different aspects may be more effective than using one fuzzer of type II that tries to do both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 18 Sulley PULSAR nyx-net IJON RESTler AFLNet LearnLib AutoFuzz GANFuzz SGFuzz SECFuzz FFuzz Sulley LearnLib AFLNet SGFuZzIJON FFuzz RESTler PULSAR ABILITYTOEXPLORESTATES nyx-net I II SECFuzz GANFuzz AutoFuzz ABILITYTOEXPLOREMESSAGES IIIFuzzers for stateful systems , , For fuzzing of non-stateful systems it has already demonstrated that using a combination of tools may be the optimal approach, especially if these tools can exchange information [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' we expect that this will be even more so for stateful systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' By providing insight into the components used in various fuzzing approaches, our research suggests several interesting directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' One direction is in trying our new combinations of approaches and components, for example, using LearnLib as a pre-processing phase may be useful get a good initial approximation of the protocol state machine, or using the SUT response as feedback in man-in-the-middle fuzzers to build a more accurate protocol state model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Some of the performance optimisations implemented by specific fuzzers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' the use of snapshots) can be applied to a broader set of fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Another 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' IEEE Software 38, 3 (2021), 79–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [12] Yuanliang Chen, Yu Jiang, Fuchen Ma, Jie Liang, Mingzhe Wang, Chijin Zhou, Xun Jiao, and Zhuo Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' EnFuzz: Ensemble Fuzzing with Seed Synchronization among Diverse Fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In USENIX Security Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' USENIX Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [13] Yurong Chen, Tian lan, and Guru Venkataramani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Exploring Effective Fuzzing Strategies to Analyze Communication Protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In Proceedings of the 3rd ACM Workshop on Forming an Ecosystem Around Software Transformation (FEAST’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' ACM, 17–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='1145/3338502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content='3359762 [14] Colin de la Higuera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Grammatical inference: learning automata and grammars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [15] Joeri de Ruiter and Erik Poll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Protocol State Fuzzing of TLS Implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In USENIX Security Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [30] Dominik Maier, Otto Bittner, Marc Munier, and Julian Beier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' FitM: Binary-Only Coverage-Guided Fuzzing for Stateful Network Protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In Workshop on Binary Analysis Research (BAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' Internet Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' boofuzz Documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' THIS REFERENCE STILL NEEDS TO BE FIXED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' [36] Van-Thuan Pham, Marcel Böhme, and Abhik Roychoudhury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' AFLNET: A Greybox Fuzzer for Network Protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQflwF_/content/2301.02490v1.pdf'} +page_content=' In 13th International Conference on 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moiré systems +Prathyush P. Poduval1 and Mathias S. Scheurer2 +1Condensed Matter Theory Center, Department of Physics, +University of Maryland, College Park, MD 20742, USA +2Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria +Motivated by the phenomenology of graphene moiré superlattices, we study a 2D model with +strong tendencies towards both magnetism and triplet superconductivity. Individually, their respec- +tive order parameters, N and d, cannot order at finite temperature. Nonetheless, the model exhibits +a variety of vestigial phases, including charge-4e superconductivity and broken time-reversal sym- +metry. Our main focus is on a phase characterized by finite d·N, which has the same symmetries as +the BCS state, a Meissner effect, and metastable supercurrents, yet rather different spectral proper- +ties: most notably, the suppression of the electronic density of states at the Fermi can resemble that +of either a fully gapped or nodal superconductor, depending on parameters. This could provide a +possible explanation for recent tunneling experiments in graphene moiré systems. +Strongly correlated systems often exhibit complex +phase diagrams with multiple phases, characterized by +long-range or quasi-long-range order (QLRO) of differ- +ent order parameters. Aside from phase competition as +a possible origin, a rich set of phases might also be un- +derstood as different manifestations of an underlying pri- +mary order—a concept often referred to as “intertwined +orders” [1]. +For instance, thermal or quantum fluctu- +ations can disorder a primary order parameter, while +higher-order composite order parameters can still sur- +vive. +An example of such a “vestigial phase” [2, 3], is +the charge-4e superconducting state that emerges when +a charge-2e pair density wave order parameter, ∆Q, itself +vanishes, yet ∆Q∆−Q does not [4]; this and other forms +of charge-4e superconductivity have attracted a lot of +attention [5–18], in particular, as a result of recent ex- +periments [19, 20]. +Another exciting recent development is the emergence +of twisted graphene moiré superlattices as versatile play- +grounds for strongly correlated physics [21, 22]. These +systems display a variety of different phases such as +nematic [23–25] and density-wave order [26–28], differ- +ent forms of magnetism [29–33], and, possibly uncon- +ventional [34, 35], superconductivity [36]; magnetism +and superconductivity appear in the same density range +[34, 35, 37–41] and recent experiments [33, 42] demon- +strate that they can coexist microscopically. Motivated +by these observations, we here study the case of two pri- +mary order parameters: a fully gapped spin-triplet su- +perconductor (d) and, in line with the conclusions of +[41, 43], magnetic order (N) with antiparallel spins in the +two valleys. At finite temperature, T > 0, it must hold +⟨d⟩ = ⟨N⟩ = 0, in two-dimensions (2D). However, there +are several different vestigial phases, see Fig. 1(a), char- +acterized by the composite order parameters φdd = d · d, +φdN = d·N, and φddN = i(d†×d)·N. These include not +only a charge-4e superconductor [44, 45], see Fig. 1(b), +but also a charge-2e state, which has the same symme- +tries as and is, hence, adiabatically connected to the BCS +state. However, it should primarily be thought as a con- +densate of three electrons and a hole, see Fig. 1(c), or, +more formally, QLRO of φdN. We develop a theory for +this state and study its spectral properties at finite T, +which are rather different from those of the BCS state. +Depending on T and φdN, we obtain a low-energy sup- +pression of the density of states (DOS) similar to a fully +gaped or nodal state. This could provide an alternative +explanation [43, 46, 47] to the tunneling data of [34, 35], +which does not require any momentum dependence in the +superconducting order parameter. +Model.—We consider a 2D model exhibiting both +triplet superconductivity and magnetism, with three- +component order parameter fields d (complex) and N +(real), respectively. Denoting the electronic field opera- +tors of spin s =↑, ↓ (Pauli matrices s) and in valley τ = ± +(Pauli matrices τ) by ck,s,τ, where k = (iωn, k) comprises +(a) +(b) +b2/b1 +c1/b1 +(A) +(C) +(B1) +(B2) +Φdd≠0 +ΦddN≠0 +Φdd,ΦdN≠0 +Φdd,ΦdN≠0 +C2z, Θ, SO(2) +C2z, Θ, no SC +C2z, Θ, SO(3) +C2z, Θ, 4e-SC +C2z, Θ, SO(2) +C2z, Θ, 2e-SC +C2z, Θ, 2e-SC +C2z, Θ, SO(3) +Φdd +0 +(A) +e– +e– +e– +e– +S=1 +S=1 +S=0 +charge-4e +(B) +e– +e– +h+ +e– +S=1 +S=1 +S=0 +charge-2e +(c) +FIG. 1: (a) Mean-field phase diagram for rd = rN, b3 = b1, +c2 = 0, where we indicate the symmetries at T = 0 (blue), +those of the resulting vestigial phases at T > 0 (red), and +which composite order parameters are finite. Solid (dashed) +orange lines are phase transitions at T = 0 and T > 0 +(become a crossover at T > 0). (b,c) illustrate the finite-T +pairing in phases (A) and (B) schematically. +arXiv:2301.01344v1 [cond-mat.supr-con] 3 Jan 2023 + +0.4 +0.2 +0.0 +-0.2 +-0.4 +-0.4 +-0.2 +0.0 +0.2 +0.42 +Matsubara frequencies and 2D momentum, they couple +as +Sc = λ +� +k,q +� +c† +k−qsN qτzck + (c† +k−qsdqisyτyc† +−k + H.c.) +� +. +Note that N couples anti-ferromagnetically in the two +valleys; while the ferromagnetic case—with τ0 instead of +τz in the first term of Sc above—can be studied similarly, +we focus on antiferromagnetism not only for concreteness +here but also because recent microwave experiments [41] +and a systematic analysis [43] of multiple other exper- +iments on graphene moiré systems favor this scenario. +The bare dynamics of d and N is governed by +Sχ = +� +q +� +χ−1 +N (q) N qN −q + χ−1 +d (q) d† +qdq +� +. +We take the susceptibilitites to be χµ(q) = χ0 +µ/(rµ+Ω2 +n+ +v2 +µq2), µ = N, d, where q = (iΩn, q) and Ωn are bosonic +Matsubara frequencies. The nature of the phase realized +in the system depends crucially on the interactions be- +tween the bosonic fields. Up to quartic order, the local +terms allowed by the symmetries listed in Table I can be +written as SV = +� +x V (d(x), N(x)) with +V = b1(d†d)2 + b2|dd|2 + b3N 4 + c1|dN|2 + c2(d†d)N 2. +Finally, the bare electronic action is given by +Se = +� +k +c† +k,τ,s (−iωn + ϵτ·k) ck,τ,s, +where we already used that the band structures in the +two valleys are related by time-reversal Θ. +Mean-field and possible phases.—To probe the possible +phases, we start with a mean-field analysis with respect +to d and N. Absorbing the impact of the coupling to the +electrons [48] by a redefinition of the parameters of V , we +obtain the four distinct zero-temperature phases labeled +(A), (B1,2), and (C) in Fig. 1(a), where we assumed that +both ⟨d⟩ and ⟨N⟩ are non-zero and homogeneous. Using +ˆe1,2,3 ∈ R3 to denote orthogonal unit vectors, we have +N = N0ˆe1 and d = d0eiαˆe2 in phase (A), which breaks +SO(3) completely, while Θ is preserved (in any gauge- +invariant observable); as for any phase with ⟨N⟩ ̸= 0, C2z +is broken. In phase (B1), N and d are aligned; we, thus, +obtain a residual spin-rotation symmetry SO(2) along +that direction and Θ is preserved too. Beyond a critical +value of b2, an additional component with relative phase +π/2 emerges in d, defining phase (B2) where N = N0ˆe1 +and d = d0eiα(ˆe1 + iηˆe2), with 0 < η < 1; this is a dis- +tinct phase as η ̸= 0 breaks both the residual SO(2) spin +symmetry and Θ. Finally, phase (C) is characterized by +N = N0ˆe1 and d = d0eiα(ˆe2 + iˆe3). Consequently, Θ is +also broken but a residual SO(2) spin-symmetry remains. +Importantly, ⟨d⟩ , ⟨N⟩ ̸= 0 is only possible and, thus, +our discussion of symmetries is only valid for T = 0 in +TABLE I: Relevant symmetries g and their action on the +field operators. Here Rϕ is the orthogonal matrix obeying +e−iϕ·sseiϕ·s = R(ϕ)s. All symmetries are linear except for +Θ which is anti-linear. +g +ck +N +d +φdd +φdN +φddN +U(1) +eiϕck +N +e−2iϕd e−4iϕφdd e−2iϕφdN +φddN +SO(3) +eiϕ·sck +RϕN +Rϕd +φdd +φdN +φddN +C2z +τxc−k +−N +−d +φdd +φdN +−φddN +Θ +isyτxc−k +N +−d∗ +φ∗ +dd +−φ∗ +dN +−φddN +2D. To analyze the resulting vestigial phases at finite +T, where SO(3) spin-rotation symmetry is preserved and +⟨d⟩ = ⟨N⟩ = 0, it is convenient to define the following +composite order parameters φdd = d·d, φdN = d·N, and +φddN = i(d† × d) · N, with symmetry properties listed in +Table I. Crucially, all of them transform trivially under +SO(3) spin-rotations and, hence, can exhibit long-range +(in case of the last one) or QLRO (in case of the former +two) at finite T. We indicate this in Fig. 1(a) for the +different phases. This immediately tells us that, in spite +of ⟨d⟩ = 0, phase (A) transitions for finite T into state +where φdd has QLRO and, thus, constitutes a charge-4e +superconductor (as φdN = 0), which does not break C2z +or Θ (as φddN = 0); intuitively, one can think of this +state as a condensate of four electrons forming a spin- +singlet out of two triplets, see Fig. 1(b). At finite T, (B1) +and (B2) will both preserve all normal-state symmetries +and become the same phase, which we denote by (B) in +the following. It is characterized by QLRO not only in +φdd but also in φdN; as the latter has charge 2e, it is a +charge-2e superconductor and adiabatically connected to +the conventional BCS state. Nonetheless, in our current +description, this state should rather be thought of as the +condensation of three electrons and a hole, see Fig. 1(c), +consisting of a pair of electrons in a triplet state forming a +singlet with a spin-1 particle-hole excitation. In fact, we +will see below that it exhibits spectral properties rather +different from those of the BCS state at finite T. Finally, +while phase (C) does not exhibit any vestigial pairing at +T > 0, it will have long-range order in φddN and, as such, +continues to break both C2z and Θ. +Theory for phase (B).—As c1 < 0 is found when the +coefficients in V are computed by integrating out elec- +trons [48], we next focus on phase (B). To obtain an +efficient description of this phase that properly captures +the preserved SO(3) symmetry at finite temperature, we +first decouple the four terms in V using four Hubbard- +Stratonovich fields, ψd for d†d, ψN for N 2, φd for d · d, +and φdN for d · N. We treat them on the saddle-point +level, which becomes exact in the limit where the num- +ber of components of d and N is taken to be infinitely + +3 +(a) +(b) +−1.0 +−0.5 +0.0 +0.5 +1.0 +ω/√rN +0.0 +0.5 +1.0 +1.5 +2.0 +DOS +(c) +Free +Σ1 +Σ2 +Σ1 + Σ2 +-17 +-11 +-5 +0 +5 +11 +17 +n (iωn = πn/β) +0.000 +0.025 +0.050 +0.075 +0.100 +(d) +ϵN +ϵ1 +∆N +∆1 +FIG. 2: Diagrams contributing to the fermionic self energy +Σ (a) in the matrix-large-N limit defined in the main text +and (b) to first order. (c) Impact of spin (Σ1) and triplet +fluctuations (Σ2) on the constant DOS (blue) of a 2D band +with finite bandwidth. (d) Comparing the first order +solution (ϵ1, ∆1) and self consistent solution (ϵN, ∆N) for +G = iω − ϵ(iω)γz + ∆(iω)γy for S2 only (both without +momentum integration). We use ϵ/√rN = 0.1, φ0/rN = 0.5. +large [49]. The saddle point values of ψd and ψN will in +general be non-zero, which we absorb into a redefinition +of rd,N. Then, the effective action for phase (B) becomes +Seff = Sχ + Se + Sc + Sφ where +Sφ = +� +q +� +φ0 +dN dq · N −q + φ0 +dd dq · d−q + H.c. +� +. +(1) +While generically both saddle point values φ0 +dN and φ0 +dd +are expected to be non-zero simultaneously in phase (B), +we take φ0 +dd → 0 and φ0 +dN ≡ φ0 ̸= 0 for the following +explicit calculations. Setting φ0 +dd = 0 does not change +any symmetries of the phase, allows for a more compact +discussion of the results, and can formally be seen as +the large b2 limit of the theory where φ0 +dd is suppressed +[cf. Fig. 1(a)]. More generally than the derivation of Seff +via Hubbard-Stratonovich transformations, it can also be +thought of as the simplest field theory capturing the key +aspects of phase (B) in Fig. 1(a) at finite T. +Electronic self energy.—To compute the spectral prop- +erties of the electrons within this model, we employ a +large-N technique similar to [50, 51]: we add extra indices +to the electrons and bosons, ck,τ,s → ck,τ,s,a, dab → dab +and similarly for N, where a, b = 1, 2, . . . , N, which are +contracted in all terms of Seff so as to ensure O(N) sym- +metry. In the limit N → ∞, the electronic self-energy +Σ is given by the “rainbow diagrams” [50, 51] shown in +Fig. 2(a). In our case, however, Σ involves both normal +and anomalous contributions as a result of the anoma- +lous bosonic term ∝ φ0 in Eq. (1). To make this more +explicit, we integrate out the bosons, yielding the effec- +tive fermionic interactions Sint = S1 + S2 with +S1 = − +� +q +λ2 +Mq +�χ−1 +d +4 Sq · S−q + χ−1 +N Dq · D† +q +� +, +(2a) +S2 = −1 +2 +� +q +λ2 +Mq +� +φ0 Sq · D† +q + φ∗ +0 Dq · S−q +� +, +(2b) +where Mq = χ−1 +d χ−1 +N − |φ0|2 and Sq = +� +k c† +k+qsτzck, +Dq = +� +k c† +k+qsisyτyc† +−k. The two terms in S1 describe +spin and superconducting triplet fluctuations, respec- +tively; their associated self-energy contributions are nor- +mal in the sense that U(1) symmetry is preserved, with +leading terms represented by the first two diagrams Σ1,2 +in Fig. 2(b). Conversely, S2 breaks U(1) symmetry, when +φ0 attains a mean-field value, and results in an anoma- +lous contribution to the self-energy, with leading term +given by the last diagram Σ3 in Fig. 2(b). +To represent the diagrams algebraically, we shift to the +Bogoliubov-de Gennes basis (cq,+, isyc† +−q,−)T , with Pauli +matrices γi acting on this space. In this basis, the free +Green’s function is G0(iω, ϵ) = iω−ϵγz. Up to first order +in λ2, the spin-spin self energy term can be written as +Σ1(k) = 3λ2 � +q +χ−1 +d +(q) +2Mq G0(iω+iΩ, ϵk+q), while the triplet- +triplet term is Σ2(k) = 12λ2 � +q +χ−1 +N (q) +Mq +G0(iω+iΩ, −ϵk+q). +After performing a gauge transformation to make φ0 real, +the anomalous term from the spin-triplet interaction is +given by +Σ3(k) = 3φ0 +� +q +λ2 +Mq +{γy, γzG0(iω + iΩ, ϵk+q)}. +(3) +For concreteness and since spin fluctuations are believed +to occur already at higher energies than superconducting +fluctuations in graphene moiré systems [37, 38], we focus +on rd > rN; we will use rd/rN = 9, v2 +d/v2 +N = 8, χ0 +N = χ0 +d, +and set χ0 +µ = 1 by rescaling of the fields. +Density of states.—Figure 2(c) shows the effect of the +normal contributions of the self energy Σ1,2 on the DOS +of a 2D parabolic band. The effect of Σ1 is to push the +peak of the free spectral function at energy ϵ away from +ω = 0. This results in the opening of a gap (which can +be soft depending on the parameter regime), very sim- +ilar to fluctuating anti-ferromagnetism discussed in the +cuprates [52–54]. Σ2 on the other hand has the oppo- +site effect, where it pushes states towards ω = 0. This +is because Σ1 and Σ2 have the exact same functional +form with one key difference: ϵk+q of Σ1 is replaced by +−ϵk+q in Σ2. The effect of the total normal self energy +Σ1 + Σ2 is to enhance the DOS in the vicinity of the +Fermi level, see Fig. 2(c). The anomalous contribution +Σ3 does not interfere in these effects since it occurs in the +γy channel. The role of Σ1 + Σ2 can, thus, be intuitively +thought of as providing a renormalized DOS in the nor- +mal state on top of which the anomalous Σ3 opens up + +V= +2. +Z1 : +Z2 ++h.c +Sa +Da +D! +Dt +q +a4 +−0.2 +0.0 +0.2 +ω/√rN +−2 +−1 +0 +1 +2 +ϵ/√rN(×10−2) +(a) +Σ3 +Free +−0.2 +0.0 +0.2 +ω/√rN +−2 +−1 +0 +1 +2 +ϵ/√rN(×10−1) +(b) +−10 +−5 +0 +5 +10 +ω/√rN(×10−2) +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +DOS +(c) +3φ0/rN +0.01 +0.2 +0.5 +1.5 +FIG. 3: Spectral weight as a function of ω with (blue) and +without (purple) Σ3 (a) close to ϵk = 0 and (b) including a +larger energy range; in both cases, we focus on the q = 0 +contribution (see text). (c) The effect of all three self energy +contributions Σ1 + Σ2 + Σ3 (including the momentum +integration) on the DOS. For small φ0, there is suppression +of the DOS at ω = 0 which resembles the V-shaped DOS of a +nodal state. For large φ0, the gap resembles a hard BCS gap. +a gap. We have checked [48] by numerically solving the +self-consistency equation for the self-energy [Fig. 2(a)] in +the limit (of large vµ) where only the q = 0 term of the +momentum sum contributes that higher-order corrections +do not change our results qualitatively for small φ0. For +instance, Fig. 2(d) shows the numerical solution for the +Green’s function G = iω − ϵ(iω)γz + ∆(iω)γy in Matsub- +ara space upon including the effect of S2; the difference +to the first-order result is small. +To gain intuition for the impact of Σ3 on the DOS, we +first focus again on the q = 0 term of the momentum +sum in Eq. (3). In this limit, one can easily see [48] that +Σ3 vanishes linearly in ϵk for small energies. Since Σ3 +is in the γy channel, the effect of any non-zero value is +to generically open a gap. As a result of the linear be- +havior, the states exactly at zero energy are unaffected, +but slightly away from it, the states get pushed away +to higher energy; this is clearly visible in Fig. 3(a). In +contrast, for large energies, Σ3 is readily seen to tend +to zero. +The spectral function, thus, remains asymp- +totically unaffected, as can be seen in Fig. 3(b). Taken +together, we expect the DOS to be reduced (but not fully +suppressed for small φ0) in an energy range around the +Fermi level, exhibiting an enhancement with respect to +its normal-state value at intermediate energies, and then +approaching the normal-state limit at larger energies. +To demonstrate this explicitly beyond the simple q = 0 +limit, we approximate ϵk+q ≃ ϵk+vF q∥+q2/(2m), where +q∥ is the component of q along k, and numerically evalu- +ate the momentum integrals to find the total self energy +Σ = Σ1 +Σ2 +Σ3. Choosing vF = 1.5vN, 2m = √rN/v2 +N +for concreteness, Fig. 3(c) shows the resulting DOS. As +expected, we see that there is a suppression of the DOS. +0 +2 +4 +6 +8 +√rN +vN |x| +0 +2 +4 +ψB(x)/φ0 +(b) +Numerical +Analytical +0 +2 +4 +6 +8 +10 +rd/rN +0.0 +0.5 +1.0 +1.5 +(c) +−rφ +v2 × 5 +ρ × 5 +0 +5 +10 +15 +� +2m +β¯h2 |x| +−1 +0 +1 +ψF (x)(×10−3) +(a) +βϵF = 5 +βϵF = 10 +βϵF = 20 +0 +2 +4 +6 +8 +10 +v2 +d/v2 +N +2 +4 +(d) +−rφ +v2 × 30 +ρ × 30 +FIG. 4: (a) The fermionic and (b) the bosonic ODLRO +“macroscopic wavefunction”. The mass rφ [in units of +r−1/2 +N +v−2 +N ], superfluid density ρ [r−3/2 +N +v−2 +N ], and velocity v2 +[r−3/2 +N +] of SGL in Eq. (5) as a function of rd and v2 +d are +shown in (c) and (d), respectively. +However, for small values of φ0, the resulting DOS has a +V-shaped behavior, which is typically only seen in nodal +states (with either nodal lines or points). +Recall that +the superconducting phase in our model is symmetry- +equivalent to a conventional BCS state and that the +triplet superconductor that arises at T = 0 in phase (B) +will be fully gapped. For larger φ0, the gap at ω = 0 in- +creases, and resembles a hard BCS gap. The suppression +of the DOS ρF at ω = 0 can be estimated analytically +at finite temperature by again taking the limit (of large +vµ) where the integration over q can be replaced by an +evaluation at q = 0; we find +ρF (φ0) +ρF (φ0 = 0) = +1 +√ +1 + α2 , +α = +3φ0λ2rN +2Tv2 +N(rdrN − φ2 +0). (4) +Note that φ2 +0 is bounded above by rdrN, at which point +the bosonic fields would condense and continuous sym- +metries would be broken, which cannot happen at finite +T. As φ0 increases, α increases the suppression of the +DOS, and near the instability point of φ2 +0 = rdrN, there +are no states near the Fermi surface. +To complement this analysis, we have also studied the +Hamiltonian associated with setting q = 0 in Eq. (2b) +within self-consistent Hartree-Fock, only allowing for +spin-rotation invariant operators to condense [48]. For +small α, one also finds only a partial suppression of the +low-energy spectral weight, akin to Eq. (4); including +higher-order corrections leads to a hard gap for α ≥ 1. +Electromagnetic response.—We will finally demon- +strate that the superconducting phase characterized +by +φ0 +̸= +0 +has +the +same +electromagnetic +phe- +nomenology as BCS superconductors, despite the un- +usual electronic spectral properties. +To this end, +we study off-diagonal long-range order (ODLRO) [55– + +5 +57] which implies the Meissner effect [58], flux quan- +tization +[59], +Josephson +effect +and +persistent +cur- +rents +[60]. +First +focusing +on +the +electrons, +we +show +that +⟨c† +s1,+(x1)c† +s2,−(x2)cs′ +2,−(x′ +2)cs′ +1,+(x′ +1)⟩ +→ +n0(Ψ∗ +F(x12))s1,s2(ΨF(x′ +12))s′ +1,s′ +2, with ΨF ̸= 0, as |xj − +x′ +j| → ∞ at finite x12 = x1 − x2 and x′ +12 = x′ +1 − x′ +2, +to leading (first) order in φ0; as non-zero ΨF to lin- +ear order in φ0 implies that it cannot vanish identi- +cally for generic φ0, this is sufficient to show the pres- +ence of ODLRO. We find the “macroscopic wave func- +tion” to be a singlet, ΨF(x) = isyψF(x), as expected +since spin-rotation symmetry is preserved at finite T, +with ψF(x) shown in Fig. 4(a). Alternatively, one can +demonstrate ODLRO to arbitrary order in φ0, by fo- +cusing on the bosons: +to zeroth order in λ, we find +⟨(d†(x1)N(x2))(d(x′ +1)N(x′ +2))⟩ → ψ∗ +B(x12)ψB(x′ +12) as +|xj − x′ +j| → ∞, with ψB(x) plotted in Fig. 4(b) along +with an analytic asymptotic form for large x; in [48], we +show that this leads to the same constraints as the con- +ventional form of bosonic ODLRO [55, 56]. Finally, the +connection to the textbook theory of superconductivity +can be made more explicit by deriving the analogue of +the time-dependent Ginzburg-Landay theory: we rein- +state fluctuations via φ0 → φ(x, τ) and integrate out all +other degrees of freedom yielding +SGL = +� +x,τ +� +ρ|Dτφ|2 + (rφ + |c1|−1)|φ|2 + v2 |Dφ|2� +(5) +to leading order in φ and gauge-covariant derivatives +(Dτ, D)µ = ∂µ − i2eAµ. For demonstration purposes, +we evaluated the coefficients in SGL to leading (zeroth) +order in Sc and find ρ, vφ > 0 and rφ < 0 for low T +[see Fig. 4(c,d)]; the state with QLRO in φ0 thus corre- +sponds, as usual, to the Higgs phase, with Meissner effect +and massive Higgs mode, but without Goldstone modes. +Conclusion.—We have studied the finite-T vestigial +phases, see Fig. 1(a), associated with two primary order +parameters, d and N, describing a fully gapped triplet +superconductor and spin magnetism, respectively. A cru- +cial result is the DOS of phase (B1,2) in Fig. 3(c): varying +φ0 changes the low-energy DOS from partial suppression, +akin to that of a nodal superconducting state, to a hard +gap. As φ0 is expected to change with electron filling, +this could explain the tunneling data in [34, 35]. We fi- +nally point out that the suppression of N would immedi- +ately also suppress φ0 in our model and could, therefore, +explain why superconductivity is connected to the reset +behavior in trilayer graphene [34, 35, 39, 40]. +Acknowledgments. +We thank Rafael Fernandes, +Victor Gurarie, Peter Orth, and Subir Sachdev for fruit- +ful discussions on the project and Jakob Wessling for a re- +lated collaboration. M.S.S. acknowledges funding by the +European Union (ERC-2021-STG, Project 101040651— +SuperCorr). Views and opinions expressed are however +those of the authors only and do not necessarily reflect +those of the European Union or the European Research +Council Executive Agency. Neither the European Union +nor the granting authority can be held responsible for +them. P 3 acknowledges support by the Laboratory for +Physical Sciences through the Condensed Matter Theory +Center. +[1] E. Fradkin, S. A. Kivelson, and J. M. 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Zhao, “Off-diagonal +long-range order: Meissner effect and flux quantization,” +Physical Review B 51, 3760 (1995). +[60] G. L. Sewell, “Off-diagonal long range order and super- +conductive electrodynamics,” Journal of Mathematical +Physics 38, 2053 (1997). +[61] R. M. Fernandes and A. J. Millis, “Nematicity as a Probe +of Superconducting Pairing in Iron-Based Superconduc- +tors,” Physical Review Letters 111, 127001 (2013). +[62] V. Kozii, H. Isobe, J. W. F. Venderbos, +and L. Fu, +“Nematic superconductivity stabilized by density wave +fluctuations: +Possible application to twisted bilayer +graphene,” Physical Review B 99, 144507 (2019). +Appendix A: Mean-field form of the bosonic interactions +In the main text, we view the field theory defined by the action S = Se + Sχ + Sc + SV as an effective low- +energy theory that arises when high-energy electronic degrees of freedom have already been integrated out. Due to +the symmetry and locality constraints, it only depends on a few parameters, rµ, vµ, b1,2,3, c1,2. As can be seen in +Fig. 1(a), in particular, (the sign of) the parameters c1 and b2 entering V crucially determine the phase of the system. +We here provide an estimate for these parameters using mean-field theory. To this end, we replace the bosonic fields +by classical homogeneous and time-independent vectors, N q → δq,0N, dq → δq,0d, in Se + Sχ + Sc; this yields +SHE = +� +k +c† +k,τ,s (−iωn + ϵτ·k) ck,τ,s + λ +� +k +� +c† +ks · Nτzck + (c† +ks · d isyτyc† +−k + H.c.) +� ++ const., +(A1) +which we now view as our full action, also containing the high-energy degrees of freedom. Integrating out the electronic +degrees of freedom and expanding the resulting action in terms of N and d to quartic order, one obtains exactly the +same terms as in V defined in the main text, as expected by symmetry. Moreover, one finds +c1 = b2 = −b1/2 < 0, +with +b1 = 32 λ4T +� +ωn +� +d2k +(2π)2 +1 +(ω2n + ϵ2 +k)2 > 0. +(A2) +As stated in the main text, this places us into phase (B). We note, however, that fluctuation corrections to mean field +can modify the values of these coupling constants significantly [45, 61, 62]. For instance, ferromagnetic fluctuations + +8 +can change the sign of b2 to positive values [45]. +Appendix B: Evaluation of the self-energies at leading order +In this section, we show the evaluation of the self energies up to first order in perturbation theory. We first evaluate +the anomalous part of the self energy, Σ3 in Fig. 2(b), which is contributed by the anomalous term of the action given +by +S2 = −1 +2 +� +q +λ2 +χ−1 +d χ−1 +N − |φ0|2 +� +φ0Sq · D† +q + φ∗ +0Dq · S−q +� +. +(B1) +In the following, we work in the +� +cq,+ isyc† +−q,− +�T +Bogoliubov-de Gennes basis, with the Pauli matrices γi acting on +it. The free Green’s function then reads as G−1 +0 (k) = iω − ϵkγz. Choosing φ0 to be real, we have +Σ3 = 3 +� +q +φ0λ2 +Mq +(γyG0,k+qγz + γzG0,k+qγy) = 6 +� +q +φ0λ2 +Mq +ϵk+q +(iω + iΩ)2 − ϵ2 +k+q +γy, +(B2) +where +Mq = χ−1 +N χ−1 +d +− φ2 +0 = +� +− (iΩ)2 + rN + v2 +Nq2� � +− (iΩ)2 + rd + v2 +dq2� +− φ2 +0 +(B3) += ((iΩ)2 − E2 ++)((iΩ)2 − E2 +−), +(B4) +with E2 +± = +gd+gN±√ +(gd−gN)2+4φ2 +0 +2 +, and gµ = rµ + v2 +µq2. Thus, +Σ3 = 6φ0λ2 +� +q +T +� +iΩ∈Bosonic +1 +� +(iΩ)2 − E+(q)2 +� � +(iΩ)2 − E−(q)2 +� +ϵk+q +(iω + iΩ)2 − ϵ2 +k+q +γy. +(B5) +The Matsubara sum can be evaluated using +f(iω, ϵ) =T +� +iΩ +1 +((iΩ)2 − E2 ++)((iΩ)2 − E2 +−) +1 +iω + iΩ − ϵ +(B6) += 1 +2 +1 +E2 ++ − E2 +− +� 1 +E+ +(K(iω, ϵ, E+) − K(iω, ϵ, −E+)) − 1 +E− +(K(iω, ϵ, E−) − K(iω, ϵ, −E−)) +� +, +(B7) +K(iω, ϵ, E) = nf(ϵ) + nB(−E) +E + ϵ − iω +, +(B8) +where nf/B(ϵ) = +1 +eβϵ±1. Thus we get, +Σ3(k) = 3φ0λ2 +� +q +(f(iω, ϵk+q) − f(iω, −ϵk+q)) γy, +(B9) +where we performed a partial fraction decomposition of +2ϵk+q +(iω+iΩ)2−ϵ2 +k+q = +1 +iω+iΩ−ϵk+q − +1 +iω+iΩ+ϵk+q to arrive at the +expression. +The normal part of the self energy, Σ1,2 in Fig. 2(b), is contributed by the following term of the action +S1 = − +� +q +λ2 +χ−1 +d χ−1 +N − |φ0|2 +�χ−1 +d +4 Sq · S−q + χ−1 +N Dq · D† +q +� +. +(B10) +Defining γ± = 1 +2 (γx ± iγy) , the corresponding contribution to the self energy is given by +Σ1 + Σ2 = +� +q +λ2 +Mq +� +6χ−1 +d (q) +4 +γzG0,k+qγz + 12χ−1 +N (q) (γ+G0,k+qγ− + γ−G0,k+qγ+) +� +(B11) + +9 += +� +q +T +� +iΩ∈Bosonic +λ2 +Mq +1 +(iω + iΩ)2 − ϵ2 +k+q +�2 +3(gd − (iΩ)2) (iω + iΩ + ϵk+qγz) + 12(gN − (iΩ)2)(iω + iΩ − ϵk+qγz) +� +. +(B12) +Note that γzG0γz = G0 = iω − ϵγz, while γ−G0γ+ + γ+G0γ− = iω + ϵγz. As a result, if we consider the self energies +as function of iω and ϵk+q, we find that Σ1 ∼ λ2 � +q +3χ−1 +d +(q) +2Mq +G0(iω, ϵk+q) while Σ2 ∼ λ2 � +q +12χ−1 +N (q) +Mq +G0(iω, −ϵk+q). This +allows us to argue the effect of Σ2 pushing high energy states towards the vicinity of ω = 0, while Σ1 pushes states +away from ω = 0. +To perform the Matsubara sums, we define +h(iω, ϵ, g) =T +� +iΩ +−(iΩ)2 + g +((iΩ)2 − E2 ++)((iΩ)2 − E2 +−) +1 +iω + iΩ − ϵ +(B13) += 1 +2 +1 +E2 ++ − E2 +− +�E2 ++ − g +E+ +(K(iω, ϵ, E+) − K(iω, ϵ, −E+)) − E2 +− − g +E− +(K(iω, ϵ, E−) − K(iω, ϵ, −E−)) +� +, +(B14) +with K(iω, ϵ, E) as defined in (B8). In terms of these functions, the self energy is given by +Σ1 = λ2 +� +q +1 +3 [(h(iω, ϵk+q, gd) + h(iω, −ϵk+q, gd)) + (h(iω, ϵk+q, gd) − h(iω, −ϵk+q, gd)) γz] , +(B15) +Σ2 = λ2 +� +q +6 [(h(iω, ϵk+q, gN) + h(iω, −ϵk+q, gN)) − (h(iω, ϵk+q, gN) − h(iω, −ϵk+q, gN)) γz] . +(B16) +We can expand the total self energy Σ = Σ1 + Σ2 + Σ3 in terms of Pauli matrices in Nambu space, +Σ(k) = ΣId(k) + Σz(k)γz + Σγy(k)γy, +(B17) +where +ΣId(k) = λ2 +� +q +�1 +3 (h(iω, ϵk+q, gd) + h(iω, −ϵk+q, gd)) + 6 (h(iω, ϵk+q, gN) + h(iω, −ϵk+q, gN)) +� +, +(B18) +Σz(k) = λ2 +� +q +�1 +3 (h(iω, ϵk+q, gd) − h(iω, −ϵk+q, gd)) − 6 (h(iω, ϵk+q, gN) − h(iω, −ϵk+q, gN)) +� +, +(B19) +Σγy(k) = 3φ0λ2 +� +q +[f(iω, ϵk+q) − f(iω, −ϵk+q)] . +(B20) +Appendix C: Suppression of DOS at ω = 0 +In this section, we derive a compact approximate analytical expression for the suppression of the density of states +(DOS) as a result of the anomalous term Σ3 = Σγyγy. To this end, we focus on the limit of large bosonic velocities +vµ in χµ and replace the q integral in Eq. (B20) with the value of the integrand at q = 0, +Σγy(ω + i0+, k) = 3φ0λ2 rN +v2 +N +� +f(ω + i0+, ϵk) − f(ω + i0+, −ϵk) +� +. +(C1) +Note that we would first need to re-parametrize the integral in terms of ˜q = q√rN/vN and then set ˜q = 0. This +approximation would then be valid in the large vd/vN limit with this re-scaling. We then Taylor expand f(z, ϵ) with +respect to ϵ, ω, at a non-zero finite T (satisfying ϵ ≪ T ≪ +4� +rdrN − φ2 +0). In this limit, we find the self energy to be +Σγy = +3φ0rNλ2 +2v2 +NT(rdrN − φ2 +0)ϵk = αϵk. +(C2) +This expression is in agreement with the result in the main text [Fig. 3(a)] which shows that as ϵ → 0, the contribution +of Σy vanishes. With such a self-energy, the spectral function is given by +A(ω) = − 1 +π Im +ω + i0+ +(ω + i0+)2 − (1 + α2)ϵ2 +k +. +(C3) +A simple way to look at this, is that the band structure is simply renormalized as ϵk → +√ +1 + α2ϵk. This reduces the +effective band mass, and thus the DOS is suppressed by a factor of +√ +1 + α2, as stated in the main text. + +10 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.095 +0.100 +0.105 +(i +n), +0 = 0.0 +Self Consistent Solution Including +3 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.05 +0.00 +0.05 +(i +n), +0 = 0.0 +Self Consistent Solution Including +3 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.1000 +0.1005 +(i +n), +0 = 0.2 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.000 +0.005 +(i +n), +0 = 0.2 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.100 +0.102 +(i +n), +0 = 0.4 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.01 +(i +n), +0 = 0.4 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.100 +0.105 +(i +n), +0 = 0.6 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.02 +(i +n), +0 = 0.6 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.10 +0.11 +(i +n), +0 = 0.8 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.02 +0.04 +(i +n), +0 = 0.8 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.10 +0.12 +(i +n), +0 = 1.0 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.000 +0.025 +0.050 +(i +n), +0 = 1.0 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.10 +0.12 +0.14 +(i +n), +0 = 1.2 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.05 +(i +n), +0 = 1.2 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.10 +0.15 +(i +n), +0 = 1.4 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.05 +0.10 +(i +n), +0 = 1.4 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.1 +0.2 +(i +n), +0 = 1.6 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.0 +0.1 +0.2 +(i +n), +0 = 1.6 +Converged +Perturbative +FIG. 5: The first order solution to ε(iω), ∆(iω) (red) and the self consistent solution (green) for the self energy in Matsubara +space. Note the offset by 0.1 in the y axis in the left column. We chose ϵk = 0.1, rd = 9, rN = 1, T = 1 +β = 0.2, λ = 1 and +measured all energies in units of √rN. +Appendix D: Higher-order corrections to electronic Green’s function +In this section, we show comparisons between the first order perturbative solution and the full self consistent solution +to the fermionic Green’s function. We define the corrected Green’s function to be G(iω, k) = iωZk(iω) − εk(iω)γz + +∆k(iω)γy. In practice, we find that Zk(iω) ≃ 1, so we focus on εk(iω) and ∆k(iω) in the following. +In Fig. 5, we show a comparison of the first order result for εk(iωn) and ∆k(iωn) after including the evaluation +of the Σ3 term of the self energy [last diagram in Fig. 2(b)] and the full self consistent solution to the self energy in +Matsubara space [obtained by summing up the diagrams in Fig. 2(a) corresponding to Σ3] at fixed k. We find that +for small values up to φ0 ∼ 0.6rN, the first order and self consistent solutions differ little. In first order, εk(iωn) does + +11 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.06 +0.08 +0.10 +(i +n), +0 = 0.0 +Self Consistent Solution Including +1 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.06 +0.08 +0.10 +(i +n), +0 = 0.2 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.06 +0.08 +0.10 +(i +n), +0 = 0.4 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.06 +0.08 +0.10 +(i +n), +0 = 0.6 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.06 +0.08 +0.10 +(i +n), +0 = 0.8 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.06 +0.08 +0.10 +(i +n), +0 = 1.0 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.050 +0.075 +0.100 +(i +n), +0 = 1.2 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.050 +0.075 +0.100 +(i +n), +0 = 1.4 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.050 +0.075 +0.100 +(i +n), +0 = 1.6 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.2 +0.4 +(i +n), +0 = 0.0 +Self Consistent Solution Including +2 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.2 +0.4 +(i +n), +0 = 0.2 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.2 +0.4 +(i +n), +0 = 0.4 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.2 +0.4 +(i +n), +0 = 0.6 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.2 +0.4 +(i +n), +0 = 0.8 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.2 +0.4 +(i +n), +0 = 1.0 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.2 +0.4 +(i +n), +0 = 1.2 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.2 +0.4 +(i +n), +0 = 1.4 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.2 +0.4 +(i +n), +0 = 1.6 +Converged +Perturbative +FIG. 6: The first order solution to ε(iω) (red) and the self consistent solution (green) after including the effects of Σ1 (left +column) and Σ2 (right column) separately. Same parameters as in Fig. 5. +not get renormalized since Σ3 acquires a γz term only if the Green’s function has a γy term. Such a γy term does +not exist in the normal state about which we perform perturbation theory. As φ0 increases, we find that the self +consistent solution is lower in magnitude that the first order solution. +In Fig. 6, we show the corrections in ε(iωn) after including the effects of Σ1 (left column) and Σ2 (right column). +As expected and argued in the main text, we find that Σ1 and Σ2 have qualitatively the opposite effects on the +renormalization of ε(iωn). In both the cases, we find that the magnitude of the self consistent solution is higher than +the perturbative corrections. However, since the fermionic Matsubara frequencies do not contain 0, we cannot directly +say what this implies for the solution on the real axis. The magnitude of φ0 has little effect on the solution since the +effect of spin and triplet fluctuations are controlled by gN and gd, respectively, which we keep constant. +In Fig. 7, we plot the corrections in ε(iωn) and ∆(iωn) after including the effects of all the self energies Σ = +Σ1 + Σ2 + Σ3. We find that the inclusion Σ1 and Σ2 together reduces the difference between the self consistent and +perturbative solution (refer to the plot near φ0 ∼ 0). As we increase φ0, the difference between the self consistent and +perturbative solution increases due to the effect of Σ3 which is controlled by φ0. + +12 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.15 +0.20 +(i +n), +0 = 0.0 +Self Consistent Solution Including +1 + +2 + +3 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.05 +0.00 +0.05 +(i +n), +0 = 0.0 +Self Consistent Solution Including +1 + +2 + +3 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.15 +0.20 +(i +n), +0 = 0.2 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.000 +0.005 +0.010 +(i +n), +0 = 0.2 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.15 +0.20 +(i +n), +0 = 0.4 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.01 +0.02 +(i +n), +0 = 0.4 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.15 +0.20 +(i +n), +0 = 0.6 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.02 +0.04 +(i +n), +0 = 0.6 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.15 +0.20 +(i +n), +0 = 0.8 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.000 +0.025 +0.050 +(i +n), +0 = 0.8 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.15 +0.20 +(i +n), +0 = 1.0 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.05 +(i +n), +0 = 1.0 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.15 +0.20 +(i +n), +0 = 1.2 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.05 +(i +n), +0 = 1.2 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.15 +0.20 +0.25 +(i +n), +0 = 1.4 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.00 +0.05 +0.10 +(i +n), +0 = 1.4 +Converged +Perturbative +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.15 +0.20 +0.25 +(i +n), +0 = 1.6 +-17 +-11 +-5 +0 +5 +11 +17 +n( +n = n/ ) +0.0 +0.1 +(i +n), +0 = 1.6 +Converged +Perturbative +FIG. 7: The first order solution to ε(iω), ∆(iω) (red) and the self consistent solution (green) after including the effects of all +the terms of the self energy Σ1 + Σ2 + Σ3. Same parameters as in Fig. 5. +Taken together, we see that the inclusion of second- and higher-order diagrams that contribute in the large-N +limit defined in the main text yields qualitatively similar behavior on the imaginary axis compared to the first-order +diagrams. We therefore expect that the qualitative picture that S1 renormalizes the DOS close to the Fermi level on +top of which S2 reduces the low-energy spectral weight still applies. Since the impact of S2 is controlled by small +φ0 and good quantitative agreement is found for φ0 up to 0.6rN, we expect that Fig. 3(c) would look similar when +higher-order corrections were included. + +13 +Appendix E: Meissner Effect From ODLRO +The consequences of ODLRO defined in terms of four-fermion or two-boson correlators are well-known [57–60]. As +a result of spin-rotation symmetry, we cannot capture ODLRO using a correlator of only two bosons. Instead, we +have to study the four-boson density matrix +ρ(x1, x2, x′ +1, x′ +2) = ⟨N(x1) · d∗(x2)N(x′ +1) · d(x′ +2)⟩. +(E1) +Although the derivation is in close analogy to the two-boson or four-fermion case, we here show explicitly how the +Meissner effect follows from +ρ(x1, x2, x′ +1, x′ +2) → φ∗ +0(x1, x2)φ0(x′ +1, x′ +2) ̸= 0, +|xj − x′ +j| → ∞. +(E2) +Let us consider the system to be in the presence of a spatially uniform orbital magnetic field of strength B = B0ˆz in +the out of plane direction. Note that an in-plane orbital magnetic field does not couple to the bosons as the spatial +motion is constrained to the two-dimensional plane of the system. The corresponding vector potential is given by +A(x) = 1 +2B × x, with x = (x, y, 0). Under an in-plane translation by a, the vector potential transforms as +A(x) → A(x − a) = A(x) − 1 +2B × a +(E3) += A(x) − 1 +2∇ [a · (x × B)] +(E4) += A(x) + ∇χa(x), +(E5) +where χa(x) = − 1 +2a·(x×B). Note that the triplet pairing field d is a charge-2e bosonic field, while the magnetization +field N is neutral. Therefore, under simultaneous gauge transformation and displacement by a in the presence of a +magnetic field, the fields transform as +d(x) → ei 2e +ℏc χa(x)d(x − a), +(E6) +N(x) → N(x − a), +(E7) +A(x) → A(x). +(E8) +As a result of gauge covariance and translational symmetry, the four-body density matrix obeys +ρ(x1, x2, x′ +1, x′ +2) = ei 2e +ℏc(χa(x′ +2)−χa(x2))ρ(x1 − a, x2 − a, x′ +1 − a, x′ +2 − a). +(E9) +Now suppose the system has ODLRO, i.e., Eq. (E2) holds. In combination with Eq. (E9), this implies +φ∗ +0(x1, x2)φ0(x′ +1, x′ +2) = ei 2e +ℏc(χa(x′ +2)−χa(x2))φ∗ +0(x1 − a, x2 − a)φ0(x′ +1 − a, x′ +2 − a) +(E10) +=⇒ φ0(x1, x2) = faei 2e +ℏc χa(x2)φ0(x1 − a, x2 − a), +(E11) +where fa is a position-independent phase factor. Now suppose we perform two different translations by a and b. We +can perform a first and then b. Alternatively, we can do b first and then a. They respectively give us +φ0(x1, x2) = fbfaei 2e +ℏc χa(x2)ei 2e +ℏc χb(x2−a)φ0(x1 − a, x2 − a), +(E12) +φ0(x1, x2) = fbfaei 2e +ℏc χb(x2)ei 2e +ℏc χa(x2−b)φ0(x1 − a, x2 − a). +(E13) +This is only consistent if +ei 2e +ℏc (χb(x2)+χa(x2−b)−χa(x2)−χb(x2−a)) = 1. +(E14) +We can evaluate χb(x2) + χa(x2 − b) − χa(x2) − χb(x2 − a) = B · (a × b). Thus, the condition for equality of phases +becomes +2e +ℏcB · (a × b) = 2πn, +(E15) +for some integer n. The only solution for arbitrary a, b is thus B = 0. + +14 +Appendix F: Demonstration of Off Diagonal Long Range Order +In this section, we calculate the ODLRO wavefunctions for both the bosons and fermions. The idea is to calculate +the 4−body correlator ⟨N(x′ +1) · d(x′ +2)∗N(x1) · d(x2)⟩ for the bosons and ⟨c† +τ ′ +1s′ +1(x′ +1)c† +τ ′ +2s′ +2(x′ +2)cτ1,s1(x1)cτ2,s2(x2)⟩ for +the fermions. +Due to the U(1) symmetry breaking mediated by N · d attaining a finite expectation value (and +correspondingly c† +τc† +−τ for the fermions), the ODLRO factorizes into a product of functions of x1 − x2 and x′ +1 − x′ +2 +in the limit x − x′ → ∞, where x = x1+x2 +2 +and x′ = x′ +1+x′ +2 +2 +, giving rise to ODLRO. These wavefunctions decay +as a function of their respective relative coordinates x1 − x2 and x′ +1 − x′ +2. We now calculate these “macroscopic +wavefunctions” explicitly for the bosonic and fermionic cases. +1. +Bosonic ODLRO +The bosonic ODLRO is given by ⟨N(x′ +1)·d(x′ +2)∗N(x1)·d(x2)⟩ ≃ ⟨N(x′ +1)·d(x′ +2)∗⟩⟨N(x1)·d(x2)⟩ as x−x′ → ∞. +All the correlators are evaluated at time t = 0. As discussed in the main text, to demonstrate ODLRO, it is sufficient +to evaluate these correlators to first non-trivial order in the coupling constants. For bosonic ODLRO it is in fact +sufficient to focus on zeroth order, i.e., neglecting the coupling to the fermions. Using the translation invariance of +the system (and summing over the Matsubara frequencies iΩ since we are evaluating the correlator at time t = 0), +we then have +ψB(x) = ⟨N(x) · d(x = 0)⟩ = +� +q +T +� +iΩ +eiq·x⟨N −q · dq⟩ +(F1) += +� +q +T +� +iΩ +eiq·r +φ0 +[(iΩ)2 − E2 ++(q)][(iΩ)2 − E2 +−(q)] +(F2) += +� +q +eiq·x +φ0 +2E+(q)E−(q)(E+(q) + E−(q)) +(F3) +≃ +� +q +eiq·x +φ0 +a + bq2 +(F4) += φ0 +b +� +q +eiq·√ a +b x +1 +1 + q2 = 2πφ0K0 +��a +b |x| +� +/b +(F5) += 2πφ0K0 (|x|/ξ) /b, +(F6) +where K0 is the zeroth modified Bessel function of second kind. In the third line, we evaluated the Matsubara sum at +T = 0, and in the fourth line we series expanded 2E+(q)E−(q) (E+(q) + E−(q)) about q = 0 up to quadratic order. +The length scale ξ = +� +b +a is determined by rµ, vµ. In the limit of |vN − vd| ≪ vN + vd, we get +ξ = 1 +2 +� +� +� +�(v2 +N + v2 +d) +�� +rNrd − φ2 +0 + rN + rd +� +rNrd − φ2 +0 +. +(F7) +In Fig. 4(b), we plot the numerical ODLRO wavefunction ψB(x) with the full functional dependence on q in +Eq. (F3) included, and compare it with the asymptotic analytical form in Eq. (F6). We find good agreement between +the numerical and analytical results. +2. +Fermionic ODLRO +Similarly, +we +can +find +the +fermionic +ODLRO, +which +in +real +space +is +generically +written +as +⟨c† +τ ′ +1s′ +1(x′ +1)c† +τ ′ +2s′ +2(x′ +2)cτ1,s1(x1)cτ2,s2(x2)⟩ ∼ ⟨c† +τ ′ +1s′ +1(x′ +1)c† +τ ′ +2s′ +2(x′ +2)⟩⟨cτ1,s1(x1)cτ2,s2(x2)⟩ in the limit xj − x′ +j +→ ∞. +Here, τ, s are the valley and spin indices respectively. +To demonstrate ODLRO, we thus have to evaluate the +2−fermion correlators, which in momentum space becomes +(Ψ∗ +F(x))s1,s2 = ⟨c† +τ1,s1(x, t = 0)c† +τ2,s2(x = 0, t = 0)⟩ = +� +k +eik·x⟨c† +k,τ1,s1c† +−k,τ2,s2⟩. +(F8) + +15 +Since the superconducting pairing takes place only between electrons between opposite valleys, we will have only +τ2 = −τ1 giving non-zero correlators. Without loss of generality we chose τ1 = +, τ2 = −. Up to first order in φ0, we +have +⟨c† +k,+,s1c† +−k,−,s2⟩ = ⟨c† +k,+,s1c† +−k,−,s2 +� +− +� +q +1 +2 +φ0λ2 +Mq +Sq · D† +q +� +⟩0, +(F9) +where ⟨...⟩ is the average with respect to the interacting and ⟨...⟩0 with respect to the non-interacting ground state. +We define G(k) = δss′δττ ′GV,k = δss′δττ′ +iωn−ϵk = −⟨cs,τc† +s′,τ ′⟩ to be the Green’s function in the fermionic basis (assuming +ϵk = ϵ−k). Equation (F9) can then be evaluated as, +−φ0λ2 +2 +� +q +1 +Mq +⟨c† +k,+,s1c† +−k,−,s2 +� +Sq · D† +q +� +⟩0 +(F10) += −φ0λ2 +2 +� +q +1 +Mq +⟨c† +k+,s1c† +−k−,s2 +� +� +� +k1,k2,p1=±,p2=± +p1p2 +� +c† +k1+q,p1sck1,p1 +� +· (ck2+q,p2isysc−k2,−p2) +� +�⟩0 +(F11) += −2φ0λ2 +2 +� +q +1 +Mq +⟨c† +k+,s1c† +−k−,s2 +� +�� +k1,p +− (c−k1,−pisys(−GV,k1+q,p)sck1,p) +� +�⟩0 +(F12) += −6φ0λ2 +2 +� +q +1 +Mq +⟨c† +k+,s1c† +−k−,s2 +� +�� +k1,p +(c−k1,−pisyGV,k1+q,pck1,p) +� +�⟩0 +(F13) += −6φ0λ2 +2 +� +q +1 +Mq +⟨c† +k+,s1c† +−k−,s2 +�� +k1 +(c−k1,−isyGV,k1+q,+ck1,+ + c−k1,+isyGV,k1+q,−ck1,−) +� +⟩0 +(F14) += −6φ0λ2 +2 +� +q +1 +Mq +(−(−GV,−k,−)(isy)s2s1GV,k+q,+GV,k,+ + (−GV,k,+)(isy)s1s2GV,−k+q,−GV,−k,−) +(F15) += −6φ0λ2 +2 +� +q +1 +Mq +GV,kGV,−k (GV,−k+q + GV,k+q) (isy)s2s1. +(F16) +We continue by calculating the Matsubara sum over iΩn and over iωn [see Eq. (F8)], +T 2 +� +iωn,iΩn +1 +(iΩ2n − E+(q)2)(iΩ2n − E−(q)2)GV,kGV,−k (GV,−k+q + GV,k+q) +(F17) += −T 2 � +iωn,iΩ +1 +(iωn)2 − ϵ2 +k +1 +((iΩn)2 − E+(q)2)((iΩn)2 − E−(q)2) +� +1 +iωn + iΩn − ϵ−k+q ++ +1 +−iωn + iΩn − ϵk+q +� +(F18) +=: X(ϵk, q). +(F19) +For simplicity, we here focus on the limit where the remaining sum over q in Eq. (F9) is determined by its q = 0 +component. With E± ≡ E±(q = 0) and vN, rN = 1, we have +ˆX(ϵ) ≡ X(ϵ, q → 0) +(F20) += nf(ϵ)2 +2ϵ +� +−2 +eβϵ +E2 ++E2 +− ++ +2 +� +E2 ++ − 4ϵ2� � +E2 +− − 4ϵ2� + +� +2ϵnB(E+) − E+nf(E+) +E+(E2 ++ − E2 +−)(E2 ++ − 4ϵ2)nf(E+)nB(2ϵ) + E+ ↔ E− +�� +, +(F21) +we can then finally write +Ψ∗ +F(x) = 3|φ0|λ2sy +� +1 +V +� +k +eik·x ˆX(ϵk) +� +, +(F22) += 3|φ0|λ2sy +2π +� ∞ +0 +dkkJ0(k · x) ˆX(ℏ2(k2 − k2 +F )/(2m)). +(F23) + +16 +In the second line, we assumed ϵk = ℏ2(k2 − k2 +F )/2m. Using this expression, we calculate the spatial profile of the +fermionic ODLRO wavefunction numerically for various values of ϵF ≡ ϵkF in Fig. 4(a). Unlike the case of the bosonic +ODLRO (which was exponentially decaying), the fermionic ODLRO has an oscillating component superimposed on +an exponentially decaying envelope. +Appendix G: Ginzburg-Landau theory +We here calculate the Landau-Ginzburg theory for the bosonic superfluid condensate parameter to leading (zeroth) +order in the fermion-boson coupling λ. To tis end, we assume that φ0 is now spatially and temporally varying. This +results in non-zero Fourier modes φq for q, iΩ ̸= 0. +In momentum space, the bosonic action is generalized according to +SB = +� +q +[χ−1 +N (q)N q · N −q + χ−1 +SC(q)d∗ +q · dq + (φ0dq · N −q + H.c.)] +(G1) += +� +q +� +N T +−q d† +q +� � +χ−1 +N (q) +φ0 +φ0 +χ−1 +d (q) +� � +N q +dq +� +(G2) +→ +� +q,k +� +N T +−q−q2 d† +q+q2 +� � +χ−1 +N (q)δq2=0 +φq2 +φ∗ +−q2 +χ−1 +d (q)δq2=0 +� � +N q +dq +� +(G3) +So after integrating out d and N, the effective action for φ reads as +Seff = 1 +2Tr ln G−1[φ], +(G4) +where +G−1[φ](q + q1, q) = G−1 +0 (q)δq1,0 + Γq+q1,q +(G5) +G−1 +0 += +� +χ−1 +N (q) +0 +0 +χ−1 +d (q) +� +(G6) +Γq+q1,q = +� +0 +φq1 +φ∗ +−q1 +0 +� +. +(G7) +To derive the Ginzburg-Landau theory for φ, we expand Tr ln G−1 upto second order in Γ, which is equivalent to +second order in φ. This gives us +SGL = Tr ln(G−1 +0 ++ Γ) ≃ TrG−1 +0 ++ TrG0Γ − 1 +2TrG0ΓG0Γ +(G8) +Because of the diagonal structure of G0, and the off diagonal structure of Γ, the linear term TrG0Γ is 0. The quadratic +term becomes +� +q′,q +TrG0(q′ + q)Γ(q′ + q, q′)G0(q′)Γ(q′, q′ + q) = +� +q′,q +Tr +� +0 +χN(q′ + q)φq +χd(q′ + q)φ∗ +−q +0 +� � +0 +χN(q′)φ−q +χd(q′)φ∗ +q +0 +� +(G9) += +� +q′,q +χN(q′ + q)χd(q′)φqφ∗ +q + χN(q′)χd(q′ + q)φ−qφ∗ +−q +(G10) += +� +q′,q +(χN(q′ + q)χd(q′) + χN(q′)χd(q′ − q)) φqφ∗ +q +(G11) += +� +q′,q +(χN(q′ + q)χd(q′) + χN(q′ + q)χd(q′)) φqφ∗ +q +(G12) += 2 +� +q′,q +χN(q′ + q)χd(q′)φqφ∗ +q +(G13) + +17 +We need to evaluate +� +q′ +χN(q′ + q)χd(q′) = +� +q′ T +� +iΩ′∈Bosonic +� +1 +((iΩ′ + iΩ)2 − rN − v2 +N(q′ + q)2)((iΩ′)2 − rd − v2 +dq′2) +� +(G14) += −1 +2 +� +q′ +� +1 +� +rN + v2 +N(q′ + q/2)2 + +1 +� +rd + v2 +d(q′ − q/2)2 +� � +� +� +1 +iΩ2 − +�� +rN + v2 +N(q′ + q/2)2 + +� +rd + v2 +d(q′ − q/2)2 +�2 +� +� +� . +(G15) +By expanding the above expression up to second order in iΩ, q, we find the effective action for the φ field to be +T +� +iΩ,q +� +rφ − ρ(iΩ)2 + v2q2� +|φ(q,iΩ)|2 +(G16) +where the coefficients are given by +rφ = − +� +q′ +π +√gd√gN +�√gd + √gN +� +(G17) +ρ = +� +q′ +π +√gd√gN +�√gd + √gN +�3 +(G18) +v2 = +� +q′ +π +� +4q′2(√gd + √gN) +� +v2 +d +g3/2 +d +− +v2 +N +g3/2 +N +� � +v2 +N +√gN − +v2 +d +√gd +� +− +�√gd + √gN +�2 +� +v2 +d(3v2 +dq′2−2gd) +g5/2 +d ++ +v2 +N(3v2 +Nq′2−2gN) +g5/2 +N +�� +16 +�√gd + √gN +�4 +(G19) +− +2π +� +1 +√gd + +1 +√gN +� � +3q′2(√gNv2 +d−√gdv2 +N) +2 +gdgN +− +�√gd + √gN +� � +v2 +d(2gd−q′2v2 +d) +g3/2 +d ++ +v2 +N(2gN−q′2v2 +N) +g3/2 +N +�� +16 +�√gd + √gN +�4 +(G20) +with gµ = rµ + v2 +µq′2. We numerically calculate the quantities rφ, ρ, v2 and plot it in Fig. 4(c,d) of the main text. +Appendix H: Self-consistent equations in special limits +In this appendix, we complement the previous analysis by studying two simple limits of the model for phase +(B)—mean-field theory and the limit of zero energy-momentum transfer of the bosons. +This allows us to study +possible non-perturbative solutions systematically. In both cases, we find that the soft gap behavior obtained within +perturbation theory is also found in these descriptions as long as T is large enough/the coupling constants, λ or φ0, +are small enough. +1. +Mean-field Theory +In this section, we consider the effective interaction contributed by the S2 part of the action between the electrons +at time t = 0, in the limit where we replace the q integral with the corresponding value of the integrand at q = 0, +and then perform a mean-field decomposition of the interaction. Defining the Bogoliubov-de Gennes basis as before, +ξk = +� +ck,+ isyc† +−k,− +�T +, with Pauli matrix γi acting on it, and ˜φ0 = φ0λ2rN/v2 +N the corresponding interaction +potential is given by +V = −1 +2 +1 +χ−1 +d χ−1 +N − |φ0|2 +� +˜φ0Sq=0 · D† +q=0 + ˜φ∗ +0Dq=0 · S−q=0 +� +|q=0 +(H1) += −1 +2 +1 +rNrd − |φ0|2 +� +k1,k2 +� +−˜φ0 +� +c† +k1sτzck1 +� +· (ck2sisyτyc−k2) + h.c +� +(H2) += − +1 +rNrd − |φ0|2 +� +k1,k2 +� +˜φ0 +� +ξ† +k1sγzξk1 +� +· +� +ξ† +k2siγ−ξk2 +� ++ h.c +� +, +(H3) + +18 +while the free Hamiltonian is given by +H0 = +� +k +ξ† +kϵkγzξk. +(H4) +We consider only the effective Hamiltonian at time t = 0, which is why there are no Matsuabra indices. +We perform a Hartree-Fock decomposition of V , which gives us +V = +1 +rNrd − |φ0|2 +� +k1,k2 +� +˜φ0 +� +ξ† +k1sγzξk1 +� +· +� +ξ† +k2siγ−ξk2 +� ++ h.c +� +(H5) +→ c +2 +� +k +ξ† +k (γyCkγz + γzCkγy) ξk, +(H6) +where Ck = −⟨ξkξ† +k⟩, c = 6 +˜φ0 +rNrd−φ2 +0 , choosing a gauge with real φ0; further take φ0 to be positive such that c > 0. +Note that this correlator is related to the Green’s function G by Ck = T � +iωn G(k). Note that all the Hartree terms +vanish since we do not allow for spontaneous breaking of spin-rotation invariance (recall we study finite T in 2D). +The effective 2−particle Hamiltonian is given by +H = +� +k +ξ† +k +� +ϵkγz + c +2γyCkγz + c +2γzCkγy +� +ξk +(H7) += +� +k +ξ† +k +� +˜ϵkγz + ˜∆kγy +� +ξk +(H8) +where ˜ϵk, ˜∆k are the self consistent band structure and gap. Making connection with the diagrammatic self consistency +relationship to be discussed below, we can foresee that the resulting self consistent equation we get will be the same +as (H18) but with ˜ϵ, ˜∆ replaced with the corresponding iωn averaged value, and the whole equation itself will be iωn +averaged. +The correlators in terms of ˜ϵ, ˜∆ are given by +Ck = T +� +iωn +1 +iωn − +� +˜ϵkγz + ˜∆kγy +� = nf(Ek) − nf(−Ek) +2Ek +� +˜ϵkγz + ˜∆kγy +� +, +(H9) +where Ek = +� +˜ϵ2 +k + ˜∆2 +k > 0. Thus, using (H7), the self consistency equations become +˜ϵk = ϵk + c ˜∆k +nf(Ek) − nf(−Ek) +2Ek +(H10) +˜∆k = c˜ϵk +nf(Ek) − nf(−Ek) +2Ek +. +(H11) +Let us define βk = c nf (−Ek)−nf (Ek) +2Ek += c +tanh +� Ek +2T +� +2Ek +and first assume βk < 1, which always holds as long as T > c/4. +The self consistency equations can then be rearranged as +˜ϵk = +1 +1 − β2 +k +ϵk +(H12a) +˜∆k = +−βk +1 − β2 +k +ϵk. +(H12b) +Using this, we find Ek = +√ +1+β2 +k +1−β2 +k ϵk. Note, however, that βk also depends on Ek and, thus, this relation should be +thought of as a self consistency equation, to be solved for βk or Ek. +Equations (H12) allow to derive asymptotic relations. In the limit ϵk → 0, we then have Ek → 0 and βk → +c +4T , +ensuring the self-consistent solutions are well controlled in the ϵk → 0 regime that we are interested in. Near ϵk = 0 +and for large T ≫ c (βk ≪ 1), the renormalized spectrum is given by Ek = +√ +1+β2 +k +1−β2 +k ϵk ≃ +� +1 + 3β2 +kϵk ≃ +� +1 + +3c2 +16T 2 ϵk. +The suppression of DOS is now given by +ρF (φ0) +ρF (φ0 = 0) = +1 +√ +1 + α′2 , +α′ = +3 +√ +3φ0λ2rN +2v2 +NT(rdrN − φ2 +0), + +19 +-0.2 +-0.1 +0.1 +0.2 +-0.4 +-0.2 +0.2 +0.4 +-0.2 +-0.1 +0.1 +0.2 +-0.4 +-0.2 +0.2 +0.4 +-0.2 +-0.1 +0.1 +0.2 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +FIG. 8: The self consistent solution for ˜ϵk, ˜∆k and Ek as a function of ϵk for various temperatures. At T/c = 1/4, the +self-consistent solutions become non-analytic having an infinite slope at ϵk = 0, and a gap opens up as the temperature +decreases. There is a discontinuity in ˜ϵk, ˜∆k at ϵk = 0, where the gap value has different signs for ϵk → 0−, 0+. +which is of the same form as Eq. (4), found through the perturbative calculation presented in the main text and +derived in Sec. C. +When T/c = 1/4, we have β2 +k = 1 for ϵk → 0, and Eq. (H12) are not valid. At this point, the self consistent +solutions open up a gap in Ek when ϵk = 0. This gap follows by solving the equation β2 +k = 1. When ϵk = 0 and +βk = 1, we also have ˜ϵk = − ˜∆k [see Eq. (H11)] which gives Ek = +√ +2˜ϵk. For T/c approaching 1/4 from below, we +find that βk ≃ +c +4T +� +1 − 1 +12 +E2 +k +T 2 +� +. Thus the condition that β2 +k = 1 gives us Ek = +√ +12T +� +1 − 4T +c . +To summarize, for T > c/4, self consistent energy and gap (˜ϵ, ˜∆) are proportional to ϵ. As T approaches c/4 from +above, the slope of proportionality approaches ∞ at ϵ = 0, and becomes non-analytic at T = c/4. Going below +T = c/4, this non-analyticity at ϵ = 0 turns into a discontinuity at ϵ = 0, with the self consistent solutions developing +a finite gap. The value of this gap at T = 0 is given as | ˜∆| = |˜ϵ| = +|c| +2 +√ +2. Figure 8 illustrates the behavior obtained by +numerical solution of the self-consistency equations. +2. +Zero energy-momentum transfer +In this section, we consider the limit where the bosonic fields N, d do not transfer any momentum or Matsubara +frequency in the interaction (q = 0 in Sc). Additionally, we consider only the effect of S2 on the self energy to study +the effect of the anomalous contribution. In this limit, we would like to analyze the self consistent solution of the +Green’s function up to all orders in λ within the large-N theory of the main text. The ansatz of the full Green’s +function is given by G−1 = iωn − ˜ϵkγz − ˜∆kγy, since Σ3 renormalizes only the anomalous term ˜∆k and the spectrum +˜ϵk. We have +G = iωn + ˜ϵkγz + ˜∆kγy +(iωn)2 − ˜ϵ2 +k − ˜∆2 +k +. +(H13) +Thus the self-consistent analogue of Σ3 in Eq. (3) becomes (where we have replaced the integration over q by the +q = 0 value of the integrand, and ˜φ0 = φ0λ2rN/v2 +N) +Σ3 = 6T +˜φ0 +rNrd − φ2 +0 +˜ϵkγy + ˜∆kγz +(iωn)2 − ˜ϵ2 +k − ˜∆2 +k +. +(H14) +From the self-energy equation we get +G−1 = G−1 +0 +− Σ3 +(H15) +iωn − ˜ϵkγz − ˜∆kγy = iωn − ϵkγz − 6T +˜φ0 +rNrd − φ2 +0 +˜ϵkγy + ˜∆kγz +(iωn)2 − ˜ϵ2 +k − ˜∆2 +k +(H16) +˜ϵk = ϵk + Tc +˜∆k +(iωn)2 − ˜ϵ2 +k − ˜∆2 +k +(H17) +˜∆k = Tc +˜ϵk +(iωn)2 − ˜ϵ2 +k − ˜∆2 +k +, +(H18) + +20 +where c = +6 ˜φ0 +rNrd−φ2 +0 . Right at the Fermi surface, ϵk = 0, the self consistency equations reduce to +˜ϵk = Tc +˜∆k +(iωn)2 − ˜ϵ2 +k − ˜∆2 +k +(H19) +˜∆k = Tc +˜ϵk +(iωn)2 − ˜ϵ2 +k − ˜∆2 +k +(H20) +=⇒ ˜ϵk = T 2c2 +˜ϵk +(ω2n + ˜ϵ2 +k + ˜∆2 +k)2 +(H21) +There are two possible solutions to Eqs. (H19) and (H20). The first is ˜ϵk = ˜∆k = 0; this is exactly what we find +within perturbation theory. For a solution with ˜ϵk ̸= 0 to exist, it must hold (assuming ˜ϵk, ˜∆k ∈ R as expected in the +gauge that we use) +1 = T 2 +c2 +(ω2n + ˜ϵ2 +k + ˜∆2 +k)2 < T 2 +c2 +π4T 4 +(H22) +Thus, a non-zero solution only exists if T < c/π2 ∼ c/9. As compared to Hartree-Fock, the critical temperature for a +non-perturbative solution is lower. + diff --git a/Y9AzT4oBgHgl3EQfY_xr/content/tmp_files/load_file.txt b/Y9AzT4oBgHgl3EQfY_xr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb44e8c618379d61cd12bb5c25458b1efae89f34 --- /dev/null +++ b/Y9AzT4oBgHgl3EQfY_xr/content/tmp_files/load_file.txt @@ -0,0 +1,1371 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf,len=1370 +page_content='Vestigial singlet pairing in a fluctuating magnetic triplet superconductor: Applications to graphene moiré systems Prathyush P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Poduval1 and Mathias S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Scheurer2 1Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD 20742, USA 2Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria Motivated by the phenomenology of graphene moiré superlattices, we study a 2D model with strong tendencies towards both magnetism and triplet superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Individually, their respec- tive order parameters, N and d, cannot order at finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Nonetheless, the model exhibits a variety of vestigial phases, including charge-4e superconductivity and broken time-reversal sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Our main focus is on a phase characterized by finite d·N, which has the same symmetries as the BCS state, a Meissner effect, and metastable supercurrents, yet rather different spectral proper- ties: most notably, the suppression of the electronic density of states at the Fermi can resemble that of either a fully gapped or nodal superconductor, depending on parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This could provide a possible explanation for recent tunneling experiments in graphene moiré systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Strongly correlated systems often exhibit complex phase diagrams with multiple phases, characterized by long-range or quasi-long-range order (QLRO) of differ- ent order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Aside from phase competition as a possible origin, a rich set of phases might also be un- derstood as different manifestations of an underlying pri- mary order—a concept often referred to as “intertwined orders” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For instance, thermal or quantum fluctu- ations can disorder a primary order parameter, while higher-order composite order parameters can still sur- vive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' An example of such a “vestigial phase” [2, 3], is the charge-4e superconducting state that emerges when a charge-2e pair density wave order parameter, ∆Q, itself vanishes, yet ∆Q∆−Q does not [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' this and other forms of charge-4e superconductivity have attracted a lot of attention [5–18], in particular, as a result of recent ex- periments [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Another exciting recent development is the emergence of twisted graphene moiré superlattices as versatile play- grounds for strongly correlated physics [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' These systems display a variety of different phases such as nematic [23–25] and density-wave order [26–28], differ- ent forms of magnetism [29–33], and, possibly uncon- ventional [34, 35], superconductivity [36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' magnetism and superconductivity appear in the same density range [34, 35, 37–41] and recent experiments [33, 42] demon- strate that they can coexist microscopically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Motivated by these observations, we here study the case of two pri- mary order parameters: a fully gapped spin-triplet su- perconductor (d) and, in line with the conclusions of [41, 43], magnetic order (N) with antiparallel spins in the two valleys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' At finite temperature, T > 0, it must hold ⟨d⟩ = ⟨N⟩ = 0, in two-dimensions (2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' However, there are several different vestigial phases, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(a), char- acterized by the composite order parameters φdd = d · d, φdN = d·N, and φddN = i(d†×d)·N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' These include not only a charge-4e superconductor [44, 45], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(b), but also a charge-2e state, which has the same symme- tries as and is, hence, adiabatically connected to the BCS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' However, it should primarily be thought as a con- densate of three electrons and a hole, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(c), or, more formally, QLRO of φdN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We develop a theory for this state and study its spectral properties at finite T, which are rather different from those of the BCS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Depending on T and φdN, we obtain a low-energy sup- pression of the density of states (DOS) similar to a fully gaped or nodal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This could provide an alternative explanation [43, 46, 47] to the tunneling data of [34, 35], which does not require any momentum dependence in the superconducting order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='—We consider a 2D model exhibiting both triplet superconductivity and magnetism, with three- component order parameter fields d (complex) and N (real), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Denoting the electronic field opera- tors of spin s =↑, ↓ (Pauli matrices s) and in valley τ = ± (Pauli matrices τ) by ck,s,τ, where k = (iωn, k) comprises (a) (b) b2/b1 c1/b1 (A) (C) (B1) (B2) Φdd≠0 ΦddN≠0 Φdd,ΦdN≠0 Φdd,ΦdN≠0 C2z, Θ, SO(2) C2z, Θ, no SC C2z, Θ, SO(3) C2z, Θ, 4e-SC C2z, Θ, SO(2) C2z, Θ, 2e-SC C2z, Θ, 2e-SC C2z, Θ, SO(3) Φdd 0 (A) e– e– e– e– S=1 S=1 S=0 charge-4e (B) e– e– h+ e– S=1 S=1 S=0 charge-2e (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1: (a) Mean-field phase diagram for rd = rN, b3 = b1, c2 = 0, where we indicate the symmetries at T = 0 (blue), those of the resulting vestigial phases at T > 0 (red), and which composite order parameters are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Solid (dashed) orange lines are phase transitions at T = 0 and T > 0 (become a crossover at T > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (b,c) illustrate the finite-T pairing in phases (A) and (B) schematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='01344v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='supr-con] 3 Jan 2023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='42 Matsubara frequencies and 2D momentum, they couple as Sc = λ � k,q � c† k−qsN qτzck + (c† k−qsdqisyτyc† −k + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=') � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Note that N couples anti-ferromagnetically in the two valleys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' while the ferromagnetic case—with τ0 instead of τz in the first term of Sc above—can be studied similarly, we focus on antiferromagnetism not only for concreteness here but also because recent microwave experiments [41] and a systematic analysis [43] of multiple other exper- iments on graphene moiré systems favor this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The bare dynamics of d and N is governed by Sχ = � q � χ−1 N (q) N qN −q + χ−1 d (q) d† qdq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We take the susceptibilitites to be χµ(q) = χ0 µ/(rµ+Ω2 n+ v2 µq2), µ = N, d, where q = (iΩn, q) and Ωn are bosonic Matsubara frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The nature of the phase realized in the system depends crucially on the interactions be- tween the bosonic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Up to quartic order, the local terms allowed by the symmetries listed in Table I can be written as SV = � x V (d(x), N(x)) with V = b1(d†d)2 + b2|dd|2 + b3N 4 + c1|dN|2 + c2(d†d)N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Finally, the bare electronic action is given by Se = � k c† k,τ,s (−iωn + ϵτ·k) ck,τ,s, where we already used that the band structures in the two valleys are related by time-reversal Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Mean-field and possible phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='—To probe the possible phases, we start with a mean-field analysis with respect to d and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Absorbing the impact of the coupling to the electrons [48] by a redefinition of the parameters of V , we obtain the four distinct zero-temperature phases labeled (A), (B1,2), and (C) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(a), where we assumed that both ⟨d⟩ and ⟨N⟩ are non-zero and homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Using ˆe1,2,3 ∈ R3 to denote orthogonal unit vectors, we have N = N0ˆe1 and d = d0eiαˆe2 in phase (A), which breaks SO(3) completely, while Θ is preserved (in any gauge- invariant observable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' as for any phase with ⟨N⟩ ̸= 0, C2z is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In phase (B1), N and d are aligned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' we, thus, obtain a residual spin-rotation symmetry SO(2) along that direction and Θ is preserved too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Beyond a critical value of b2, an additional component with relative phase π/2 emerges in d, defining phase (B2) where N = N0ˆe1 and d = d0eiα(ˆe1 + iηˆe2), with 0 < η < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' this is a dis- tinct phase as η ̸= 0 breaks both the residual SO(2) spin symmetry and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Finally, phase (C) is characterized by N = N0ˆe1 and d = d0eiα(ˆe2 + iˆe3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Consequently, Θ is also broken but a residual SO(2) spin-symmetry remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Importantly, ⟨d⟩ , ⟨N⟩ ̸= 0 is only possible and, thus, our discussion of symmetries is only valid for T = 0 in TABLE I: Relevant symmetries g and their action on the field operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Here Rϕ is the orthogonal matrix obeying e−iϕ·sseiϕ·s = R(ϕ)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' All symmetries are linear except for Θ which is anti-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' g ck N d φdd φdN φddN U(1) eiϕck N e−2iϕd e−4iϕφdd e−2iϕφdN φddN SO(3) eiϕ·sck RϕN Rϕd φdd φdN φddN C2z τxc−k −N −d φdd φdN −φddN Θ isyτxc−k N −d∗ φ∗ dd −φ∗ dN −φddN 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To analyze the resulting vestigial phases at finite T, where SO(3) spin-rotation symmetry is preserved and ⟨d⟩ = ⟨N⟩ = 0, it is convenient to define the following composite order parameters φdd = d·d, φdN = d·N, and φddN = i(d† × d) · N, with symmetry properties listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Crucially, all of them transform trivially under SO(3) spin-rotations and, hence, can exhibit long-range (in case of the last one) or QLRO (in case of the former two) at finite T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We indicate this in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(a) for the different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This immediately tells us that, in spite of ⟨d⟩ = 0, phase (A) transitions for finite T into state where φdd has QLRO and, thus, constitutes a charge-4e superconductor (as φdN = 0), which does not break C2z or Θ (as φddN = 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' intuitively, one can think of this state as a condensate of four electrons forming a spin- singlet out of two triplets, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' At finite T, (B1) and (B2) will both preserve all normal-state symmetries and become the same phase, which we denote by (B) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' It is characterized by QLRO not only in φdd but also in φdN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' as the latter has charge 2e, it is a charge-2e superconductor and adiabatically connected to the conventional BCS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Nonetheless, in our current description, this state should rather be thought of as the condensation of three electrons and a hole, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(c), consisting of a pair of electrons in a triplet state forming a singlet with a spin-1 particle-hole excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In fact, we will see below that it exhibits spectral properties rather different from those of the BCS state at finite T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Finally, while phase (C) does not exhibit any vestigial pairing at T > 0, it will have long-range order in φddN and, as such, continues to break both C2z and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Theory for phase (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='—As c1 < 0 is found when the coefficients in V are computed by integrating out elec- trons [48], we next focus on phase (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To obtain an efficient description of this phase that properly captures the preserved SO(3) symmetry at finite temperature, we first decouple the four terms in V using four Hubbard- Stratonovich fields, ψd for d†d, ψN for N 2, φd for d · d, and φdN for d · N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We treat them on the saddle-point level, which becomes exact in the limit where the num- ber of components of d and N is taken to be infinitely 3 (a) (b) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 ω/√rN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 DOS (c) Free Σ1 Σ2 Σ1 + Σ2 17 11 5 0 5 11 17 n (iωn = πn/β) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='100 (d) ϵN ϵ1 ∆N ∆1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2: Diagrams contributing to the fermionic self energy Σ (a) in the matrix-large-N limit defined in the main text and (b) to first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (c) Impact of spin (Σ1) and triplet fluctuations (Σ2) on the constant DOS (blue) of a 2D band with finite bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (d) Comparing the first order solution (ϵ1, ∆1) and self consistent solution (ϵN, ∆N) for G = iω − ϵ(iω)γz + ∆(iω)γy for S2 only (both without momentum integration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We use ϵ/√rN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1, φ0/rN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' large [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The saddle point values of ψd and ψN will in general be non-zero, which we absorb into a redefinition of rd,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Then, the effective action for phase (B) becomes Seff = Sχ + Se + Sc + Sφ where Sφ = � q � φ0 dN dq · N −q + φ0 dd dq · d−q + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (1) While generically both saddle point values φ0 dN and φ0 dd are expected to be non-zero simultaneously in phase (B), we take φ0 dd → 0 and φ0 dN ≡ φ0 ̸= 0 for the following explicit calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Setting φ0 dd = 0 does not change any symmetries of the phase, allows for a more compact discussion of the results, and can formally be seen as the large b2 limit of the theory where φ0 dd is suppressed [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' More generally than the derivation of Seff via Hubbard-Stratonovich transformations, it can also be thought of as the simplest field theory capturing the key aspects of phase (B) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(a) at finite T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Electronic self energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='—To compute the spectral prop- erties of the electrons within this model, we employ a large-N technique similar to [50, 51]: we add extra indices to the electrons and bosons, ck,τ,s → ck,τ,s,a, dab → dab and similarly for N, where a, b = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' , N, which are contracted in all terms of Seff so as to ensure O(N) sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In the limit N → ∞, the electronic self-energy Σ is given by the “rainbow diagrams” [50, 51] shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In our case, however, Σ involves both normal and anomalous contributions as a result of the anoma- lous bosonic term ∝ φ0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To make this more explicit, we integrate out the bosons, yielding the effec- tive fermionic interactions Sint = S1 + S2 with S1 = − � q λ2 Mq �χ−1 d 4 Sq · S−q + χ−1 N Dq · D† q � , (2a) S2 = −1 2 � q λ2 Mq � φ0 Sq · D† q + φ∗ 0 Dq · S−q � , (2b) where Mq = χ−1 d χ−1 N − |φ0|2 and Sq = � k c† k+qsτzck, Dq = � k c† k+qsisyτyc† −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The two terms in S1 describe spin and superconducting triplet fluctuations, respec- tively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' their associated self-energy contributions are nor- mal in the sense that U(1) symmetry is preserved, with leading terms represented by the first two diagrams Σ1,2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Conversely, S2 breaks U(1) symmetry, when φ0 attains a mean-field value, and results in an anoma- lous contribution to the self-energy, with leading term given by the last diagram Σ3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To represent the diagrams algebraically, we shift to the Bogoliubov-de Gennes basis (cq,+, isyc† −q,−)T , with Pauli matrices γi acting on this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In this basis, the free Green’s function is G0(iω, ϵ) = iω−ϵγz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Up to first order in λ2, the spin-spin self energy term can be written as Σ1(k) = 3λ2 � q χ−1 d (q) 2Mq G0(iω+iΩ, ϵk+q), while the triplet- triplet term is Σ2(k) = 12λ2 � q χ−1 N (q) Mq G0(iω+iΩ, −ϵk+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' After performing a gauge transformation to make φ0 real, the anomalous term from the spin-triplet interaction is given by Σ3(k) = 3φ0 � q λ2 Mq {γy, γzG0(iω + iΩ, ϵk+q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (3) For concreteness and since spin fluctuations are believed to occur already at higher energies than superconducting fluctuations in graphene moiré systems [37, 38], we focus on rd > rN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' we will use rd/rN = 9, v2 d/v2 N = 8, χ0 N = χ0 d, and set χ0 µ = 1 by rescaling of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='—Figure 2(c) shows the effect of the normal contributions of the self energy Σ1,2 on the DOS of a 2D parabolic band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The effect of Σ1 is to push the peak of the free spectral function at energy ϵ away from ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This results in the opening of a gap (which can be soft depending on the parameter regime), very sim- ilar to fluctuating anti-ferromagnetism discussed in the cuprates [52–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Σ2 on the other hand has the oppo- site effect, where it pushes states towards ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This is because Σ1 and Σ2 have the exact same functional form with one key difference: ϵk+q of Σ1 is replaced by −ϵk+q in Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The effect of the total normal self energy Σ1 + Σ2 is to enhance the DOS in the vicinity of the Fermi level, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The anomalous contribution Σ3 does not interfere in these effects since it occurs in the γy channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The role of Σ1 + Σ2 can, thus, be intuitively thought of as providing a renormalized DOS in the nor- mal state on top of which the anomalous Σ3 opens up V= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Z1 : Z2 +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='c Sa Da D!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Dt q a4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 ω/√rN −2 −1 0 1 2 ϵ/√rN(×10−2) (a) Σ3 Free −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 ω/√rN −2 −1 0 1 2 ϵ/√rN(×10−1) (b) −10 −5 0 5 10 ω/√rN(×10−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 DOS (c) 3φ0/rN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 3: Spectral weight as a function of ω with (blue) and without (purple) Σ3 (a) close to ϵk = 0 and (b) including a larger energy range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' in both cases, we focus on the q = 0 contribution (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (c) The effect of all three self energy contributions Σ1 + Σ2 + Σ3 (including the momentum integration) on the DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For small φ0, there is suppression of the DOS at ω = 0 which resembles the V-shaped DOS of a nodal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For large φ0, the gap resembles a hard BCS gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We have checked [48] by numerically solving the self-consistency equation for the self-energy [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(a)] in the limit (of large vµ) where only the q = 0 term of the momentum sum contributes that higher-order corrections do not change our results qualitatively for small φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For instance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(d) shows the numerical solution for the Green’s function G = iω − ϵ(iω)γz + ∆(iω)γy in Matsub- ara space upon including the effect of S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' the difference to the first-order result is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To gain intuition for the impact of Σ3 on the DOS, we first focus again on the q = 0 term of the momentum sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In this limit, one can easily see [48] that Σ3 vanishes linearly in ϵk for small energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Since Σ3 is in the γy channel, the effect of any non-zero value is to generically open a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As a result of the linear be- havior, the states exactly at zero energy are unaffected, but slightly away from it, the states get pushed away to higher energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' this is clearly visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In contrast, for large energies, Σ3 is readily seen to tend to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The spectral function, thus, remains asymp- totically unaffected, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Taken together, we expect the DOS to be reduced (but not fully suppressed for small φ0) in an energy range around the Fermi level, exhibiting an enhancement with respect to its normal-state value at intermediate energies, and then approaching the normal-state limit at larger energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To demonstrate this explicitly beyond the simple q = 0 limit, we approximate ϵk+q ≃ ϵk+vF q∥+q2/(2m), where q∥ is the component of q along k, and numerically evalu- ate the momentum integrals to find the total self energy Σ = Σ1 +Σ2 +Σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Choosing vF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5vN, 2m = √rN/v2 N for concreteness, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 3(c) shows the resulting DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As expected, we see that there is a suppression of the DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 0 2 4 6 8 √rN vN |x| 0 2 4 ψB(x)/φ0 (b) Numerical Analytical 0 2 4 6 8 10 rd/rN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5 (c) −rφ v2 × 5 ρ × 5 0 5 10 15 � 2m β¯h2 |x| −1 0 1 ψF (x)(×10−3) (a) βϵF = 5 βϵF = 10 βϵF = 20 0 2 4 6 8 10 v2 d/v2 N 2 4 (d) −rφ v2 × 30 ρ × 30 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 4: (a) The fermionic and (b) the bosonic ODLRO “macroscopic wavefunction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The mass rφ [in units of r−1/2 N v−2 N ], superfluid density ρ [r−3/2 N v−2 N ], and velocity v2 [r−3/2 N ] of SGL in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (5) as a function of rd and v2 d are shown in (c) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' However, for small values of φ0, the resulting DOS has a V-shaped behavior, which is typically only seen in nodal states (with either nodal lines or points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Recall that the superconducting phase in our model is symmetry- equivalent to a conventional BCS state and that the triplet superconductor that arises at T = 0 in phase (B) will be fully gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For larger φ0, the gap at ω = 0 in- creases, and resembles a hard BCS gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The suppression of the DOS ρF at ω = 0 can be estimated analytically at finite temperature by again taking the limit (of large vµ) where the integration over q can be replaced by an evaluation at q = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' we find ρF (φ0) ρF (φ0 = 0) = 1 √ 1 + α2 , α = 3φ0λ2rN 2Tv2 N(rdrN − φ2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (4) Note that φ2 0 is bounded above by rdrN, at which point the bosonic fields would condense and continuous sym- metries would be broken, which cannot happen at finite T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As φ0 increases, α increases the suppression of the DOS, and near the instability point of φ2 0 = rdrN, there are no states near the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To complement this analysis, we have also studied the Hamiltonian associated with setting q = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (2b) within self-consistent Hartree-Fock, only allowing for spin-rotation invariant operators to condense [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For small α, one also finds only a partial suppression of the low-energy spectral weight, akin to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' including higher-order corrections leads to a hard gap for α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Electromagnetic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='—We will finally demon- strate that the superconducting phase characterized by φ0 ̸= 0 has the same electromagnetic phe- nomenology as BCS superconductors, despite the un- usual electronic spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To this end, we study off-diagonal long-range order (ODLRO) [55– 5 57] which implies the Meissner effect [58], flux quan- tization [59], Josephson effect and persistent cur- rents [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' First focusing on the electrons, we show that ⟨c† s1,+(x1)c† s2,−(x2)cs′ 2,−(x′ 2)cs′ 1,+(x′ 1)⟩ → n0(Ψ∗ F(x12))s1,s2(ΨF(x′ 12))s′ 1,s′ 2, with ΨF ̸= 0, as |xj − x′ j| → ∞ at finite x12 = x1 − x2 and x′ 12 = x′ 1 − x′ 2, to leading (first) order in φ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' as non-zero ΨF to lin- ear order in φ0 implies that it cannot vanish identi- cally for generic φ0, this is sufficient to show the pres- ence of ODLRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We find the “macroscopic wave func- tion” to be a singlet, ΨF(x) = isyψF(x), as expected since spin-rotation symmetry is preserved at finite T, with ψF(x) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Alternatively, one can demonstrate ODLRO to arbitrary order in φ0, by fo- cusing on the bosons: to zeroth order in λ, we find ⟨(d†(x1)N(x2))(d(x′ 1)N(x′ 2))⟩ → ψ∗ B(x12)ψB(x′ 12) as |xj − x′ j| → ∞, with ψB(x) plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 4(b) along with an analytic asymptotic form for large x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' in [48], we show that this leads to the same constraints as the con- ventional form of bosonic ODLRO [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Finally, the connection to the textbook theory of superconductivity can be made more explicit by deriving the analogue of the time-dependent Ginzburg-Landay theory: we rein- state fluctuations via φ0 → φ(x, τ) and integrate out all other degrees of freedom yielding SGL = � x,τ � ρ|Dτφ|2 + (rφ + |c1|−1)|φ|2 + v2 |Dφ|2� (5) to leading order in φ and gauge-covariant derivatives (Dτ, D)µ = ∂µ − i2eAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For demonstration purposes, we evaluated the coefficients in SGL to leading (zeroth) order in Sc and find ρ, vφ > 0 and rφ < 0 for low T [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 4(c,d)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' the state with QLRO in φ0 thus corre- sponds, as usual, to the Higgs phase, with Meissner effect and massive Higgs mode, but without Goldstone modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='—We have studied the finite-T vestigial phases, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(a), associated with two primary order parameters, d and N, describing a fully gapped triplet superconductor and spin magnetism, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' A cru- cial result is the DOS of phase (B1,2) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 3(c): varying φ0 changes the low-energy DOS from partial suppression, akin to that of a nodal superconducting state, to a hard gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As φ0 is expected to change with electron filling, this could explain the tunneling data in [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We fi- nally point out that the suppression of N would immedi- ately also suppress φ0 in our model and could, therefore, explain why superconductivity is connected to the reset behavior in trilayer graphene [34, 35, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We thank Rafael Fernandes, Victor Gurarie, Peter Orth, and Subir Sachdev for fruit- ful discussions on the project and Jakob Wessling for a re- lated collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' acknowledges funding by the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Fernandes and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Millis, “Nematicity as a Probe of Superconducting Pairing in Iron-Based Superconduc- tors,” Physical Review Letters 111, 127001 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' [62] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Kozii, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Isobe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Venderbos, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Fu, “Nematic superconductivity stabilized by density wave fluctuations: Possible application to twisted bilayer graphene,” Physical Review B 99, 144507 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Appendix A: Mean-field form of the bosonic interactions In the main text, we view the field theory defined by the action S = Se + Sχ + Sc + SV as an effective low- energy theory that arises when high-energy electronic degrees of freedom have already been integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Due to the symmetry and locality constraints, it only depends on a few parameters, rµ, vµ, b1,2,3, c1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1(a), in particular, (the sign of) the parameters c1 and b2 entering V crucially determine the phase of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We here provide an estimate for these parameters using mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To this end, we replace the bosonic fields by classical homogeneous and time-independent vectors, N q → δq,0N, dq → δq,0d, in Se + Sχ + Sc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' this yields SHE = � k c† k,τ,s (−iωn + ϵτ·k) ck,τ,s + λ � k � c† ks · Nτzck + (c† ks · d isyτyc† −k + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=') � + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=', (A1) which we now view as our full action, also containing the high-energy degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Integrating out the electronic degrees of freedom and expanding the resulting action in terms of N and d to quartic order, one obtains exactly the same terms as in V defined in the main text, as expected by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Moreover, one finds c1 = b2 = −b1/2 < 0, with b1 = 32 λ4T � ωn � d2k (2π)2 1 (ω2n + ϵ2 k)2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (A2) As stated in the main text, this places us into phase (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We note, however, that fluctuation corrections to mean field can modify the values of these coupling constants significantly [45, 61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For instance, ferromagnetic fluctuations 8 can change the sign of b2 to positive values [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Appendix B: Evaluation of the self-energies at leading order In this section, we show the evaluation of the self energies up to first order in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We first evaluate the anomalous part of the self energy, Σ3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(b), which is contributed by the anomalous term of the action given by S2 = −1 2 � q λ2 χ−1 d χ−1 N − |φ0|2 � φ0Sq · D† q + φ∗ 0Dq · S−q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (B1) In the following, we work in the � cq,+ isyc† −q,− �T Bogoliubov-de Gennes basis, with the Pauli matrices γi acting on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The free Green’s function then reads as G−1 0 (k) = iω − ϵkγz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Choosing φ0 to be real, we have Σ3 = 3 � q φ0λ2 Mq (γyG0,k+qγz + γzG0,k+qγy) = 6 � q φ0λ2 Mq ϵk+q (iω + iΩ)2 − ϵ2 k+q γy, (B2) where Mq = χ−1 N χ−1 d − φ2 0 = � − (iΩ)2 + rN + v2 Nq2� � − (iΩ)2 + rd + v2 dq2� − φ2 0 (B3) = ((iΩ)2 − E2 +)((iΩ)2 − E2 −), (B4) with E2 ± = gd+gN±√ (gd−gN)2+4φ2 0 2 , and gµ = rµ + v2 µq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Thus, Σ3 = 6φ0λ2 � q T � iΩ∈Bosonic 1 � (iΩ)2 − E+(q)2 � � (iΩ)2 − E−(q)2 � ϵk+q (iω + iΩ)2 − ϵ2 k+q γy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (B5) The Matsubara sum can be evaluated using f(iω, ϵ) =T � iΩ 1 ((iΩ)2 − E2 +)((iΩ)2 − E2 −) 1 iω + iΩ − ϵ (B6) = 1 2 1 E2 + − E2 − � 1 E+ (K(iω, ϵ, E+) − K(iω, ϵ, −E+)) − 1 E− (K(iω, ϵ, E−) − K(iω, ϵ, −E−)) � , (B7) K(iω, ϵ, E) = nf(ϵ) + nB(−E) E + ϵ − iω , (B8) where nf/B(ϵ) = 1 eβϵ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Thus we get, Σ3(k) = 3φ0λ2 � q (f(iω, ϵk+q) − f(iω, −ϵk+q)) γy, (B9) where we performed a partial fraction decomposition of 2ϵk+q (iω+iΩ)2−ϵ2 k+q = 1 iω+iΩ−ϵk+q − 1 iω+iΩ+ϵk+q to arrive at the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The normal part of the self energy, Σ1,2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(b), is contributed by the following term of the action S1 = − � q λ2 χ−1 d χ−1 N − |φ0|2 �χ−1 d 4 Sq · S−q + χ−1 N Dq · D† q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (B10) Defining γ± = 1 2 (γx ± iγy) , the corresponding contribution to the self energy is given by Σ1 + Σ2 = � q λ2 Mq � 6χ−1 d (q) 4 γzG0,k+qγz + 12χ−1 N (q) (γ+G0,k+qγ− + γ−G0,k+qγ+) � (B11) 9 = � q T � iΩ∈Bosonic λ2 Mq 1 (iω + iΩ)2 − ϵ2 k+q �2 3(gd − (iΩ)2) (iω + iΩ + ϵk+qγz) + 12(gN − (iΩ)2)(iω + iΩ − ϵk+qγz) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (B12) Note that γzG0γz = G0 = iω − ϵγz, while γ−G0γ+ + γ+G0γ− = iω + ϵγz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As a result, if we consider the self energies as function of iω and ϵk+q, we find that Σ1 ∼ λ2 � q 3χ−1 d (q) 2Mq G0(iω, ϵk+q) while Σ2 ∼ λ2 � q 12χ−1 N (q) Mq G0(iω, −ϵk+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This allows us to argue the effect of Σ2 pushing high energy states towards the vicinity of ω = 0, while Σ1 pushes states away from ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To perform the Matsubara sums, we define h(iω, ϵ, g) =T � iΩ −(iΩ)2 + g ((iΩ)2 − E2 +)((iΩ)2 − E2 −) 1 iω + iΩ − ϵ (B13) = 1 2 1 E2 + − E2 − �E2 + − g E+ (K(iω, ϵ, E+) − K(iω, ϵ, −E+)) − E2 − − g E− (K(iω, ϵ, E−) − K(iω, ϵ, −E−)) � , (B14) with K(iω, ϵ, E) as defined in (B8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In terms of these functions, the self energy is given by Σ1 = λ2 � q 1 3 [(h(iω, ϵk+q, gd) + h(iω, −ϵk+q, gd)) + (h(iω, ϵk+q, gd) − h(iω, −ϵk+q, gd)) γz] , (B15) Σ2 = λ2 � q 6 [(h(iω, ϵk+q, gN) + h(iω, −ϵk+q, gN)) − (h(iω, ϵk+q, gN) − h(iω, −ϵk+q, gN)) γz] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (B16) We can expand the total self energy Σ = Σ1 + Σ2 + Σ3 in terms of Pauli matrices in Nambu space, Σ(k) = ΣId(k) + Σz(k)γz + Σγy(k)γy, (B17) where ΣId(k) = λ2 � q �1 3 (h(iω, ϵk+q, gd) + h(iω, −ϵk+q, gd)) + 6 (h(iω, ϵk+q, gN) + h(iω, −ϵk+q, gN)) � , (B18) Σz(k) = λ2 � q �1 3 (h(iω, ϵk+q, gd) − h(iω, −ϵk+q, gd)) − 6 (h(iω, ϵk+q, gN) − h(iω, −ϵk+q, gN)) � , (B19) Σγy(k) = 3φ0λ2 � q [f(iω, ϵk+q) − f(iω, −ϵk+q)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (B20) Appendix C: Suppression of DOS at ω = 0 In this section, we derive a compact approximate analytical expression for the suppression of the density of states (DOS) as a result of the anomalous term Σ3 = Σγyγy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To this end, we focus on the limit of large bosonic velocities vµ in χµ and replace the q integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (B20) with the value of the integrand at q = 0, Σγy(ω + i0+, k) = 3φ0λ2 rN v2 N � f(ω + i0+, ϵk) − f(ω + i0+, −ϵk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (C1) Note that we would first need to re-parametrize the integral in terms of ˜q = q√rN/vN and then set ˜q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This approximation would then be valid in the large vd/vN limit with this re-scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We then Taylor expand f(z, ϵ) with respect to ϵ, ω, at a non-zero finite T (satisfying ϵ ≪ T ≪ 4� rdrN − φ2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In this limit, we find the self energy to be Σγy = 3φ0rNλ2 2v2 NT(rdrN − φ2 0)ϵk = αϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (C2) This expression is in agreement with the result in the main text [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 3(a)] which shows that as ϵ → 0, the contribution of Σy vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' With such a self-energy, the spectral function is given by A(ω) = − 1 π Im ω + i0+ (ω + i0+)2 − (1 + α2)ϵ2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (C3) A simple way to look at this, is that the band structure is simply renormalized as ϵk → √ 1 + α2ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This reduces the effective band mass, and thus the DOS is suppressed by a factor of √ 1 + α2, as stated in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 10 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='105 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Self Consistent Solution Including 3 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='05 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Self Consistent Solution Including 3 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1005 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='005 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='102 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='01 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='105 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='02 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='11 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='8 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='04 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='8 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='12 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='050 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='14 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='05 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 Converged Perturbative FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 5: The first order solution to ε(iω), ∆(iω) (red) and the self consistent solution (green) for the self energy in Matsubara space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Note the offset by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1 in the y axis in the left column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We chose ϵk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1, rd = 9, rN = 1, T = 1 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2, λ = 1 and measured all energies in units of √rN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Appendix D: Higher-order corrections to electronic Green’s function In this section, we show comparisons between the first order perturbative solution and the full self consistent solution to the fermionic Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We define the corrected Green’s function to be G(iω, k) = iωZk(iω) − εk(iω)γz + ∆k(iω)γy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In practice, we find that Zk(iω) ≃ 1, so we focus on εk(iω) and ∆k(iω) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 5, we show a comparison of the first order result for εk(iωn) and ∆k(iωn) after including the evaluation of the Σ3 term of the self energy [last diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(b)] and the full self consistent solution to the self energy in Matsubara space [obtained by summing up the diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2(a) corresponding to Σ3] at fixed k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We find that for small values up to φ0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6rN, the first order and self consistent solutions differ little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In first order, εk(iωn) does 11 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Self Consistent Solution Including 1 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='8 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='100 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='100 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='100 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Self Consistent Solution Including 2 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='8 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 Converged Perturbative FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 6: The first order solution to ε(iω) (red) and the self consistent solution (green) after including the effects of Σ1 (left column) and Σ2 (right column) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Same parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' not get renormalized since Σ3 acquires a γz term only if the Green’s function has a γy term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Such a γy term does not exist in the normal state about which we perform perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As φ0 increases, we find that the self consistent solution is lower in magnitude that the first order solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 6, we show the corrections in ε(iωn) after including the effects of Σ1 (left column) and Σ2 (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As expected and argued in the main text, we find that Σ1 and Σ2 have qualitatively the opposite effects on the renormalization of ε(iωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In both the cases, we find that the magnitude of the self consistent solution is higher than the perturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' However, since the fermionic Matsubara frequencies do not contain 0, we cannot directly say what this implies for the solution on the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The magnitude of φ0 has little effect on the solution since the effect of spin and triplet fluctuations are controlled by gN and gd, respectively, which we keep constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 7, we plot the corrections in ε(iωn) and ∆(iωn) after including the effects of all the self energies Σ = Σ1 + Σ2 + Σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We find that the inclusion Σ1 and Σ2 together reduces the difference between the self consistent and perturbative solution (refer to the plot near φ0 ∼ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As we increase φ0, the difference between the self consistent and perturbative solution increases due to the effect of Σ3 which is controlled by φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 12 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='20 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Self Consistent Solution Including 1 + 2 + 3 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='05 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Self Consistent Solution Including 1 + 2 + 3 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='20 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='010 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='20 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='02 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='20 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='04 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='20 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='8 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='050 (i n), 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='8 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='20 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='05 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='20 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='05 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='25 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='10 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 Converged Perturbative 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='25 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 17 11 5 0 5 11 17 n( n = n/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1 (i n), 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 Converged Perturbative FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 7: The first order solution to ε(iω), ∆(iω) (red) and the self consistent solution (green) after including the effects of all the terms of the self energy Σ1 + Σ2 + Σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Same parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Taken together, we see that the inclusion of second- and higher-order diagrams that contribute in the large-N limit defined in the main text yields qualitatively similar behavior on the imaginary axis compared to the first-order diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We therefore expect that the qualitative picture that S1 renormalizes the DOS close to the Fermi level on top of which S2 reduces the low-energy spectral weight still applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Since the impact of S2 is controlled by small φ0 and good quantitative agreement is found for φ0 up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6rN, we expect that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 3(c) would look similar when higher-order corrections were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 13 Appendix E: Meissner Effect From ODLRO The consequences of ODLRO defined in terms of four-fermion or two-boson correlators are well-known [57–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As a result of spin-rotation symmetry, we cannot capture ODLRO using a correlator of only two bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Instead, we have to study the four-boson density matrix ρ(x1, x2, x′ 1, x′ 2) = ⟨N(x1) · d∗(x2)N(x′ 1) · d(x′ 2)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (E1) Although the derivation is in close analogy to the two-boson or four-fermion case, we here show explicitly how the Meissner effect follows from ρ(x1, x2, x′ 1, x′ 2) → φ∗ 0(x1, x2)φ0(x′ 1, x′ 2) ̸= 0, |xj − x′ j| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (E2) Let us consider the system to be in the presence of a spatially uniform orbital magnetic field of strength B = B0ˆz in the out of plane direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Note that an in-plane orbital magnetic field does not couple to the bosons as the spatial motion is constrained to the two-dimensional plane of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The corresponding vector potential is given by A(x) = 1 2B × x, with x = (x, y, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Under an in-plane translation by a, the vector potential transforms as A(x) → A(x − a) = A(x) − 1 2B × a (E3) = A(x) − 1 2∇ [a · (x × B)] (E4) = A(x) + ∇χa(x), (E5) where χa(x) = − 1 2a·(x×B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Note that the triplet pairing field d is a charge-2e bosonic field, while the magnetization field N is neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Therefore, under simultaneous gauge transformation and displacement by a in the presence of a magnetic field, the fields transform as d(x) → ei 2e ℏc χa(x)d(x − a), (E6) N(x) → N(x − a), (E7) A(x) → A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (E8) As a result of gauge covariance and translational symmetry, the four-body density matrix obeys ρ(x1, x2, x′ 1, x′ 2) = ei 2e ℏc(χa(x′ 2)−χa(x2))ρ(x1 − a, x2 − a, x′ 1 − a, x′ 2 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (E9) Now suppose the system has ODLRO, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (E2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In combination with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (E9), this implies φ∗ 0(x1, x2)φ0(x′ 1, x′ 2) = ei 2e ℏc(χa(x′ 2)−χa(x2))φ∗ 0(x1 − a, x2 − a)φ0(x′ 1 − a, x′ 2 − a) (E10) =⇒ φ0(x1, x2) = faei 2e ℏc χa(x2)φ0(x1 − a, x2 − a), (E11) where fa is a position-independent phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Now suppose we perform two different translations by a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We can perform a first and then b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Alternatively, we can do b first and then a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' They respectively give us φ0(x1, x2) = fbfaei 2e ℏc χa(x2)ei 2e ℏc χb(x2−a)φ0(x1 − a, x2 − a), (E12) φ0(x1, x2) = fbfaei 2e ℏc χb(x2)ei 2e ℏc χa(x2−b)φ0(x1 − a, x2 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (E13) This is only consistent if ei 2e ℏc (χb(x2)+χa(x2−b)−χa(x2)−χb(x2−a)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (E14) We can evaluate χb(x2) + χa(x2 − b) − χa(x2) − χb(x2 − a) = B · (a × b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Thus, the condition for equality of phases becomes 2e ℏcB · (a × b) = 2πn, (E15) for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The only solution for arbitrary a, b is thus B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 14 Appendix F: Demonstration of Off Diagonal Long Range Order In this section, we calculate the ODLRO wavefunctions for both the bosons and fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The idea is to calculate the 4−body correlator ⟨N(x′ 1) · d(x′ 2)∗N(x1) · d(x2)⟩ for the bosons and ⟨c† τ ′ 1s′ 1(x′ 1)c† τ ′ 2s′ 2(x′ 2)cτ1,s1(x1)cτ2,s2(x2)⟩ for the fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Due to the U(1) symmetry breaking mediated by N · d attaining a finite expectation value (and correspondingly c† τc† −τ for the fermions), the ODLRO factorizes into a product of functions of x1 − x2 and x′ 1 − x′ 2 in the limit x − x′ → ∞, where x = x1+x2 2 and x′ = x′ 1+x′ 2 2 , giving rise to ODLRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' These wavefunctions decay as a function of their respective relative coordinates x1 − x2 and x′ 1 − x′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We now calculate these “macroscopic wavefunctions” explicitly for the bosonic and fermionic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Bosonic ODLRO The bosonic ODLRO is given by ⟨N(x′ 1)·d(x′ 2)∗N(x1)·d(x2)⟩ ≃ ⟨N(x′ 1)·d(x′ 2)∗⟩⟨N(x1)·d(x2)⟩ as x−x′ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' All the correlators are evaluated at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As discussed in the main text, to demonstrate ODLRO, it is sufficient to evaluate these correlators to first non-trivial order in the coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For bosonic ODLRO it is in fact sufficient to focus on zeroth order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=', neglecting the coupling to the fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Using the translation invariance of the system (and summing over the Matsubara frequencies iΩ since we are evaluating the correlator at time t = 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' we then have ψB(x) = ⟨N(x) · d(x = 0)⟩ = � q T � iΩ eiq·x⟨N −q · dq⟩ (F1) = � q T � iΩ eiq·r φ0 [(iΩ)2 − E2 +(q)][(iΩ)2 − E2 −(q)] (F2) = � q eiq·x φ0 2E+(q)E−(q)(E+(q) + E−(q)) (F3) ≃ � q eiq·x φ0 a + bq2 (F4) = φ0 b � q eiq·√ a b x 1 1 + q2 = 2πφ0K0 ��a b |x| � /b (F5) = 2πφ0K0 (|x|/ξ) /b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F6) where K0 is the zeroth modified Bessel function of second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In the third line, we evaluated the Matsubara sum at T = 0, and in the fourth line we series expanded 2E+(q)E−(q) (E+(q) + E−(q)) about q = 0 up to quadratic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The length scale ξ = � b a is determined by rµ, vµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In the limit of |vN − vd| ≪ vN + vd, we get ξ = 1 2 � � � �(v2 N + v2 d) �� rNrd − φ2 0 + rN + rd � rNrd − φ2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F7) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 4(b), we plot the numerical ODLRO wavefunction ψB(x) with the full functional dependence on q in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F3) included, and compare it with the asymptotic analytical form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We find good agreement between the numerical and analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Fermionic ODLRO Similarly, we can find the fermionic ODLRO, which in real space is generically written as ⟨c† τ ′ 1s′ 1(x′ 1)c† τ ′ 2s′ 2(x′ 2)cτ1,s1(x1)cτ2,s2(x2)⟩ ∼ ⟨c† τ ′ 1s′ 1(x′ 1)c† τ ′ 2s′ 2(x′ 2)⟩⟨cτ1,s1(x1)cτ2,s2(x2)⟩ in the limit xj − x′ j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Here, τ, s are the valley and spin indices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To demonstrate ODLRO, we thus have to evaluate the 2−fermion correlators, which in momentum space becomes (Ψ∗ F(x))s1,s2 = ⟨c† τ1,s1(x, t = 0)c† τ2,s2(x = 0, t = 0)⟩ = � k eik·x⟨c† k,τ1,s1c† −k,τ2,s2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F8) 15 Since the superconducting pairing takes place only between electrons between opposite valleys, we will have only τ2 = −τ1 giving non-zero correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Without loss of generality we chose τ1 = +, τ2 = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Up to first order in φ0, we have ⟨c† k,+,s1c† −k,−,s2⟩ = ⟨c† k,+,s1c† −k,−,s2 � − � q 1 2 φ0λ2 Mq Sq · D† q � ⟩0, (F9) where ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='⟩ is the average with respect to the interacting and ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='⟩0 with respect to the non-interacting ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We define G(k) = δss′δττ ′GV,k = δss′δττ′ iωn−ϵk = −⟨cs,τc† s′,τ ′⟩ to be the Green’s function in the fermionic basis (assuming ϵk = ϵ−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Equation (F9) can then be evaluated as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' −φ0λ2 2 � q 1 Mq ⟨c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s1c† −k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s2 � Sq · D† q � ⟩0 (F10) = −φ0λ2 2 � q 1 Mq ⟨c† k+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s1c† −k−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s2 � � � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p1=±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p2=± p1p2 � c† k1+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p1sck1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p1 � (ck2+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p2isysc−k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−p2) � �⟩0 (F11) = −2φ0λ2 2 � q 1 Mq ⟨c† k+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s1c† −k−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s2 � �� k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p − (c−k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−pisys(−GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='k1+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p)sck1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p) � �⟩0 (F12) = −6φ0λ2 2 � q 1 Mq ⟨c† k+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s1c† −k−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s2 � �� k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p (c−k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−pisyGV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='k1+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='pck1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='p) � �⟩0 (F13) = −6φ0λ2 2 � q 1 Mq ⟨c† k+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s1c† −k−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='s2 �� k1 (c−k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−isyGV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='k1+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='+ck1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='+ + c−k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='+isyGV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='k1+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−ck1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−) � ⟩0 (F14) = −6φ0λ2 2 � q 1 Mq (−(−GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−)(isy)s2s1GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='k+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='+GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='+ + (−GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='+)(isy)s1s2GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−k+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−) (F15) = −6φ0λ2 2 � q 1 Mq GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='kGV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−k (GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='−k+q + GV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='k+q) (isy)s2s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F16) We continue by calculating the Matsubara sum over iΩn and over iωn [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F8)], T 2 � iωn,iΩn 1 (iΩ2n − E+(q)2)(iΩ2n − E−(q)2)GV,kGV,−k (GV,−k+q + GV,k+q) (F17) = −T 2 � iωn,iΩ 1 (iωn)2 − ϵ2 k 1 ((iΩn)2 − E+(q)2)((iΩn)2 − E−(q)2) � 1 iωn + iΩn − ϵ−k+q + 1 −iωn + iΩn − ϵk+q � (F18) =: X(ϵk, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F19) For simplicity, we here focus on the limit where the remaining sum over q in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F9) is determined by its q = 0 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' With E± ≡ E±(q = 0) and vN, rN = 1, we have ˆX(ϵ) ≡ X(ϵ, q → 0) (F20) = nf(ϵ)2 2ϵ � −2 eβϵ E2 +E2 − + 2 � E2 + − 4ϵ2� � E2 − − 4ϵ2� + � 2ϵnB(E+) − E+nf(E+) E+(E2 + − E2 −)(E2 + − 4ϵ2)nf(E+)nB(2ϵ) + E+ ↔ E− �� , (F21) we can then finally write Ψ∗ F(x) = 3|φ0|λ2sy � 1 V � k eik·x ˆX(ϵk) � , (F22) = 3|φ0|λ2sy 2π � ∞ 0 dkkJ0(k · x) ˆX(ℏ2(k2 − k2 F )/(2m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (F23) 16 In the second line, we assumed ϵk = ℏ2(k2 − k2 F )/2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Using this expression, we calculate the spatial profile of the fermionic ODLRO wavefunction numerically for various values of ϵF ≡ ϵkF in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Unlike the case of the bosonic ODLRO (which was exponentially decaying), the fermionic ODLRO has an oscillating component superimposed on an exponentially decaying envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Appendix G: Ginzburg-Landau theory We here calculate the Landau-Ginzburg theory for the bosonic superfluid condensate parameter to leading (zeroth) order in the fermion-boson coupling λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To tis end, we assume that φ0 is now spatially and temporally varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This results in non-zero Fourier modes φq for q, iΩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In momentum space, the bosonic action is generalized according to SB = � q [χ−1 N (q)N q · N −q + χ−1 SC(q)d∗ q · dq + (φ0dq · N −q + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=')] (G1) = � q � N T −q d† q � � χ−1 N (q) φ0 φ0 χ−1 d (q) � � N q dq � (G2) → � q,k � N T −q−q2 d† q+q2 � � χ−1 N (q)δq2=0 φq2 φ∗ −q2 χ−1 d (q)δq2=0 � � N q dq � (G3) So after integrating out d and N, the effective action for φ reads as Seff = 1 2Tr ln G−1[φ], (G4) where G−1[φ](q + q1, q) = G−1 0 (q)δq1,0 + Γq+q1,q (G5) G−1 0 = � χ−1 N (q) 0 0 χ−1 d (q) � (G6) Γq+q1,q = � 0 φq1 φ∗ −q1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (G7) To derive the Ginzburg-Landau theory for φ, we expand Tr ln G−1 upto second order in Γ, which is equivalent to second order in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This gives us SGL = Tr ln(G−1 0 + Γ) ≃ TrG−1 0 + TrG0Γ − 1 2TrG0ΓG0Γ (G8) Because of the diagonal structure of G0, and the off diagonal structure of Γ, the linear term TrG0Γ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The quadratic term becomes � q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='q TrG0(q′ + q)Γ(q′ + q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' q′)G0(q′)Γ(q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' q′ + q) = � q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='q Tr � 0 χN(q′ + q)φq χd(q′ + q)φ∗ −q 0 � � 0 χN(q′)φ−q χd(q′)φ∗ q 0 � (G9) = � q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='q χN(q′ + q)χd(q′)φqφ∗ q + χN(q′)χd(q′ + q)φ−qφ∗ −q (G10) = � q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='q (χN(q′ + q)χd(q′) + χN(q′)χd(q′ − q)) φqφ∗ q (G11) = � q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='q (χN(q′ + q)χd(q′) + χN(q′ + q)χd(q′)) φqφ∗ q (G12) = 2 � q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='q χN(q′ + q)χd(q′)φqφ∗ q (G13) 17 We need to evaluate � q′ χN(q′ + q)χd(q′) = � q′ T � iΩ′∈Bosonic � 1 ((iΩ′ + iΩ)2 − rN − v2 N(q′ + q)2)((iΩ′)2 − rd − v2 dq′2) � (G14) = −1 2 � q′ � 1 � rN + v2 N(q′ + q/2)2 + 1 � rd + v2 d(q′ − q/2)2 � � � � 1 iΩ2 − �� rN + v2 N(q′ + q/2)2 + � rd + v2 d(q′ − q/2)2 �2 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (G15) By expanding the above expression up to second order in iΩ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' we find the effective action for the φ field to be T � iΩ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='q � rφ − ρ(iΩ)2 + v2q2� |φ(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='iΩ)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='(G16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='where the coefficients are given by ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='N) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='g3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='�√gd + √gN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='�4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='(G20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='with gµ = rµ + v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='µq′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We numerically calculate the quantities rφ, ρ, v2 and plot it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 4(c,d) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Appendix H: Self-consistent equations in special limits In this appendix, we complement the previous analysis by studying two simple limits of the model for phase (B)—mean-field theory and the limit of zero energy-momentum transfer of the bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This allows us to study possible non-perturbative solutions systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In both cases, we find that the soft gap behavior obtained within perturbation theory is also found in these descriptions as long as T is large enough/the coupling constants, λ or φ0, are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Mean-field Theory In this section, we consider the effective interaction contributed by the S2 part of the action between the electrons at time t = 0, in the limit where we replace the q integral with the corresponding value of the integrand at q = 0, and then perform a mean-field decomposition of the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Defining the Bogoliubov-de Gennes basis as before, ξk = � ck,+ isyc† −k,− �T , with Pauli matrix γi acting on it, and ˜φ0 = φ0λ2rN/v2 N the corresponding interaction potential is given by V = −1 2 1 χ−1 d χ−1 N − |φ0|2 � ˜φ0Sq=0 · D† q=0 + ˜φ∗ 0Dq=0 · S−q=0 � |q=0 (H1) = −1 2 1 rNrd − |φ0|2 � k1,k2 � −˜φ0 � c† k1sτzck1 � (ck2sisyτyc−k2) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='c � (H2) = − 1 rNrd − |φ0|2 � k1,k2 � ˜φ0 � ξ† k1sγzξk1 � � ξ† k2siγ−ξk2 � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='c � , (H3) 18 while the free Hamiltonian is given by H0 = � k ξ† kϵkγzξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (H4) We consider only the effective Hamiltonian at time t = 0, which is why there are no Matsuabra indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We perform a Hartree-Fock decomposition of V , which gives us V = 1 rNrd − |φ0|2 � k1,k2 � ˜φ0 � ξ† k1sγzξk1 � � ξ† k2siγ−ξk2 � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='c � (H5) → c 2 � k ξ† k (γyCkγz + γzCkγy) ξk, (H6) where Ck = −⟨ξkξ† k⟩, c = 6 ˜φ0 rNrd−φ2 0 , choosing a gauge with real φ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' further take φ0 to be positive such that c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Note that this correlator is related to the Green’s function G by Ck = T � iωn G(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Note that all the Hartree terms vanish since we do not allow for spontaneous breaking of spin-rotation invariance (recall we study finite T in 2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The effective 2−particle Hamiltonian is given by H = � k ξ† k � ϵkγz + c 2γyCkγz + c 2γzCkγy � ξk (H7) = � k ξ† k � ˜ϵkγz + ˜∆kγy � ξk (H8) where ˜ϵk, ˜∆k are the self consistent band structure and gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Making connection with the diagrammatic self consistency relationship to be discussed below, we can foresee that the resulting self consistent equation we get will be the same as (H18) but with ˜ϵ, ˜∆ replaced with the corresponding iωn averaged value, and the whole equation itself will be iωn averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The correlators in terms of ˜ϵ, ˜∆ are given by Ck = T � iωn 1 iωn − � ˜ϵkγz + ˜∆kγy � = nf(Ek) − nf(−Ek) 2Ek � ˜ϵkγz + ˜∆kγy � , (H9) where Ek = � ˜ϵ2 k + ˜∆2 k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Thus, using (H7), the self consistency equations become ˜ϵk = ϵk + c ˜∆k nf(Ek) − nf(−Ek) 2Ek (H10) ˜∆k = c˜ϵk nf(Ek) − nf(−Ek) 2Ek .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (H11) Let us define βk = c nf (−Ek)−nf (Ek) 2Ek = c tanh � Ek 2T � 2Ek and first assume βk < 1, which always holds as long as T > c/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The self consistency equations can then be rearranged as ˜ϵk = 1 1 − β2 k ϵk (H12a) ˜∆k = −βk 1 − β2 k ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (H12b) Using this, we find Ek = √ 1+β2 k 1−β2 k ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Note, however, that βk also depends on Ek and, thus, this relation should be thought of as a self consistency equation, to be solved for βk or Ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Equations (H12) allow to derive asymptotic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In the limit ϵk → 0, we then have Ek → 0 and βk → c 4T , ensuring the self-consistent solutions are well controlled in the ϵk → 0 regime that we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Near ϵk = 0 and for large T ≫ c (βk ≪ 1), the renormalized spectrum is given by Ek = √ 1+β2 k 1−β2 k ϵk ≃ � 1 + 3β2 kϵk ≃ � 1 + 3c2 16T 2 ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The suppression of DOS is now given by ρF (φ0) ρF (φ0 = 0) = 1 √ 1 + α′2 , α′ = 3 √ 3φ0λ2rN 2v2 NT(rdrN − φ2 0), 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content='6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 8: The self consistent solution for ˜ϵk, ˜∆k and Ek as a function of ϵk for various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' At T/c = 1/4, the self-consistent solutions become non-analytic having an infinite slope at ϵk = 0, and a gap opens up as the temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' There is a discontinuity in ˜ϵk, ˜∆k at ϵk = 0, where the gap value has different signs for ϵk → 0−, 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' which is of the same form as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (4), found through the perturbative calculation presented in the main text and derived in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' When T/c = 1/4, we have β2 k = 1 for ϵk → 0, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (H12) are not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' At this point, the self consistent solutions open up a gap in Ek when ϵk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' This gap follows by solving the equation β2 k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' When ϵk = 0 and βk = 1, we also have ˜ϵk = − ˜∆k [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (H11)] which gives Ek = √ 2˜ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For T/c approaching 1/4 from below, we find that βk ≃ c 4T � 1 − 1 12 E2 k T 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Thus the condition that β2 k = 1 gives us Ek = √ 12T � 1 − 4T c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' To summarize, for T > c/4, self consistent energy and gap (˜ϵ, ˜∆) are proportional to ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As T approaches c/4 from above, the slope of proportionality approaches ∞ at ϵ = 0, and becomes non-analytic at T = c/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Going below T = c/4, this non-analyticity at ϵ = 0 turns into a discontinuity at ϵ = 0, with the self consistent solutions developing a finite gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The value of this gap at T = 0 is given as | ˜∆| = |˜ϵ| = |c| 2 √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Figure 8 illustrates the behavior obtained by numerical solution of the self-consistency equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Zero energy-momentum transfer In this section, we consider the limit where the bosonic fields N, d do not transfer any momentum or Matsubara frequency in the interaction (q = 0 in Sc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Additionally, we consider only the effect of S2 on the self energy to study the effect of the anomalous contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' In this limit, we would like to analyze the self consistent solution of the Green’s function up to all orders in λ within the large-N theory of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The ansatz of the full Green’s function is given by G−1 = iωn − ˜ϵkγz − ˜∆kγy, since Σ3 renormalizes only the anomalous term ˜∆k and the spectrum ˜ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' We have G = iωn + ˜ϵkγz + ˜∆kγy (iωn)2 − ˜ϵ2 k − ˜∆2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (H13) Thus the self-consistent analogue of Σ3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (3) becomes (where we have replaced the integration over q by the q = 0 value of the integrand, and ˜φ0 = φ0λ2rN/v2 N) Σ3 = 6T ˜φ0 rNrd − φ2 0 ˜ϵkγy + ˜∆kγz (iωn)2 − ˜ϵ2 k − ˜∆2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (H14) From the self-energy equation we get G−1 = G−1 0 − Σ3 (H15) iωn − ˜ϵkγz − ˜∆kγy = iωn − ϵkγz − 6T ˜φ0 rNrd − φ2 0 ˜ϵkγy + ˜∆kγz (iωn)2 − ˜ϵ2 k − ˜∆2 k (H16) ˜ϵk = ϵk + Tc ˜∆k (iωn)2 − ˜ϵ2 k − ˜∆2 k (H17) ˜∆k = Tc ˜ϵk (iωn)2 − ˜ϵ2 k − ˜∆2 k , (H18) 20 where c = 6 ˜φ0 rNrd−φ2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' Right at the Fermi surface, ϵk = 0, the self consistency equations reduce to ˜ϵk = Tc ˜∆k (iωn)2 − ˜ϵ2 k − ˜∆2 k (H19) ˜∆k = Tc ˜ϵk (iωn)2 − ˜ϵ2 k − ˜∆2 k (H20) =⇒ ˜ϵk = T 2c2 ˜ϵk (ω2n + ˜ϵ2 k + ˜∆2 k)2 (H21) There are two possible solutions to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' (H19) and (H20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' The first is ˜ϵk = ˜∆k = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' this is exactly what we find within perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' For a solution with ˜ϵk ̸= 0 to exist, it must hold (assuming ˜ϵk, ˜∆k ∈ R as expected in the gauge that we use) 1 = T 2 c2 (ω2n + ˜ϵ2 k + ˜∆2 k)2 < T 2 c2 π4T 4 (H22) Thus, a non-zero solution only exists if T < c/π2 ∼ c/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} +page_content=' As compared to Hartree-Fock, the critical temperature for a non-perturbative solution is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AzT4oBgHgl3EQfY_xr/content/2301.01344v1.pdf'} diff --git a/YNFJT4oBgHgl3EQf6S0s/vector_store/index.faiss b/YNFJT4oBgHgl3EQf6S0s/vector_store/index.faiss new file mode 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/dev/null +++ b/ZNFQT4oBgHgl3EQfejY8/content/tmp_files/2301.13336v1.pdf.txt @@ -0,0 +1,1802 @@ +1 +The Fair Value of Data Under Heterogeneous +Privacy Constraints +Justin Kang, Ramtin Pedarsani, Kannan Ramchandran +Abstract +Modern data aggregation often takes the form of a platform collecting data from a network of users. +More than ever, these users are now requesting that the data they provide is protected with a guarantee of +privacy. This has led to the study of optimal data acquisition frameworks, where the optimality criterion +is typically the maximization of utility for the agent trying to acquire the data. This involves determining +how to allocate payments to users for the purchase of their data at various privacy levels. The main goal +of this paper is to characterize a fair amount to pay users for their data at a given privacy level. We +propose an axiomatic definition of fairness, analogous to the celebrated Shapley value. Two concepts for +fairness are introduced. The first treats the platform and users as members of a common coalition and +provides a complete description of how to divide the utility among the platform and users. In the second +concept, fairness is defined only among users, leading to a potential fairness-constrained mechanism +design problem for the platform. We consider explicit examples involving private heterogeneous data +and show how these notions of fairness can be applied. To the best of our knowledge, these are the first +fairness concepts for data that explicitly consider privacy constraints. +I. INTRODUCTION +From media to healthcare to transportation, the vast amount of data generated by people living +their everyday lives has been used to great effect to solve difficult problems across many domains. +For example, nearly all machine learning algorithms, including those based on deep learning rely +heavily on data. Many of the largest companies to ever exist center their business around the +precious resource of data. This includes directly selling access to data to others for profit, selling +targeted advertisements based on data, or by exploiting data through data-driven engineering, +to better develop and market products. In this sense, data has many of the features of other +commodities. It is a raw material that is processed and exploited to generate products of greater +value. Unlike other commodities, however, modern data markets are largely informal, where in +arXiv:2301.13336v1 [cs.LG] 30 Jan 2023 + +2 +many cases individuals sell their data in exchange for access to services, as depicted in Fig.1. +Due to the lack of a formal exchange, it can be difficult to understand if such an exchange is fair. +Unlike a barrel of oil, one cannot simply put a fair price on a gigabyte of data. The development +of an economic theory for the value of data is still nascent [Ghorbani and Zou, 2019], [Jia +et al., 2019], [Acemoglu et al., 2019], and the dynamics of formal data markets are largely not +understood. A major shortcoming of the current understanding of data value is that in many cases, +it fails to explicitly consider a critical factor in an individual’s decision to share data—privacy. +This work puts forth two rigorous notions of the fair value of data in Section III that explicitly +include privacy, which has become as important area of concern as discussed below. +With the rise of large-scale data collection, many high-profile data breaches have occurred. +Consumers, governments and online platforms have become aware of the negative aspects of +the way data is used and the lack of privacy, which has led to an array of responses from +stakeholders. For example, there is significant discourse among consumers about the addictive +properties of content serving algorithms powered by their data [Hou et al., 2019]. Governments +have set regulations such as the General Data Protection Regulation (GDPR) [European Union, +2016] of the European Union, one of the first pieces of legislation aimed at regulating online +platforms to protect data. Platforms that collect and utilize data have pursued frameworks for +privacy such as Federated Learning (FL) [Kairouz et al., 2021], proposed and implemented by +Google [McMahan et al., 2017] and Differential Privacy (DP) [Dwork, 2008], which is used +by Apple in their operating systems [Tang et al., 2017] and recently in the US 2020 Census +[US Census Bureau, 2021]. These approaches allow platforms to utilize distributed data from +individuals while providing a level of privacy to the individuals that participate, potentially at +the cost of decreased data utility. +Compelled by the importance of data in our modern economy and a growing social concern +about privacy, this paper presents frameworks for quantifying the fair value of private data +to better understand the relationship between those that generate data and the platforms that +collect and benefit from data. More specifically, we consider a setting where users are willing to +provide their data to a platform in exchange for some sort of payment and under some privacy +guarantees depending on their level of privacy requirements. The platform is responsible to run +the private learning algorithm on the gathered data and make the fair payments with the objective +of maximizing its utility including statistical accuracy and total amount of payments. Our goal is +to design a fair mechanism for this procedure as depicted in Fig. 1. + +3 +Fig. 1. Diagram depiction of interactions between platform and users. Users generate data by using devices like phones, cameras, +autonomous vehicles and drones. They provide this data to the platform, potentially requesting some level of privacy. The +platform uses this data to generate utility, often by using the data for learning tasks. In return, the platform may provide the +users with payments in the form of access to services, discounts on products, or monetary compensation. +A. Related Work +With the widespread use of the internet and data-driven methods, interactions involving those +that have data and those that seek to acquire it have become an important area of theoretical +study [Balazinska et al., 2011], but also a practical necessity [Spiekermann et al., 2015b]. Among +these interactions, the economics of data from privacy conscious users has received significant +attention [Acquisti et al., 2016], [Wieringa et al., 2021]. DP [Dwork, 2008] and its variations, +[Bun and Steinke, 2016] are widely studied as a formal framework for privacy. [Ghosh and +Roth, 2015] studies the purchase of private data, where privacy is quantified under DP. [Ghosh +and Roth, 2015] assumes that each player has binary data and its own heterogeneous privacy +sensitivity parameter that they report, potentially strategically. The goal is to design a dominant +strategy truthful mechanism to acquire data and estimate the sum of users’ data. In [Fallah et al., +2022], the authors consider an optimal data acquisition problem in the context of private mean +estimation in two different heterogeneous DP settings. It is assumed that players consider both +the estimation error of the common estimator generated by the platform and any payments made +to them by the platform in their decision making. By assuming that the privacy sensitivities are +represented by scalars drawn from a distribution, they devise a mechanism for computing the +optimal privacy levels to provide to the players. +In [Hu and Gong, 2020] the federated learning setting is considered, where each play + +4 +has a unique privacy sensitivity function parameterized by a scalar variable. Players report +their sensitivity parameter and the platform assigns each user a privacy level. A heuristic +proportional payment mechanism is considered. In the case of linear privacy sensitivity functions, +a computationally efficient way to compute the Nash Equilibrium is derived. In [Oh et al., 2020], +a multi-stage data market is studied where data-brokers acquire data from users, competing to +sell data to a platform that further sells services based on the data. The economic and social +implications of privacy and data markets are considered in [Spiekermann et al., 2015a]. In +[Acemoglu et al., 2019] the impact of data externalities is investigated. In particular the leakage +of data leading to the suppression of its market value is considered. In [Jia et al., 2019], [Ghorbani +and Zou, 2019] and [Ghorbani et al., 2020] a framework for determining the fair value of data is +proposed. These works extend the foundational principles of the Shapley Value [Shapley, 1952], +which was originally proposed as a concept for utility division in coalitional games to the setting +of data. Our work takes this idea further and explicitly includes privacy in the definition of the +fair value of data. +Finally, we note that we consider the concept of fairness in data valuation, not algorithmic +fairness, which relates to the systematic failure of machine learning systems to account for data +imbalances. +B. Main Contributions +The main contribution of this work is in the development of a rigorous notion of fairness in +the context of user data acquisition with privacy. While the existing literature has investigated +how a platform should design incentives for users in order to optimize its utility, the definitions +of fairness that we propose in this work offer another way to evaluate these mechanisms. We +summarize the main contributions as follows: +• We present an axiomatic notion of fairness that is inclusive of the platforms and the users +in Theorem 1. The amount of utility that should be awarded to each user and the platform +is uniquely determined. This unique allocation can be used as a benchmark to determine +how fair an optimal data acquisition procedure is. +• In the realistic scenario that fairness is only considered between users, Theorem 2 defines a +notion of fairness based on axioms, but only places restriction on the relative amount of +utility distributed to the players. This creates an opportunity for the platform to optimize its +utility under a constraint of fairness. + +5 +• We provide two important applications of our framework: differentially private algorithms +and federated learning. We fully characterize the mechanism design problem in a two-user +symmetric binary privacy game in Section IV. We further demonstrate an application of our +framework in the problem of privacy-preserving federated mean estimation in Section V. +C. Notation +Lowercase boldface x and uppercase boldface X symbols denote vectors and matrices +respectively. R+ denotes the set of non-negative reals. In addition, we take R+ = R+ ∪ {∞}. +The notation [N] represents the set {1, 2, . . . , N}. NE(·) represents the set of Nash Equilibrium +actions of players, potentially parameterized by an input. +II. PROBLEM SETTING +A. Privacy Levels and Utility Functions +Consider the setting depicted by Fig. 2, where a platform collects data xi ∈ X and privacy +levels ϵi ∈ E from each user (player) i ∈ [N]. We restrict the space of privacy levels to E ⊆ R+, +adopting the notation of DP, where ϵi = 0 means the data cannot be used by the platform (full +privacy), and ϵi = ∞ means no restriction on data usage (no privacy). If ϵi > ϵj, we say ϵi is a +lower privacy level than ϵj. +The platform applies an ϵ-private algorithm Aϵ : X N �→ Y to process the data. An ϵ-private +algorithm is one that provides privacy level ϵi to data xi. In the privacy literature, Aϵ is known +as a private mechanism, but in this work, we avoid this terminology to reduce confusion with the +game-theoretic notion of mechanism design. The output of the algorithm y = Aϵ(x) is used by +the platform to derive utility U. In this work, we describe the utility as a function of privacy level +U(ϵ). For example, if the platform is trying to estimate a mean from a population, as may be the +case in a census, the utility could depend on the mean square-error of the private estimator. This +is a valid model if the platform is concerned about the statistical performance of the algorithm +at the given privacy level, such as risk minimization in a learning problem. U(ϵ) then does not +depend directly on xi, which we treat as measurements from some trusted system that cannot be +strategically altered. We make no assumptions about the mechanism Aϵ, but in many cases it is +reasonable to assume that U(ϵ) is an increasing monotone function in each coordinate. +Note that this formulation differs from typical formulations in the literature of optimal data +acquisition, where some privacy sensitivity is instead reported by users, and the platform then + +6 +Fig. 2. Depiction of interactions between the platform and individual players. Players send their data xi and a privacy level +ϵi to the central platform in exchange for payments ti(ϵi; ϵ−i). The central platform extracts utility from the data at a given +privacy level and optimizes incentives to maximize the difference between the utility and the sum of payments U(ϵ) − 1T t(ϵ). +chooses the privacy level ϵi based on this sensitivity. This typical formulation is beneficial in +several ways. It allows for the relatively straightforward application of notions such as incentive +compatibility and individual rationality from mechanism design theory. In this work, however, +we wish to emphasize the fact that the utility U depends on the privacy levels ϵ directly, so +considering actions in the space of privacy levels E is natural. Despite this difference, the notions +of fairness described in the following section can be applied more broadly, so long as U is a +function of the privacy constraints ϵ. +1) Example: In the following sections we will consider a common example to elucidate the +application of our theory. Let Xi represent the independent and identically distributed data of user +i respectively, with Pr(Xi = 1/2) = p and Pr(Xi = −1/2) = 1 − p. Assume that the platform +has the prior p ∼ Unif(0, 1). As our formal notion of privacy, we will consider a pure ϵ-DP +framework, defined below. +Definition 1. A random function A : X N → Y is ϵi-DP, ϵi ∈ R+ in coordinate i if for any +x′ ∈ X N that differs from x ∈ X N only in coordinate i, for all measurable sets S ∈ Y we have: +Pr(A(x) ∈ S) ≤ eϵiPr(A(x′) ∈ S). +(1) +Definition 2. A random function A : X N → Y is ϵ-DP if A is ϵi-DP in coordinate i for all +i ∈ [N]. +The goal of the platform is to construct an ϵ-DP estimator for µ ≜ E[Xi] = p − 1/2 that + +7 +minimizes the Bayes risk. A general procedure for finding the Bayes optimal ϵ-DP estimator +does not exist. We restrict our attention to ϵ-DP linear-Laplace estimators of the form: +A(X) = w(ϵ)TX + Z, +(2) +where Z ∼ Laplace(1/η(ϵ)). In [Fallah et al., 2022] the authors argue that unbiased linear +estimators are nearly optimal in a minimax sense for bounded random variables. We assume a +squared error loss L(a, µ) = (a − µ)2 and let A(ϵ) be the set of ϵ-DP estimators satisfying (2). +Then, we define: +Aϵ = arg min +A∈A(ϵ) +E[L(A(X), µ)] +(3) +r(ϵ) = E[L(Aϵ(X), µ)]. +(4) +In words, Aϵ is an ϵ-DP estimator of the form (2), where w(ϵ) and η(ϵ) are chosen to minimize +the Bayes risk of the estimator, and r(ϵ) is the risk achieved by Aϵ. Since the platform’s goal +is to accurately estimate the mean of the data, it is natural for the utility U(ϵ) to depend on ϵ +through the risk function r(ϵ). Note that if U is monotone decreasing in r(ϵ), then U is monotone +increasing in ϵ. +Let us now consider the case of N = 2 users, choosing from an action space of E = {0, ϵ′}, +for some ϵ′ ̸= 0. Furthermore, take U to be an affine function of r(ϵ): U(ϵ) = c1r(ϵ) + c2. For +concreteness, take U(0) = 0 and maxϵ∈R U(ϵ) = 1. Note that this ensures that U is monotone +increasing in ϵ. The utility corresponding to possible user actions can be succinctly represented +in matrix form as: +U = +� +�U([0, 0]T) +U([0, ϵ′]T) +U([ϵ′, 0]T) +U([ϵ′, ϵ′]T) +� +� +(5) += +� +� +0 +2 +� +3 + +24 +(ϵ′)2 +�−1 +2 +� +3 + +24 +(ϵ′)2 +�−1 +� +1 + +3 +(ϵ′)2 +�−1 +� +� +(6) +Details of the above calculations can be found in the appendix. +B. The Data Acquisition Problem +Central to this work is the payments users receive for providing their data to a platform. As +described previously, the platform generates utility U(ϵ) from the user data. We assume this + +8 +utility is transferable and divisible. In exchange for data, the platform distributes some portion +of the utility ti(ϵi; ϵ−i) to user i, where ϵ−i denotes the vector of privacy levels ϵ with the ith +coordinate deleted. Note that we could also write ti(ϵ), but we choose the former because it +makes explicit which parameter user i has control over. These incentives in turn may motivate +users to lower their privacy level. The behavior of users can also be modelled with the help of a +utility function. In general, each user will have some sensitivity to their data being shared. This +can be modelled by a sensitivity function ci : E → R+, ci(0) = 0 resulting in the utility: +ui(ϵ) = ti(ϵi, ϵ−i) − ci(ϵi). +(7) +The payment user i receives from the platform will tend to increase with a lower privacy level, +as the platform is able to better exploit the data, while their sensitivity will increase. Thus, by +specifying a set of ti(ϵi; ϵ−i), the platform effectively creates a game among the users. Each +user’s action in this game is the level of privacy that they request for the data they share. Users +(players) select their privacy level ϵi by considering their utility function ui and the potential +actions of the other players. From the perspective of the platform, the goal is to design the +payments ti(ϵi; ϵ−i) such that it maximizes the difference between the utility it receives and the +payments made to the players. One way to formulate this problem is to consider maximizing +this difference at equilibrium points: +max +t(·), P +U(P) − 1Tt(P) +s.t. +P ∈ NE(t). +(8) +In (8), NE(t) denotes the set of Nash Equilibrium strategies induced by the payment function +t, which is the vector with payment function ti at index i. Recall that the Nash Equilibrium is +a stable state of a system such that no user can gain by a unilateral change of strategy if the +strategies of the other users remain unchanged. We allow these equilibrium points to be mixed +strategies over the probability space, such that P represents a distribution over the privacy space +E. In addition, we have used the shorthand f(P) = Eϵ∼P [f(ϵ)]. Note that in order to solve +(8), the platform requires knowledge of the privacy sensitivity ci of each user. This can be a +reasonable assumption when the platform has interacted with the users many times in the past +and has learned ci. +Note that some restrictions must be placed on t, as it can otherwise be made arbitrarily negative. + +9 +Individual rationality is common condition in mechanism design that says that a user can be +made no worse off by participating in the game. +As a final comment, we note that the compensation ti(ϵi; ϵ−i) may not be a direct monetary +transfer. Individuals are often compensated for data through discounts or access to services. A +shortcoming of our model is that we assume a divisible and transferable utility, which may fail +to capture these nuances of compensation. +III. AXIOMATIC FAIRNESS WITH PRIVACY +Somewhat in contrast to the resource allocation view just described, we can also view the +interaction between users and the platform as a coalition where many users and the platform +come together and pool their resources to generate utility. A natural question to ask is: How +should the utility be divided fairly among members of this coalition? +The answer to this question turns out to be connected to the celebrated Shapley value [Shapley, +1952]. Shapley value is one of the most important normative utility division schemes for coalitional +games. Following an axiomatic approach to fairness, the Shapley value describes how to fairly +divide utility between members of a coalition. In the following section we develop an axiomatic +approach to defining fairness analogous to the Shapley value for the context of users providing +private data to platforms. +A. Platform as a Coalition Member +We define a coalition of users and a platform as a collection of s users, with 0 ≤ s ≤ N +and up to 1 platform. Let a ∈ {0, 1} represent the action of the platform. Let a = 1 when the +platform chooses to join the coalition, and a = 0 otherwise. Let U(ϵ) be as defined in Section II. +We augment the utility to include the platform as follows: +U(a, ϵ) ≜ +� +� +� +� +� +U(ϵ) +a = 1 +0 +a = 0 +, +(9) +to indicate that no utility is generated if the platform does not participate. We also define ϵS +such that: +[ϵS]i = +� +� +� +� +� +ϵi +i ∈ S +0 +else +. +(10) + +10 +Let φp(a, ϵ) and φi(a, ϵ), i ∈ [N] represent the “fair” amount of utility awarded to the platform +and each user i respectively, given a and ϵ, otherwise described as the “value” of a user. Note +that these values depend implicitly on both the private algorithm Aϵ and the utility function U, +but for brevity, we avoid writing this dependence explicitly. The result of [Hart and Mas-Colell, +1989] show that these values are unique and well defined if they satisfy the following three +axioms: +A.i) (Fairness) For any two users i, j ∈ [N]: +U(a, ϵS∪{i}) = U(a, ϵS∪{j}) ∀S ⊂ [N]\{i, j} =⇒ φi(a, ϵ) = φj(a, ϵ). +(11) +In addition, for any user i ∈ [N], U(1, ϵS∪{i}) − U(1, ϵS) = 0 +∀S ⊂ [N]\{i} +=⇒ +φi(a, ϵ) = 0. +A.ii) (Efficiency) The sum of values is the total utility U(a, ϵ) = φp(a, ϵ) + � +i φi(a, ϵ). +A.iii) (Additivity) Let φp(a, ϵ) and φi(a, ϵ) be the value of the platform and users respectively +for the utility function U, under the ϵ-private Aϵ. Let V be a separate utility function, +also based on the output of Aϵ, and let φ′ +p(a, ϵ) and φ′ +i(a, ϵ) be the utility of the platform +and individuals with respect to V . Then under the utility U + V , the value of user i is +φi(a, ϵ) + φ′ +i(a, ϵ) and the value of the platform is φp(a, ϵ) + φ′ +p(a, ϵ). +Theorem 1. Let φp(a, ϵ) and φi(a, ϵ) satisfying axioms (A.i-iii) represent the portion of total +utility awarded to the platform and each user i from utility U(a, ϵ). Then they are unique and +take the form: +φp(a, ϵ) = +1 +N + 1 +� +S⊆[N] +1 +� N +|S| +�U(a, ϵS), +(12) +φi(a, ϵ) = +1 +N + 1 +� +S⊆[N]\{i} +1 +� +N +|S|+1 +� � +U(a, ϵS∪{i}) − U(a, ϵS) +� +. +(13) +The Proof of Theorem 1 can be found in the appendix. +1) Example: Consider the example from Section II-A1 with binary privacy space E = {0, ∞}. +By (5), the utility can be written in matrix form as: +U = +� +� 0 +2/3 +2/3 +1 +� +� . +(14) + +11 +Note from (12) and (13), it is clear that φp(0, ϵ) = φi(0, ϵ) = 0. Let Φp and Φ(1) +i +represent the +functions φp(1, ϵ) and φi(1, ϵ) in matrix form akin to U. Then using (12) and (13), we find that +the fair allocations of the utility are given by: +Φp = +� +� 0 +1/3 +1/3 +5/9 +� +� , Φ(1) +1 += +� +� 0 +1/3 +0 +2/9 +� +� , Φ(1) +2 += +� +� 0 +0 +1/3 +2/9 +� +� . +(15) +B. Fairness Among Users +Though we can view the interactions between the platform and the users as a coalition, due to +the asymmetry that exists between the platform and the users, it also makes sense to discuss +fairness among the users alone. In this case, we can consider an analogous set of axioms that +involve only the users. +B.i) (Fairness) For any two users i, j ∈ [N]: +U(ϵS∪{i}) = U(ϵS∪{j}) ∀S ⊂ [N]\{i, j} =⇒ φi(ϵ) = φj(ϵ). +(16) +In addition, for any user i ∈ [N], U(ϵS∪{i}) − U(ϵS) = 0 ∀S ⊂ [N]\{i} =⇒ φi(ϵ) = 0. +B.ii) (Pseudo-Efficiency) The sum of values is the total utility α(ϵ)U(ϵ) = � +i φi(ϵ). Where if +U(ϵ) = U(˜ϵ) then α(ϵ) = α(˜ϵ) and 0 ≤ α(ϵ) ≤ 1. +B.iii) (Additivity) Let φi(ϵ) be the value of users for the utility function U, under the ϵ-private +algorithm Aϵ. Let V be a separate utility function, also based on the output of the algorithm +Aϵ, and let φ′ +i(ϵ) be the utility of the users with respect to V . Then under the utility U + V , +the value of user i is φi(ϵ) + φ′ +i(ϵ). +The most notable difference between these axioms and (A.i-iii) is that the efficiency condition +is replaced with a pseudo-efficiency condition. Under this condition, the platform may determine +the sum of payments awarded to the players, but this condition should in general depend only +on the utility itself, and not on how that utility is achieved. +Theorem 2. Let φi(ϵ) satisfying axioms (B.i-iii) represent the portion of total utility awarded to +each user i from utility U(ϵ). Then they must take the form: +φi(ϵ) = α(ϵ) +N +� +S⊆[N]\{i} +1 +�N−1 +|S| +� � +U(ϵS∪{i}) − U(ϵS) +� +, +(17) +where α(ϵ) satisfies axiom (B.ii). + +12 +The proof of Theorem 2 can be found in the appendix. +1) Example: Consider the utility function defined in (14), for the N = 2 user mean estimation +problem with E = {0, ∞}. By Theorem 2 the fair allocation satisfying (B.i-iii) must be of the +form: +Φ(2) +1 += A ⊙ +� +�0 +2/3 +0 +1/2 +� +� , +Φ(2) +2 += A ⊙ +� +� 0 +0 +2/3 +1/2 +� +� , +(18) +where A = AT and 0 ≤ [A]ij ≤ 1. +IV. TWO USER SYMMETRIC BINARY PRIVACY GAME +In this section, we return to our example of a mean estimation problem with N = 2 users +and E ∈ {0, ∞} to investigate what happens when our proposed notions of fairness are applied +to a data acquisition problem of the form (8). For simplicity, we assume both users have the +same privacy sensitivity function. Simulations for the case where sensitivities are asymmetric are +included in the appendix. +A. Fairness from Theorem 1 +In Section III-A1, we showed that the fair values for both the platform and the players satisfying +(A.i-iii) are unique. Thus, for a platform constrained to making payments satisfying (A.i-iii), the +data acquisition problem (8), reduces to a maximum over the equilibrium points: +max +p1, p2 +pT +1 Φpp2 +s.t. +(p1, p2) ∈ NE, +(19) +where we have used pi = [p, (1 − p)]T to represent the mixed strategy where the probability of +choosing ϵi = 0 is p. The equilibrium points themselves depend on the objective of the users. +The users’ objectives are: +ui(p1, p2) = pT +1 Φ(1) +i p2 − [0 c]Tp1. +(20) +Since we have assumed both players have the same privacy sensitivity, and we have Φ(1) +2 += +� +Φ(1) +1 +�T +, the game between the two users reduces to a 2 × 2 symmetric game, meaning the +equilibria can be found analytically. Figure 3, plots U(p∗ +1, p∗ +2) = p∗ +1 +TUp∗ +2, where (p∗ +1, p∗ +2) +maximize (19) as well as the payment to players t(p∗ +1, p∗ +2) = p∗ +1 +TΦ(1) +i p∗ +2 for a range of different +c values. + +13 +Fig. 3. Plot of optimal utility U and payments to users ti at (p∗ +1, p∗ +2) solving (19) (dashed lines) and at (˜p1, ˜p2) solving (21) +(solid lines) for a range of symmetric privacy levels c. +B. Fairness from Theorem 2 +In Section III-B1 we determined that the form of fair payments to the players satisfying (B.i-iii), +are parameterized by a matrix A ∈ A, where A = +� +A ∈ R2×2 : 0 ≤ [A]ij ≤ 1, A = AT� +. Thus, +as opposed to the previous example, the data acquisition problem (8), requires us to optimize +over the space of fair payments: +max +A ∈ A +pT +1 Up2 − pT +1 (A ⊙ U)p2 +s.t. +(p1, p2) ∈ NE(A). +(21) +Just as before, the fairness and symmetric privacy sensitivity means that the game between users +is a symmetric 2 × 2 matrix game and the equilibrium set is easily characterized analytically. +Figure 3, plots U(˜p1, ˜p2) = ˜pT +1 U˜p2, where (˜p1, ˜p2) maximize (21) as well as the payment to +players t(˜p1, ˜p2) = ˜pT +1 Φ(1) +i ˜p2 for a range of different c values. +C. Comparison +Fig. 3 plots the results of both fairness constrained data acquisition problems. First consider +the dashed lines, depicting the solution to (19). When c < 2/9, a utility of 1 is achieved, meaning +users choose ϵi = ∞ with probability 1. In this region users and the platform receive the same +amount of payment regardless of c. As c ≥ 2/9 the optimal equilibrium point is some mixed +strategy, and the utility begins to decrease. When c > 1/3 the privacy sensitivity becomes too + +14 +large and the payment become insufficient compared to the privacy cost, so the users choose +ϵi = 0 with probability 1. +The plots depicting the solution to (21) show that when c < 1/3, a utility of 1 is achieved, so +the achieves this maximum utility for a larger range of c. In contrast to the previous case, as +c increases, the amount that the players are paid increases. This is because the platform pays +the minimum possible amount to ensure users choose ϵi = 1 with probability 1. Note that the +platform forces the user utility to 0, since the payment is exactly c, which is also the privacy cost +the platform pays. For c > 1/3 the cost of paying the user to maintain ϵi = ∞ with probability +1 is too large, and it instead becomes optimal to pay users less, such that they choose some +optimal mixed strategy. Finally, for c > 2/3 the users choose ϵi = 0 with probability 1, as the +privacy loss becomes too big to compensate for. +One feature common in both cases is that the optimal solution is characterized by three +distinct regions. When c is small, users choose ϵi = ∞ with probability 1, followed by a region +where users choose a mixed strategy, and finally, when c is too large, users choose ϵi = 0 with +probability 1. In the appendix we discuss this phenomenon more, and show that if we have +symmetric privacy sensitivity and α(ϵ) in the statement of Theorem 2 is taken to be constant, +then this 3-region behavior can be proved for a class of utility functions. +V. AN APPLICATION IN FEDERATED LEARNING +FL is a distributed learning process used when data is either too large or too sensitive to +be directly transferred in full to the platform. Instead of combining all the data together and +learning at the platform, each user performs learning locally and the results are aggregated at the +platform. When training is done in this federated way, we can view this as an increased level of +privacy for the user, though perhaps in a less rigorous way than DP. In this section, we apply +our definition of fairness in the context of a federated mean estimation problem. +Let each user i ∈ [N] have some mean and variance (θi, σ2 +i ) ∼ Θ, where Θ is some global +joint distribution. Let t2 = Var(θi) and s2 = E[σ2 +i ]. User i draws n samples i.i.d. from its local +distribution Di(θi, σ2 +i ), that is, some distribution with mean θi and variance σ2 +i . The goal of the +platform is to construct estimators ˆθp +i that minimize the expected mean squared-error: +E +� +MSE(ˆθ +p, θ) +� +≜ +N +� +i=1 +E +�� +ˆθp +i − θi +�2� +. +(22) + +15 +We note the similarity of this formulation to that of [Donahue and Kleinberg, 2021], from which +we have drawn inspiration. +Fig. 4 summarizes our FL formulation. Users can choose from a 3-level privacy space +E = {0, 1, 2}. Let Nj be the number of users that choose privacy level j. When ϵi = 2, user i +provides its local estimator ˆθi directly to the platform. When ϵi = 1, user i’s local estimator is +securely aggregated with other users that choose this level such that +ˆθf = 1 +N1 +� +i:ϵi=1 +ˆθi, +(23) +and the platform receives access to ˆθf, rather than the local estimators. As before, ϵi = 0 means +user i chooses not to provide any information to the platform. Let the users be ordered such that +ϵi is a non-increasing sequence. Then for each i the platform constructs estimators of the form: +ˆθp +i = wi0ˆθf + +N2 +� +j=1 +wij ˆθj, +(24) +where, � +j wij = 1 for all i. In our model, from these estimators, the platform generates +utility U. In practice, for example, θi may represent some variable that describes what products +are marketable to user i, so a better estimate of this quantity will result in more profit from +advertisements served to that user. We can consider utilities of the form: U = c−E +� +MSE(ˆθ +p, θ) +� +. +In order to apply our definition of fairness, we must write U as a function of the privacy level +vector ϵ. The following lemma establishes that for a platform that chooses the optimal estimators +ˆθp +i it is possible to write U as a function of ϵ alone. +Lemma 1. For the platform that optimally chooses parameters wi0, and wij in (24), EMSE ≜ +E +� +MSE(ˆθ +p, θ) +� +can be written as a function of ϵ if ϵi > 0 for some i. If we define EMSE(0) ≜ +N(t2 + 2s2), then we can define: +U(ϵ) ≜ N(t2 + 2s2) − EMSE(ϵ), +(25) +which satisfies U(ϵ) ≥ 0. +The proof of Lemma 1 and a description of U can be found in the appendix. + +16 +A. Example Fair Payments +We now consider an example. Let there be N = 10 users. N1 = 5 of these users opt for +federating (ϵi = 1), N2 = 4 directly provide their data to the platform (ϵi = 2), and finally, +N0 = 1 users chooses to not participate (ϵi = 0). Each user has n = 20 samples, and t2 = 1 +and s2 = 10. Table I provides a breakdown of the division of utility when users are paid fairly +according to Theorem 1. In this case, the platform keeps roughly 64% of the total utility, with +the remaining 36% being distributed between the 9 users that choose a non-zero privacy level. +Among these users, users with privacy level ϵi = 2 earn roughly 1.6 times more than those that +choose ϵi = 1. Table II show the fair payment values according to Theorem 2, normalized by +Amount of Utility Paid +Total Utility +181.9 +User with ϵi = 0 +0 +User with ϵi = 1 +5.7 +User with ϵi = 2 +9.4 +Platform +115.9 +TABLE I +PAYMENT ALLOCATIONS FOR FAIRNESS BETWEEN ALL USERS AND PLATFORM. +α(ϵ), which is controlled by the platform. Interestingly in this case, users with ϵi = 2 earn only +roughly 1.4 times as much as users with ϵi = 1. +φi(ϵ)/α(ϵ) +User with ϵi = 0 +0 +User with ϵi = 1 +17.3 +User with ϵi = 2 +23.8 +TABLE II +PAYMENT WITH FAIRNESS AMONG USERS. +VI. CONCLUSION +This paper introduces two formal definitions of fair payments in the context of acquisition of +private data. The first treats the users and the platform together and uses axioms like those of the +Shapley value to determine a unique fair distribution of utility. In the second, we define a notion + +17 +Fig. 4. In the FL setting, users have a choice between three levels of privacy. If ϵi = 0, users send no data to the platform. +If ϵi = 1, a user’s model is securely combined with other users who also choose ϵi = 1, and the platform receives only the +combined model. If ϵi = 2, users send their model directly to the platform. +of fairness between the users only, leading to a definition of fairness that admits a range of values, +of which the platform is free to choose the most favorable. We consider examples of mechanisms +where both notions of fairness are enforced. While previous literature has investigated how +platforms should design incentives for users in order to optimize its utility, the definitions of +fairness we propose offers another important way to evaluate the fairness of these mechanisms. +This is a critical step towards future research in ensuring that data acquisition mechanisms are +both fair for users and efficient for platforms. +While we present results for fairness in a mechanism with N = 2 users with symmetric privacy +sensitivity, many open questions remain. For example, designing mechanisms that consider +fairness with heterogeneous privacy sensitivities, and the case of an arbitrary number of users +N are important question that remains, since in practice the platform interacts with large and +diverse groups of users. 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CoRR, abs/1709.02753. +[US Census Bureau, 2021] US +Census +Bureau +(2021). +Understanding +the +April +2021 +Demonstration +Data. +https://www2.census.gov/about/training-workshops/2021/2021-04-30-das-presentation.pdf. +[Wieringa et al., 2021] Wieringa, J., Kannan, P., Ma, X., Reutterer, T., Risselada, H., and Skiera, B. (2021). Data analytics in a +privacy-concerned world. Journal of Business Research, 122:915–925. +APPENDIX +A. Proof of Equation (6) +In this section, we present the calculations required to arrive at the utility values in (6). First +let’s treat the trivial case of ϵ1 = 0, ϵ2 = 0. The optimal ϵ-DP estimator is simply the optimal +Bayes estimator with no data, i.e., the prior mean. Let us define this estimator as ˆµ(0,0) = 0. Its +risk function is +R(µ, ˆµ(0,0)) = E +� +L(ˆµ(0,0), µ) | µ +� += µ2. +(26) +The Bayes risk of ˆµ(0,0) is the expectation of this quantity taken using our prior: +r([0, 0]) = E +� +µ2� += 1 +12. +(27) +Next, consider the case where user i chooses privacy level ϵ1 = ϵ′ > 0, and the other user chooses +ϵ2 = 0. In this case the estimator depends on X1, ˆµ(ϵ′,0) = w1X1 + Z. Then the risk function is: +R(µ, ˆµ(ϵ′,0)) = E +� +(w1X1 + Z − µ)2 | µ +� += +� +µ + 1 +2 +� � +µ − w1 +2 +�2 ++ +� +−µ + 1 +2 +� � +µ + w1 +2 +�2 ++ 2 +η2. +(28) + +20 +Now taking the expectation with respect to our prior over µ, we have: +E +� +R(µ, ˆµ(ϵ′,0)) +� += 1 +12 +� +3w2 +1 − 2w1 + 1 +� ++ 2 +η2, +(29) +here η is the inverse scale parameter for Z. Note that (29) is minimized when η is maximized. +The ϵ-DP condition enforces the constraint η ≤ +ϵ′ +w1. This constraint will be met with equality for +the optimal w1. The optimal w∗ +1 = +1 +3+ 24 +ϵ′2 . Thus, we have: +ˆµ(ϵ′,0) = +1 +3 + 24 +ϵ′2 +X1 + Z, +Z ∼ Laplace +� +ϵ′ +3ϵ′2 + 24 +� +, +(30) +and the resulting Bayes risk is: +r([ϵ′, 0]) = r([0, ϵ′]) = 1 +12 +� +1 − +1 +3 + 24 +ϵ′2 +� +. +(31) +For the case with ϵ1 = ϵ2 = ϵ′ we can repeat the same process by defining ˆµ(ϵ′,ϵ′) = w1X1 + +w2X2 + Z. By symmetry, we must have w1 = w2, so we drop the index. Then the risk function +and its expectation are: +R(µ, ˆµ(ϵ′,ϵ′)) = 2 +� +µ + 1 +2 +� � +−µ + 1 +2 +� +µ2 + +� +µ + 1 +2 +�2 +(w − µ)2 + +� +−µ + 1 +2 +�2 +(µ + w)2 + 2 +η +2 +(32) +E +� +R(µ, ˆµ(ϵ′,ϵ′)) +� += 1 +12(8w2 − 4w + 1) + 2 +η2. +(33) +By a similar argument to the previous case, the Bayes optimal estimator and the corresponding +Bayes risk is: +ˆµ(ϵ′,ϵ′) = +1 +4 + 12 +ϵ′2 +(X1 + X2) + Z, +Z ∼ Laplace +� +ϵ′ +4ϵ′2 + 12 +� +, +(34) +r([ϵ′, ϵ′]) = 1 +12 +� +1 − +1 +2 + +6 +ϵ′2 +� +. +(35) +Finally letting U(ϵ) = c1r(ϵ) + c2. Take U(0) = 0 =⇒ c1 = −12c2. And maxϵ U(ϵ) = 1 =⇒ +c1 = 24(1 − c2). Simplifying gives us our desired result. + +21 +B. Proof of Theorem 1 and Theorem 2 +We will begin with the proof of Theorem 2, which is standard and follows the typical proof of +the Shapley value. We begin by proving φi(ϵ) as defined in (17) satisfies axioms (B.i-iii). First +assume U(ϵS∪{i}) = U(ϵS∪{j}) ∀S ⊂ [N]\{i, j}, then: +φi(ϵ) = α(ϵ) +N +� +S⊆[N]\{i} +U(ϵS∪{i}) − U(ϵS) +�N−1 +|S| +� +(36) += α(ϵ) +N +� +� +� +S⊆[N]\{i,j} +U(ϵS∪{i}) − U(ϵS) +�N−1 +|S| +� ++ +� +S⊆[N]\{i,j} +� +U(ϵS∪{j}∪{i}) − U(ϵS∪{j}) +� +� N−1 +|S|+1 +� +� +�(37) += α(ϵ) +N +� +� +� +S⊆[N]\{i,j} +U(ϵS∪{j}) − U(ϵS) +�N−1 +|S| +� ++ +� +S⊆[N]\{i,j} +� +U(ϵS∪{i}∪{j}) − U(ϵS∪{i}) +� +� N−1 +|S|+1 +� +� +�(38) += φj(ϵ), +(39) +proving axiom (B.i) is satisfied. For the proof that axiom (B.ii) is satisfied, we write: +� +i +φi(ϵ) = α(ϵ) +N +� +i +� +S⊆[N]\{i} +U(ϵS∪{i}) − U(ϵS) +�N−1 +|S| +� +(40) += α(ϵ) +N +� +�� +i +� +S⊆[N]\{i} +U(ϵS∪{i}) +�N−1 +|S| +� +− +� +i +� +S⊆[N]\{i} +U(ϵS) +�N−1 +|S| +� +� +� +(41) += α(ϵ)U(ϵ) + α(ϵ) +N +� +� +� +� +� +i +� +S⊆[N]\{i} +|S| N1 + N2, we have vij = 0. Thus the first term can be written as: +t2 +n +N +� +j=1 +v2 +ij = t2 +n +� N2 +� +j=1 +wij + +N2+N1 +� +j=N2+1 +�wi0 +N1 +�2� +(55) += t2 +n +� N2 +� +j=1 +w2 +ij + 1 +N1 +w2 +i0 +� +. +(56) +Making these same substitutions to � +j̸=i v2 +ij and � +j̸=i vij yields the desired result. +Proposition 2. The error expression (53) is minimized if ϵi = 0 with weights: +wi0 = +N1 +N1 + N2 +, wij = +1 +N1 + N2 +. +(57) +If ϵi = 1 (53) is minimized by: +wi0 = +N1 +N1 + N2 ++ +N2 +N1 + N2 +s2 +V , +(58) + +24 +wij = +1 +N1 + N2 +− +1 +N1 + N2 +s2 +V +(59) +where V = s2 + t2 +n . Finally, if ϵi = 2, (53) is minimized by: +wi0 = +N1 +N1 + N2 +− +N1 +N1 + N2 +s2 +V , +(60) +wij = +1 +N1 + N2 +− +1 +N1 + N2 +s2 +V +(61) +wii = +1 +N1 + N2 ++ N1 + N2 − 1 +N1 + N2 +s2 +V +(62) +Proof. First we will consider the case where ϵi = 1. Considering the point where the derivative +of (53) with respect to wik, k ≥ 1 is equal to zero gives: +2t2 +n wik− 2t2 +nN1 +� +1 − +N2 +� +j=1 +wij +� ++s2 +� +2wik − 2N1 − 1 +N 2 +1 +� +1 − +N2 +� +j=1 +wij +� ++ 2 +N 2 +1 +� +N1 − 1 + +N2 +� +j=1 +wij +�� += 0, +(63) +�t2 +n + s2 +� +wik = +�t2 +n + s2 +� wi0 +N1 +− s2 +N1 +. +(64) +It is easily verified from the second derivative that solving this equation gives us the unique +minimum of (53). For ease of notation, define V ≜ +� +t2 +n + s2� +. Thus, we have: +wik = +V wi0 +N1 − s2 +N1 +V +. +(65) +Noting that wi0 + �N2 +j=1 wij = 1, we have: +wi0 + N2 +N1 +wi0 − N2 +N1 +s2 +V = 1, +(66) +wi0 = +N1 +N1 + N2 ++ +N2 +N1 + N2 +s2 +V , +(67) +wij = +1 +N1 + N2 +− +1 +N1 + N2 +s2 +V . +(68) + +25 +This completes the proof for those users i such that ϵi = 1. The case of ϵi = 2 can be found in +Lemma 7.1 of [Donahue and Kleinberg, 2021], and ϵi = 0 can be proved from a straightforward, +simpler version of the proof above. +D. N-User Uniform Privacy Sensitivity +Consider a setup with N > 2 users, each with privacy sensitivity c and identical statistical +marginal contribution, i.e., for any S ⊆ [N]\{i, j}, U(ϵS∪{i}) = U(ϵS∪{j}). The platform is +restricted to making fair payments satisfying axioms (B.i-iii) with the additional constraint that +α(ϵ) = α ∈ [0, 1] is a constant with respect to ϵ. Let p = [p, (1 − p)]T represent a mixed-strategy. +We can write the utility of the user i as +u(si, s−i) = αφ(si; s−i) − c1 {si = ϵ′ +2} , +(69) +where s ∈ EN, with E = {ϵ′ +1, ϵ′ +2} with ϵ′ +2 > ϵ′ +1. Note that we can drop the index of φi due to +the assumption of equal marginal contribution. It is helpful to define the expected player utility, +where the expectation is taken with respect to the actions of the other players. That is, the +expected utility of user i, when other users play the symmetric strategy p is: +u(si, p) = αφ(si; p) − c1 {si = ϵi} = αEsj∼p +j̸=i [φ(si; s−i)] − c1 {si = ϵ′ +2} . +(70) +The symmetric Nash equilibrium of a such a game is characterized by [Cheng et al., 2004] to be +the the minimizers of +min +p +� +s∈E +[u(s, p) − u(p, p)]2 ++ , +(71) +where u(p, p) = Es∼p [u(s, p)]. Since our action space is binary, there are only two terms in this +sum. Applying the definition of u and writing out both terms of this sum yields: +� +s∈E +[u(s, p) − u(p, p)]2 ++ = [u(ϵ1, p) − u(p, p)]2 ++ + [u(ϵ2, p) − u(p, p)]2 ++ +(72) += [c(1 − p) − α(φ(p, p) − φ(ϵ1, p))]2 ++ + [c(1 − p) − α(φ(p, p) − φ(ϵ2, p))]2 ++ +(73) += [(1 − p)(c − αγ(p))]2 ++ + [−p(c − αγ(p))]2 ++ , +(74) +where we define γ(p) ≜ φ(ϵ2, p) − φ(ϵ1, p). γ is an important quantity in this problem that +described the relative increase in payment a user receives for choosing a higher privacy level +when the other users choose mixed strategy p. In general to say something about the equilibria, + +26 +we must make some assumptions about γ. Two reasonable assumptions are γ(p) ≥ 0 and +γ′(p) ≥ 0. γ(p) ≥ 0 implies that choosing a lower level of privacy should increase φ(·, p), while +γ′(p) ≥ 0 implies that as p increases, (more users choose a higher level of privacy on average), +γ(p) increases. Making these assumptions, we can calculate p∗ for three distinct cases. Defining +γmax ≜ maxp γ(p) and γmin = minp γ(p), we have: +a) Case 1:: c − αγmax > 0: +� +s∈E +[u(s, p) − u(p, p)]2 ++ = [(1 − p)(c − αγ(p))]2 ++ +(75) +Since this quantity is non-negative, it is clearly minimized when p∗ = 1, where it is exactly 0. +Furthermore, since c − αγmax > 0 is satisfied with strict inequality, it is the unique minimizer. +b) Case 2:: c/α ∈ [γmin, γmax]: +� +s∈E +[u(s, p) − u(p, p)]2 ++ = [(1 − p)(c − αγ(p))]2 ++ + [−p(c − αγ(p))]2 ++ , +(76) +In the above case, this is minimized when p∗ ∈ γ−1(c/α). +c) Case 3:: c − αγmin < 0: +� +s∈E +[u(s, p) − u(p, p)]2 ++ = [−p(c − αγ(p))]2 ++ , +(77) +In the above case, the expression is minimized when p∗ = 0. To summarize, we have: +p∗(α) = +� +� +� +� +� +� +� +� +� +� +� +1 +if α < +c +γmax +γ−1(c/α) +if α ∈ [ +c +γmax, +c +γmin] +0 +if α > +c +γmin +. +(78) +We may consider a platform that solves the following problem, where we define U(p) ≜ +Eϵi∼p [U(ϵ)]: +min +α (1 − α)U(p∗(α)), +(79) +which is analogous to (8), but with the fairness constraint and optimization over symmetric NEs. +If U is decreasing in p, which is reasonable since increasing p corresponds to a higher average +user privacy level, we can conclude that the optimal α∗, has 3 phases: (1) a region where c is +small, and α∗ is the smallest α such that p∗ = 0, (2) an intermediate region where a symmetric +mixed strategy is played, and (3) a region where c is too large, and α∗ = 0, p∗ = 1. + +27 +E. Simulations With N = 2 Users Asymmetric Privacy Sensitivity +In this section, we provide further discussion to the setting of Section IV. In that section, we +focused on solving a problem where the platform chooses payments to users to maximize its +objective under the constraint of fairness as in (21). We now discuss the case where privacy +sensitivities c1 ̸= c2. In particular, we focus on fairness as defined by the axioms (B.i-iii) in +Theorem 2 and present plots of the payments when the platform solves the problem: +max +A ∈ A +pT +1 Up2 − pT +1 (A ⊙ U)p2 +s.t. +(p1, p2) ∈ NE(A). +(80) +This problem differs from (21) only in the fact that the equilibrium is governed by different user +utility functions that make the game asymmetric: +u1(p1, p2) = pT +1 Φ(2) +1 p2 − [0 c1]Tp1. +(81) +u2(p1, p2) = pT +1 Φ(2) +2 p2 − [0 c2]Tp2. +(82) +Fig. 5 plots the results of this simulation. It shows that there is one region when c1 and c2 are +both small and close together (< 1/3), the platform chooses A to collect data from both users. +If the difference is large, even in this region, the users may be asymmetrically engaged. When +c1 > c2 > 1/3, the platform chooses A such that only user 2 chooses to participate, even if +the difference is very small, and vice versa if c2 > c1 > 1/3, as before, when c1, c2 > 2/3 the +sensitivity to too high and the platform can no longer offer enough payment to the users. +F. Additional Numerical Examples in Federated Learning +Recall the FL example of Section V. In this section, we present an additional numerical +example experiment that could not be included in the main manuscript due to space constraints. +Let there be N = 10 users. N1 = 2 of these users opt for federating (ϵi = 1), N2 = 2 directly +provide their data to the platform (ϵi = 2), and finally, N0 = 6 users chooses to not participate +(ϵi = 0). Each user has n = 20 samples, and t2 = 1 and s2 = 10. The fair values for this +configuration are computed in Table III. Now consider the case where N1 = 8, N2 = 2 and +N0 = 0. The fair values for this configuration are computed in Table IV. + +28 +Fig. 5. (Left) The payments to User 2 from the platform for a range of c1, c2. (Right) The objective of the platform for the +optimal A∗ payments for a range of values c1, c2. +φi(ϵ)/α(ϵ) +User with ϵi = 0 +0 +User with ϵi = 1 +28.2 +User with ϵi = 2 +32.1 +Amount of Utility Paid +Total Utility +120.6 +User with ϵi = 0 +0 +User with ϵi = 1 +12.0 +User with ϵi = 2 +14.6 +Platform +115.9 +TABLE III +PAYMENT ALLOCATIONS FOR BOTH NOTIONS OF FAIRNESS (THEOREM 2 ON THE LEFT AND THEOREM 1 IN THE RIGHT), WITH +N2 = 2, N1 = 2, AND N0 = 6 +When N1 = 2, there is less privacy advantage to federating, since the if a user were to choose +this option, their data would be federated with only a small number of users. Interestingly, our +notion of fairness accounts for this. When N1 = 2 and N2 = 2 Table III indicates that the value +of users with ϵi = 1 and ϵi = 2 are very similar. This is reasonable, because with only 2 users +federating, there is little difference between the two privacy options. In contrast, when N1 = 8, +the difference between the two privacy levels is more significant. This is reflected in the relative +value of the data, as shown in Table IV, where the value of users with ϵi = 1 drops significantly +compared to the case where ϵi = 2. The value of those users with ϵi = 2 also goes down, due to +the fact there much more data is available to the platform, so its average marginal contribution +decreases. + +0.9 +0.6 +0.8 +0.5 +0.7 +0.6 +0.4 +0.5 +0.3 +0.4 +0.3 +0.2 +0.2 +0.1 +0.1 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +C10.9 +0.9 +0.8 +0.8 +0.7 +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +C129 +φi(ϵ)/α(ϵ) +User with ϵi = 1 +14.0 +User with ϵi = 2 +25.7 +Amount of Utility Paid +Total Utility +149.4 +User with ϵi = 1 +4.1 +User with ϵi = 2 +11.1 +Platform +98.6 +TABLE IV +PAYMENT ALLOCATIONS FOR BOTH NOTIONS OF FAIRNESS (THEOREM 2 ON THE LEFT AND THEOREM 1 IN THE RIGHT), WITH +N2 = 2, N1 = 8, AND N0 = 0 + diff --git a/ZNFQT4oBgHgl3EQfejY8/content/tmp_files/load_file.txt b/ZNFQT4oBgHgl3EQfejY8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33a78302bcd8500242e0b1e722818654aa627581 --- /dev/null +++ b/ZNFQT4oBgHgl3EQfejY8/content/tmp_files/load_file.txt @@ -0,0 +1,969 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf,len=968 +page_content='1 The Fair Value of Data Under Heterogeneous Privacy Constraints Justin Kang, Ramtin Pedarsani, Kannan Ramchandran Abstract Modern data aggregation often takes the form of a platform collecting data from a network of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' More than ever, these users are now requesting that the data they provide is protected with a guarantee of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This has led to the study of optimal data acquisition frameworks, where the optimality criterion is typically the maximization of utility for the agent trying to acquire the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This involves determining how to allocate payments to users for the purchase of their data at various privacy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The main goal of this paper is to characterize a fair amount to pay users for their data at a given privacy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We propose an axiomatic definition of fairness, analogous to the celebrated Shapley value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Two concepts for fairness are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The first treats the platform and users as members of a common coalition and provides a complete description of how to divide the utility among the platform and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In the second concept, fairness is defined only among users, leading to a potential fairness-constrained mechanism design problem for the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We consider explicit examples involving private heterogeneous data and show how these notions of fairness can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' To the best of our knowledge, these are the first fairness concepts for data that explicitly consider privacy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' INTRODUCTION From media to healthcare to transportation, the vast amount of data generated by people living their everyday lives has been used to great effect to solve difficult problems across many domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' For example, nearly all machine learning algorithms, including those based on deep learning rely heavily on data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Many of the largest companies to ever exist center their business around the precious resource of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This includes directly selling access to data to others for profit, selling targeted advertisements based on data, or by exploiting data through data-driven engineering, to better develop and market products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In this sense, data has many of the features of other commodities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' It is a raw material that is processed and exploited to generate products of greater value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Unlike other commodities, however, modern data markets are largely informal, where in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='13336v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='LG] 30 Jan 2023 2 many cases individuals sell their data in exchange for access to services, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Due to the lack of a formal exchange, it can be difficult to understand if such an exchange is fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Unlike a barrel of oil, one cannot simply put a fair price on a gigabyte of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The development of an economic theory for the value of data is still nascent [Ghorbani and Zou, 2019], [Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2019], [Acemoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2019], and the dynamics of formal data markets are largely not understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A major shortcoming of the current understanding of data value is that in many cases, it fails to explicitly consider a critical factor in an individual’s decision to share data—privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This work puts forth two rigorous notions of the fair value of data in Section III that explicitly include privacy, which has become as important area of concern as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' With the rise of large-scale data collection, many high-profile data breaches have occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Consumers, governments and online platforms have become aware of the negative aspects of the way data is used and the lack of privacy, which has led to an array of responses from stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' For example, there is significant discourse among consumers about the addictive properties of content serving algorithms powered by their data [Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Governments have set regulations such as the General Data Protection Regulation (GDPR) [European Union, 2016] of the European Union, one of the first pieces of legislation aimed at regulating online platforms to protect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Platforms that collect and utilize data have pursued frameworks for privacy such as Federated Learning (FL) [Kairouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2021], proposed and implemented by Google [McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2017] and Differential Privacy (DP) [Dwork, 2008], which is used by Apple in their operating systems [Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2017] and recently in the US 2020 Census [US Census Bureau, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' These approaches allow platforms to utilize distributed data from individuals while providing a level of privacy to the individuals that participate, potentially at the cost of decreased data utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Compelled by the importance of data in our modern economy and a growing social concern about privacy, this paper presents frameworks for quantifying the fair value of private data to better understand the relationship between those that generate data and the platforms that collect and benefit from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' More specifically, we consider a setting where users are willing to provide their data to a platform in exchange for some sort of payment and under some privacy guarantees depending on their level of privacy requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The platform is responsible to run the private learning algorithm on the gathered data and make the fair payments with the objective of maximizing its utility including statistical accuracy and total amount of payments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Our goal is to design a fair mechanism for this procedure as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Diagram depiction of interactions between platform and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Users generate data by using devices like phones, cameras, autonomous vehicles and drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' They provide this data to the platform, potentially requesting some level of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The platform uses this data to generate utility, often by using the data for learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In return, the platform may provide the users with payments in the form of access to services, discounts on products, or monetary compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Related Work With the widespread use of the internet and data-driven methods, interactions involving those that have data and those that seek to acquire it have become an important area of theoretical study [Balazinska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2011], but also a practical necessity [Spiekermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2015b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Among these interactions, the economics of data from privacy conscious users has received significant attention [Acquisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2016], [Wieringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' DP [Dwork, 2008] and its variations, [Bun and Steinke, 2016] are widely studied as a formal framework for privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' [Ghosh and Roth, 2015] studies the purchase of private data, where privacy is quantified under DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' [Ghosh and Roth, 2015] assumes that each player has binary data and its own heterogeneous privacy sensitivity parameter that they report, potentially strategically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The goal is to design a dominant strategy truthful mechanism to acquire data and estimate the sum of users’ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In [Fallah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2022], the authors consider an optimal data acquisition problem in the context of private mean estimation in two different heterogeneous DP settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' It is assumed that players consider both the estimation error of the common estimator generated by the platform and any payments made to them by the platform in their decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' By assuming that the privacy sensitivities are represented by scalars drawn from a distribution, they devise a mechanism for computing the optimal privacy levels to provide to the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In [Hu and Gong, 2020] the federated learning setting is considered, where each play 4 has a unique privacy sensitivity function parameterized by a scalar variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Players report their sensitivity parameter and the platform assigns each user a privacy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A heuristic proportional payment mechanism is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In the case of linear privacy sensitivity functions, a computationally efficient way to compute the Nash Equilibrium is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In [Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2020], a multi-stage data market is studied where data-brokers acquire data from users, competing to sell data to a platform that further sells services based on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The economic and social implications of privacy and data markets are considered in [Spiekermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2015a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In [Acemoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2019] the impact of data externalities is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In particular the leakage of data leading to the suppression of its market value is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In [Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2019], [Ghorbani and Zou, 2019] and [Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2020] a framework for determining the fair value of data is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' These works extend the foundational principles of the Shapley Value [Shapley, 1952], which was originally proposed as a concept for utility division in coalitional games to the setting of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Our work takes this idea further and explicitly includes privacy in the definition of the fair value of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Finally, we note that we consider the concept of fairness in data valuation, not algorithmic fairness, which relates to the systematic failure of machine learning systems to account for data imbalances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Main Contributions The main contribution of this work is in the development of a rigorous notion of fairness in the context of user data acquisition with privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' While the existing literature has investigated how a platform should design incentives for users in order to optimize its utility, the definitions of fairness that we propose in this work offer another way to evaluate these mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We summarize the main contributions as follows: We present an axiomatic notion of fairness that is inclusive of the platforms and the users in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The amount of utility that should be awarded to each user and the platform is uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This unique allocation can be used as a benchmark to determine how fair an optimal data acquisition procedure is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In the realistic scenario that fairness is only considered between users, Theorem 2 defines a notion of fairness based on axioms, but only places restriction on the relative amount of utility distributed to the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This creates an opportunity for the platform to optimize its utility under a constraint of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 5 We provide two important applications of our framework: differentially private algorithms and federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We fully characterize the mechanism design problem in a two-user symmetric binary privacy game in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We further demonstrate an application of our framework in the problem of privacy-preserving federated mean estimation in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Notation Lowercase boldface x and uppercase boldface X symbols denote vectors and matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' R+ denotes the set of non-negative reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In addition, we take R+ = R+ ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The notation [N] represents the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' NE(·) represents the set of Nash Equilibrium actions of players, potentially parameterized by an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' PROBLEM SETTING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Privacy Levels and Utility Functions Consider the setting depicted by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 2, where a platform collects data xi ∈ X and privacy levels ϵi ∈ E from each user (player) i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We restrict the space of privacy levels to E ⊆ R+, adopting the notation of DP, where ϵi = 0 means the data cannot be used by the platform (full privacy), and ϵi = ∞ means no restriction on data usage (no privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' If ϵi > ϵj, we say ϵi is a lower privacy level than ϵj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The platform applies an ϵ-private algorithm Aϵ : X N �→ Y to process the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' An ϵ-private algorithm is one that provides privacy level ϵi to data xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In the privacy literature, Aϵ is known as a private mechanism, but in this work, we avoid this terminology to reduce confusion with the game-theoretic notion of mechanism design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The output of the algorithm y = Aϵ(x) is used by the platform to derive utility U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In this work, we describe the utility as a function of privacy level U(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' For example, if the platform is trying to estimate a mean from a population, as may be the case in a census, the utility could depend on the mean square-error of the private estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This is a valid model if the platform is concerned about the statistical performance of the algorithm at the given privacy level, such as risk minimization in a learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' U(ϵ) then does not depend directly on xi, which we treat as measurements from some trusted system that cannot be strategically altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We make no assumptions about the mechanism Aϵ, but in many cases it is reasonable to assume that U(ϵ) is an increasing monotone function in each coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Note that this formulation differs from typical formulations in the literature of optimal data acquisition, where some privacy sensitivity is instead reported by users, and the platform then 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Depiction of interactions between the platform and individual players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Players send their data xi and a privacy level ϵi to the central platform in exchange for payments ti(ϵi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' ϵ−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The central platform extracts utility from the data at a given privacy level and optimizes incentives to maximize the difference between the utility and the sum of payments U(ϵ) − 1T t(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' chooses the privacy level ϵi based on this sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This typical formulation is beneficial in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' It allows for the relatively straightforward application of notions such as incentive compatibility and individual rationality from mechanism design theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In this work, however, we wish to emphasize the fact that the utility U depends on the privacy levels ϵ directly, so considering actions in the space of privacy levels E is natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Despite this difference, the notions of fairness described in the following section can be applied more broadly, so long as U is a function of the privacy constraints ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 1) Example: In the following sections we will consider a common example to elucidate the application of our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let Xi represent the independent and identically distributed data of user i respectively, with Pr(Xi = 1/2) = p and Pr(Xi = −1/2) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Assume that the platform has the prior p ∼ Unif(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' As our formal notion of privacy, we will consider a pure ϵ-DP framework, defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A random function A : X N → Y is ϵi-DP, ϵi ∈ R+ in coordinate i if for any x′ ∈ X N that differs from x ∈ X N only in coordinate i, for all measurable sets S ∈ Y we have: Pr(A(x) ∈ S) ≤ eϵiPr(A(x′) ∈ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (1) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A random function A : X N → Y is ϵ-DP if A is ϵi-DP in coordinate i for all i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The goal of the platform is to construct an ϵ-DP estimator for µ ≜ E[Xi] = p − 1/2 that 7 minimizes the Bayes risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A general procedure for finding the Bayes optimal ϵ-DP estimator does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We restrict our attention to ϵ-DP linear-Laplace estimators of the form: A(X) = w(ϵ)TX + Z, (2) where Z ∼ Laplace(1/η(ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In [Fallah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2022] the authors argue that unbiased linear estimators are nearly optimal in a minimax sense for bounded random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We assume a squared error loss L(a, µ) = (a − µ)2 and let A(ϵ) be the set of ϵ-DP estimators satisfying (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Then, we define: Aϵ = arg min A∈A(ϵ) E[L(A(X), µ)] (3) r(ϵ) = E[L(Aϵ(X), µ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (4) In words, Aϵ is an ϵ-DP estimator of the form (2), where w(ϵ) and η(ϵ) are chosen to minimize the Bayes risk of the estimator, and r(ϵ) is the risk achieved by Aϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Since the platform’s goal is to accurately estimate the mean of the data, it is natural for the utility U(ϵ) to depend on ϵ through the risk function r(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Note that if U is monotone decreasing in r(ϵ), then U is monotone increasing in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let us now consider the case of N = 2 users, choosing from an action space of E = {0, ϵ′}, for some ϵ′ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Furthermore, take U to be an affine function of r(ϵ): U(ϵ) = c1r(ϵ) + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' For concreteness, take U(0) = 0 and maxϵ∈R U(ϵ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Note that this ensures that U is monotone increasing in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The utility corresponding to possible user actions can be succinctly represented in matrix form as: U = � �U([0, 0]T) U([0, ϵ′]T) U([ϵ′, 0]T) U([ϵ′, ϵ′]T) � � (5) = � � 0 2 � 3 + 24 (ϵ′)2 �−1 2 � 3 + 24 (ϵ′)2 �−1 � 1 + 3 (ϵ′)2 �−1 � � (6) Details of the above calculations can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The Data Acquisition Problem Central to this work is the payments users receive for providing their data to a platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' As described previously, the platform generates utility U(ϵ) from the user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We assume this 8 utility is transferable and divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In exchange for data, the platform distributes some portion of the utility ti(ϵi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' ϵ−i) to user i, where ϵ−i denotes the vector of privacy levels ϵ with the ith coordinate deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Note that we could also write ti(ϵ), but we choose the former because it makes explicit which parameter user i has control over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' These incentives in turn may motivate users to lower their privacy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The behavior of users can also be modelled with the help of a utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In general, each user will have some sensitivity to their data being shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This can be modelled by a sensitivity function ci : E → R+, ci(0) = 0 resulting in the utility: ui(ϵ) = ti(ϵi, ϵ−i) − ci(ϵi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (7) The payment user i receives from the platform will tend to increase with a lower privacy level, as the platform is able to better exploit the data, while their sensitivity will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Thus, by specifying a set of ti(ϵi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' ϵ−i), the platform effectively creates a game among the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Each user’s action in this game is the level of privacy that they request for the data they share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Users (players) select their privacy level ϵi by considering their utility function ui and the potential actions of the other players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' From the perspective of the platform, the goal is to design the payments ti(ϵi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' ϵ−i) such that it maximizes the difference between the utility it receives and the payments made to the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' One way to formulate this problem is to consider maximizing this difference at equilibrium points: max t(·), P U(P) − 1Tt(P) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' P ∈ NE(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (8) In (8), NE(t) denotes the set of Nash Equilibrium strategies induced by the payment function t, which is the vector with payment function ti at index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Recall that the Nash Equilibrium is a stable state of a system such that no user can gain by a unilateral change of strategy if the strategies of the other users remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We allow these equilibrium points to be mixed strategies over the probability space, such that P represents a distribution over the privacy space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In addition, we have used the shorthand f(P) = Eϵ∼P [f(ϵ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Note that in order to solve (8), the platform requires knowledge of the privacy sensitivity ci of each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This can be a reasonable assumption when the platform has interacted with the users many times in the past and has learned ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Note that some restrictions must be placed on t, as it can otherwise be made arbitrarily negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 9 Individual rationality is common condition in mechanism design that says that a user can be made no worse off by participating in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' As a final comment, we note that the compensation ti(ϵi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' ϵ−i) may not be a direct monetary transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Individuals are often compensated for data through discounts or access to services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A shortcoming of our model is that we assume a divisible and transferable utility, which may fail to capture these nuances of compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' AXIOMATIC FAIRNESS WITH PRIVACY Somewhat in contrast to the resource allocation view just described, we can also view the interaction between users and the platform as a coalition where many users and the platform come together and pool their resources to generate utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A natural question to ask is: How should the utility be divided fairly among members of this coalition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The answer to this question turns out to be connected to the celebrated Shapley value [Shapley, 1952].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Shapley value is one of the most important normative utility division schemes for coalitional games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Following an axiomatic approach to fairness, the Shapley value describes how to fairly divide utility between members of a coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In the following section we develop an axiomatic approach to defining fairness analogous to the Shapley value for the context of users providing private data to platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Platform as a Coalition Member We define a coalition of users and a platform as a collection of s users, with 0 ≤ s ≤ N and up to 1 platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let a ∈ {0, 1} represent the action of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let a = 1 when the platform chooses to join the coalition, and a = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let U(ϵ) be as defined in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We augment the utility to include the platform as follows: U(a, ϵ) ≜ � � � � � U(ϵ) a = 1 0 a = 0 , (9) to indicate that no utility is generated if the platform does not participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We also define ϵS such that: [ϵS]i = � � � � � ϵi i ∈ S 0 else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (10) 10 Let φp(a, ϵ) and φi(a, ϵ), i ∈ [N] represent the “fair” amount of utility awarded to the platform and each user i respectively, given a and ϵ, otherwise described as the “value” of a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Note that these values depend implicitly on both the private algorithm Aϵ and the utility function U, but for brevity, we avoid writing this dependence explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The result of [Hart and Mas-Colell, 1989] show that these values are unique and well defined if they satisfy the following three axioms: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i) (Fairness) For any two users i, j ∈ [N]: U(a, ϵS∪{i}) = U(a, ϵS∪{j}) ∀S ⊂ [N]\\{i, j} =⇒ φi(a, ϵ) = φj(a, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (11) In addition, for any user i ∈ [N], U(1, ϵS∪{i}) − U(1, ϵS) = 0 ∀S ⊂ [N]\\{i} =⇒ φi(a, ϵ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='ii) (Efficiency) The sum of values is the total utility U(a, ϵ) = φp(a, ϵ) + � i φi(a, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='iii) (Additivity) Let φp(a, ϵ) and φi(a, ϵ) be the value of the platform and users respectively for the utility function U, under the ϵ-private Aϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let V be a separate utility function, also based on the output of Aϵ, and let φ′ p(a, ϵ) and φ′ i(a, ϵ) be the utility of the platform and individuals with respect to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Then under the utility U + V , the value of user i is φi(a, ϵ) + φ′ i(a, ϵ) and the value of the platform is φp(a, ϵ) + φ′ p(a, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let φp(a, ϵ) and φi(a, ϵ) satisfying axioms (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i-iii) represent the portion of total utility awarded to the platform and each user i from utility U(a, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Then they are unique and take the form: φp(a, ϵ) = 1 N + 1 � S⊆[N] 1 � N |S| �U(a, ϵS), (12) φi(a, ϵ) = 1 N + 1 � S⊆[N]\\{i} 1 � N |S|+1 � � U(a, ϵS∪{i}) − U(a, ϵS) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (13) The Proof of Theorem 1 can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 1) Example: Consider the example from Section II-A1 with binary privacy space E = {0, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' By (5), the utility can be written in matrix form as: U = � � 0 2/3 2/3 1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (14) 11 Note from (12) and (13), it is clear that φp(0, ϵ) = φi(0, ϵ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let Φp and Φ(1) i represent the functions φp(1, ϵ) and φi(1, ϵ) in matrix form akin to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Then using (12) and (13), we find that the fair allocations of the utility are given by: Φp = � � 0 1/3 1/3 5/9 � � , Φ(1) 1 = � � 0 1/3 0 2/9 � � , Φ(1) 2 = � � 0 0 1/3 2/9 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (15) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Fairness Among Users Though we can view the interactions between the platform and the users as a coalition, due to the asymmetry that exists between the platform and the users, it also makes sense to discuss fairness among the users alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In this case, we can consider an analogous set of axioms that involve only the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i) (Fairness) For any two users i, j ∈ [N]: U(ϵS∪{i}) = U(ϵS∪{j}) ∀S ⊂ [N]\\{i, j} =⇒ φi(ϵ) = φj(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (16) In addition, for any user i ∈ [N], U(ϵS∪{i}) − U(ϵS) = 0 ∀S ⊂ [N]\\{i} =⇒ φi(ϵ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='ii) (Pseudo-Efficiency) The sum of values is the total utility α(ϵ)U(ϵ) = � i φi(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Where if U(ϵ) = U(˜ϵ) then α(ϵ) = α(˜ϵ) and 0 ≤ α(ϵ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='iii) (Additivity) Let φi(ϵ) be the value of users for the utility function U, under the ϵ-private algorithm Aϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let V be a separate utility function, also based on the output of the algorithm Aϵ, and let φ′ i(ϵ) be the utility of the users with respect to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Then under the utility U + V , the value of user i is φi(ϵ) + φ′ i(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The most notable difference between these axioms and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i-iii) is that the efficiency condition is replaced with a pseudo-efficiency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Under this condition, the platform may determine the sum of payments awarded to the players, but this condition should in general depend only on the utility itself, and not on how that utility is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let φi(ϵ) satisfying axioms (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i-iii) represent the portion of total utility awarded to each user i from utility U(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Then they must take the form: φi(ϵ) = α(ϵ) N � S⊆[N]\\{i} 1 �N−1 |S| � � U(ϵS∪{i}) − U(ϵS) � , (17) where α(ϵ) satisfies axiom (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 12 The proof of Theorem 2 can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 1) Example: Consider the utility function defined in (14), for the N = 2 user mean estimation problem with E = {0, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' By Theorem 2 the fair allocation satisfying (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i-iii) must be of the form: Φ(2) 1 = A ⊙ � �0 2/3 0 1/2 � � , Φ(2) 2 = A ⊙ � � 0 0 2/3 1/2 � � , (18) where A = AT and 0 ≤ [A]ij ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' TWO USER SYMMETRIC BINARY PRIVACY GAME In this section, we return to our example of a mean estimation problem with N = 2 users and E ∈ {0, ∞} to investigate what happens when our proposed notions of fairness are applied to a data acquisition problem of the form (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' For simplicity, we assume both users have the same privacy sensitivity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Simulations for the case where sensitivities are asymmetric are included in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Fairness from Theorem 1 In Section III-A1, we showed that the fair values for both the platform and the players satisfying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i-iii) are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Thus, for a platform constrained to making payments satisfying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i-iii), the data acquisition problem (8), reduces to a maximum over the equilibrium points: max p1, p2 pT 1 Φpp2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (p1, p2) ∈ NE, (19) where we have used pi = [p, (1 − p)]T to represent the mixed strategy where the probability of choosing ϵi = 0 is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The equilibrium points themselves depend on the objective of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The users’ objectives are: ui(p1, p2) = pT 1 Φ(1) i p2 − [0 c]Tp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (20) Since we have assumed both players have the same privacy sensitivity, and we have Φ(1) 2 = � Φ(1) 1 �T , the game between the two users reduces to a 2 × 2 symmetric game, meaning the equilibria can be found analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Figure 3, plots U(p∗ 1, p∗ 2) = p∗ 1 TUp∗ 2, where (p∗ 1, p∗ 2) maximize (19) as well as the payment to players t(p∗ 1, p∗ 2) = p∗ 1 TΦ(1) i p∗ 2 for a range of different c values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Plot of optimal utility U and payments to users ti at (p∗ 1, p∗ 2) solving (19) (dashed lines) and at (˜p1, ˜p2) solving (21) (solid lines) for a range of symmetric privacy levels c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Fairness from Theorem 2 In Section III-B1 we determined that the form of fair payments to the players satisfying (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i-iii), are parameterized by a matrix A ∈ A, where A = � A ∈ R2×2 : 0 ≤ [A]ij ≤ 1, A = AT� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Thus, as opposed to the previous example, the data acquisition problem (8), requires us to optimize over the space of fair payments: max A ∈ A pT 1 Up2 − pT 1 (A ⊙ U)p2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (p1, p2) ∈ NE(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (21) Just as before, the fairness and symmetric privacy sensitivity means that the game between users is a symmetric 2 × 2 matrix game and the equilibrium set is easily characterized analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Figure 3, plots U(˜p1, ˜p2) = ˜pT 1 U˜p2, where (˜p1, ˜p2) maximize (21) as well as the payment to players t(˜p1, ˜p2) = ˜pT 1 Φ(1) i ˜p2 for a range of different c values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Comparison Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 3 plots the results of both fairness constrained data acquisition problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' First consider the dashed lines, depicting the solution to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' When c < 2/9, a utility of 1 is achieved, meaning users choose ϵi = ∞ with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In this region users and the platform receive the same amount of payment regardless of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' As c ≥ 2/9 the optimal equilibrium point is some mixed strategy, and the utility begins to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' When c > 1/3 the privacy sensitivity becomes too 14 large and the payment become insufficient compared to the privacy cost, so the users choose ϵi = 0 with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The plots depicting the solution to (21) show that when c < 1/3, a utility of 1 is achieved, so the achieves this maximum utility for a larger range of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In contrast to the previous case, as c increases, the amount that the players are paid increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This is because the platform pays the minimum possible amount to ensure users choose ϵi = 1 with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Note that the platform forces the user utility to 0, since the payment is exactly c, which is also the privacy cost the platform pays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' For c > 1/3 the cost of paying the user to maintain ϵi = ∞ with probability 1 is too large, and it instead becomes optimal to pay users less, such that they choose some optimal mixed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Finally, for c > 2/3 the users choose ϵi = 0 with probability 1, as the privacy loss becomes too big to compensate for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' One feature common in both cases is that the optimal solution is characterized by three distinct regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' When c is small, users choose ϵi = ∞ with probability 1, followed by a region where users choose a mixed strategy, and finally, when c is too large, users choose ϵi = 0 with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In the appendix we discuss this phenomenon more, and show that if we have symmetric privacy sensitivity and α(ϵ) in the statement of Theorem 2 is taken to be constant, then this 3-region behavior can be proved for a class of utility functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' AN APPLICATION IN FEDERATED LEARNING FL is a distributed learning process used when data is either too large or too sensitive to be directly transferred in full to the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Instead of combining all the data together and learning at the platform, each user performs learning locally and the results are aggregated at the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' When training is done in this federated way, we can view this as an increased level of privacy for the user, though perhaps in a less rigorous way than DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In this section, we apply our definition of fairness in the context of a federated mean estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let each user i ∈ [N] have some mean and variance (θi, σ2 i ) ∼ Θ, where Θ is some global joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let t2 = Var(θi) and s2 = E[σ2 i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' User i draws n samples i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' from its local distribution Di(θi, σ2 i ), that is, some distribution with mean θi and variance σ2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The goal of the platform is to construct estimators ˆθp i that minimize the expected mean squared-error: E � MSE(ˆθ p, θ) � ≜ N � i=1 E �� ˆθp i − θi �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (22) 15 We note the similarity of this formulation to that of [Donahue and Kleinberg, 2021], from which we have drawn inspiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 4 summarizes our FL formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Users can choose from a 3-level privacy space E = {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let Nj be the number of users that choose privacy level j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' When ϵi = 2, user i provides its local estimator ˆθi directly to the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' When ϵi = 1, user i’s local estimator is securely aggregated with other users that choose this level such that ˆθf = 1 N1 � i:ϵi=1 ˆθi, (23) and the platform receives access to ˆθf, rather than the local estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' As before, ϵi = 0 means user i chooses not to provide any information to the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let the users be ordered such that ϵi is a non-increasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Then for each i the platform constructs estimators of the form: ˆθp i = wi0ˆθf + N2 � j=1 wij ˆθj, (24) where, � j wij = 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In our model, from these estimators, the platform generates utility U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In practice, for example, θi may represent some variable that describes what products are marketable to user i, so a better estimate of this quantity will result in more profit from advertisements served to that user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We can consider utilities of the form: U = c−E � MSE(ˆθ p, θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In order to apply our definition of fairness, we must write U as a function of the privacy level vector ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The following lemma establishes that for a platform that chooses the optimal estimators ˆθp i it is possible to write U as a function of ϵ alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' For the platform that optimally chooses parameters wi0, and wij in (24), EMSE ≜ E � MSE(ˆθ p, θ) � can be written as a function of ϵ if ϵi > 0 for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' If we define EMSE(0) ≜ N(t2 + 2s2), then we can define: U(ϵ) ≜ N(t2 + 2s2) − EMSE(ϵ), (25) which satisfies U(ϵ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The proof of Lemma 1 and a description of U can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Example Fair Payments We now consider an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let there be N = 10 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' N1 = 5 of these users opt for federating (ϵi = 1), N2 = 4 directly provide their data to the platform (ϵi = 2), and finally, N0 = 1 users chooses to not participate (ϵi = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Each user has n = 20 samples, and t2 = 1 and s2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Table I provides a breakdown of the division of utility when users are paid fairly according to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In this case, the platform keeps roughly 64% of the total utility, with the remaining 36% being distributed between the 9 users that choose a non-zero privacy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Among these users, users with privacy level ϵi = 2 earn roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='6 times more than those that choose ϵi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Table II show the fair payment values according to Theorem 2, normalized by Amount of Utility Paid Total Utility 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='9 User with ϵi = 0 0 User with ϵi = 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='7 User with ϵi = 2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='4 Platform 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='9 TABLE I PAYMENT ALLOCATIONS FOR FAIRNESS BETWEEN ALL USERS AND PLATFORM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' α(ϵ), which is controlled by the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Interestingly in this case, users with ϵi = 2 earn only roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='4 times as much as users with ϵi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' φi(ϵ)/α(ϵ) User with ϵi = 0 0 User with ϵi = 1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='3 User with ϵi = 2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='8 TABLE II PAYMENT WITH FAIRNESS AMONG USERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' CONCLUSION This paper introduces two formal definitions of fair payments in the context of acquisition of private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The first treats the users and the platform together and uses axioms like those of the Shapley value to determine a unique fair distribution of utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In the second, we define a notion 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In the FL setting, users have a choice between three levels of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' If ϵi = 0, users send no data to the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' If ϵi = 1, a user’s model is securely combined with other users who also choose ϵi = 1, and the platform receives only the combined model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' If ϵi = 2, users send their model directly to the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' of fairness between the users only, leading to a definition of fairness that admits a range of values, of which the platform is free to choose the most favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We consider examples of mechanisms where both notions of fairness are enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' While previous literature has investigated how platforms should design incentives for users in order to optimize its utility, the definitions of fairness we propose offers another important way to evaluate the fairness of these mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This is a critical step towards future research in ensuring that data acquisition mechanisms are both fair for users and efficient for platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' While we present results for fairness in a mechanism with N = 2 users with symmetric privacy sensitivity, many open questions remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' For example, designing mechanisms that consider fairness with heterogeneous privacy sensitivities, and the case of an arbitrary number of users N are important question that remains, since in practice the platform interacts with large and diverse groups of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In the appendix, we present preliminary work that begins to address these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Furthermore, there is subjectivity in the choice of axioms, and other choices may lead to meaningful notions of fairness worthy of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' RAND Corporation, Santa Monica, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' [Spiekermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2015a] Spiekermann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', Acquisti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', B¨ohme, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', and Hui, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The challenges of personal data markets and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Electronic Markets, 25(2):161–167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' [Spiekermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2015b] Spiekermann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', B¨ohme, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', Acquisti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', and Hui, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Personal data markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Electronic Markets, 25(2):91–93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' [Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2017] Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', Korolova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', and Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Privacy loss in apple’s implementation of differential privacy on macos 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' CoRR, abs/1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='02753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' [US Census Bureau, 2021] US Census Bureau (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Understanding the April 2021 Demonstration Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' https://www2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='census.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='gov/about/training-workshops/2021/2021-04-30-das-presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' [Wieringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', 2021] Wieringa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', Kannan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', Reutterer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', Risselada, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', and Skiera, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Data analytics in a privacy-concerned world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Journal of Business Research, 122:915–925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Proof of Equation (6) In this section, we present the calculations required to arrive at the utility values in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' First let’s treat the trivial case of ϵ1 = 0, ϵ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The optimal ϵ-DP estimator is simply the optimal Bayes estimator with no data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=', the prior mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Let us define this estimator as ˆµ(0,0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Its risk function is R(µ, ˆµ(0,0)) = E � L(ˆµ(0,0), µ) | µ � = µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (26) The Bayes risk of ˆµ(0,0) is the expectation of this quantity taken using our prior: r([0, 0]) = E � µ2� = 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (27) Next, consider the case where user i chooses privacy level ϵ1 = ϵ′ > 0, and the other user chooses ϵ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' In this case the estimator depends on X1, ˆµ(ϵ′,0) = w1X1 + Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Then the risk function is: R(µ, ˆµ(ϵ′,0)) = E � (w1X1 + Z − µ)2 | µ � = � µ + 1 2 � � µ − w1 2 �2 + � −µ + 1 2 � � µ + w1 2 �2 + 2 η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (28) 20 Now taking the expectation with respect to our prior over µ, we have: E � R(µ, ˆµ(ϵ′,0)) � = 1 12 � 3w2 1 − 2w1 + 1 � + 2 η2, (29) here η is the inverse scale parameter for Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Note that (29) is minimized when η is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The ϵ-DP condition enforces the constraint η ≤ ϵ′ w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' This constraint will be met with equality for the optimal w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' The optimal w∗ 1 = 1 3+ 24 ϵ′2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Thus, we have: ˆµ(ϵ′,0) = 1 3 + 24 ϵ′2 X1 + Z, Z ∼ Laplace � ϵ′ 3ϵ′2 + 24 � , (30) and the resulting Bayes risk is: r([ϵ′, 0]) = r([0, ϵ′]) = 1 12 � 1 − 1 3 + 24 ϵ′2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (31) For the case with ϵ1 = ϵ2 = ϵ′ we can repeat the same process by defining ˆµ(ϵ′,ϵ′) = w1X1 + w2X2 + Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' By symmetry, we must have w1 = w2, so we drop the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Then the risk function and its expectation are: R(µ, ˆµ(ϵ′,ϵ′)) = 2 � µ + 1 2 � � −µ + 1 2 � µ2 + � µ + 1 2 �2 (w − µ)2 + � −µ + 1 2 �2 (µ + w)2 + 2 η 2 (32) E � R(µ, ˆµ(ϵ′,ϵ′)) � = 1 12(8w2 − 4w + 1) + 2 η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (33) By a similar argument to the previous case, the Bayes optimal estimator and the corresponding Bayes risk is: ˆµ(ϵ′,ϵ′) = 1 4 + 12 ϵ′2 (X1 + X2) + Z, Z ∼ Laplace � ϵ′ 4ϵ′2 + 12 � , (34) r([ϵ′, ϵ′]) = 1 12 � 1 − 1 2 + 6 ϵ′2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' (35) Finally letting U(ϵ) = c1r(ϵ) + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Take U(0) = 0 =⇒ c1 = −12c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' And maxϵ U(ϵ) = 1 =⇒ c1 = 24(1 − c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Simplifying gives us our desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' 21 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' Proof of Theorem 1 and Theorem 2 We will begin with the proof of Theorem 2, which is standard and follows the typical proof of the Shapley value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' We begin by proving φi(ϵ) as defined in (17) satisfies axioms (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i-iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' First assume U(ϵS∪{i}) = U(ϵS∪{j}) ∀S ⊂ [N]\\{i, j}, then: φi(ϵ) = α(ϵ) N � S⊆[N]\\{i} U(ϵS∪{i}) − U(ϵS) �N−1 |S| � (36) = α(ϵ) N � � � S⊆[N]\\{i,j} U(ϵS∪{i}) − U(ϵS) �N−1 |S| � + � S⊆[N]\\{i,j} � U(ϵS∪{j}∪{i}) − U(ϵS∪{j}) � � N−1 |S|+1 � � �(37) = α(ϵ) N � � � S⊆[N]\\{i,j} U(ϵS∪{j}) − U(ϵS) �N−1 |S| � + � S⊆[N]\\{i,j} � U(ϵS∪{i}∪{j}) − U(ϵS∪{i}) � � N−1 |S|+1 � � �(38) = φj(ϵ), (39) proving axiom (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' For the proof that axiom (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='ii) is satisfied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content=' we write: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='φi(ϵ) = α(ϵ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='S⊆[N]\\{i} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='U(ϵS∪{i}) − U(ϵS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='�N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='|S| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='(40) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFQT4oBgHgl3EQfejY8/content/2301.13336v1.pdf'} +page_content='= α(ϵ) ' metadata={'source': 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+t+1 +� +si +t+1, ai +t+1, aˆi +t+1; θ∗ +i +� +18: +Calculate the loss L: +19: +L = +�� +r + γnashQi +t+1 +� +− Q(si +t, ai +t, a +ˆi +t+1; θi)]2. +20: +Update θi with ∇L using RMS optimizer. +21: +Assign θi to θ∗ +i . +22: +if the number of vehicles N = 0 then +23: +break +24: +end if +25: +end if +26: +end while +27: end for +As illustrated in Fig. 1, we took four intersections as an +example to show the training process of an agent i (red +box). We can see that the intersection with red box has three +neighbors (blue box). The intersections can transmit action +information to each other, and the agent can receive the action +that the adjacent intersections want to change. After train step +number k greater than or equal to the pre-train steps value tp +is completed, the parameter of agent i is trained in the training +model. We use evaluation network and the target network with +the same fully connected network for the DQN used in this +paper. Each network contains three hidden layers with sizes + +4 +Fig. 1: Overall architecture of the OPNDQN approach. +Fig. 2: A road network with 25 traffic signal control agents. +of 32, 64, 64, and each layer selects the relu function as the +activation function. The output layer outputs the Q-value of +the action. Root Mean Square (RMS) with learning rate of +0.0001 was selected as the gradient optimizer. In training, we +utilize a mini-bacth method with a batchsize of 64. +III. SIMULATION ENVIRONMENT +In this section, we will describe the applied simulation +environment for this research. We will firstly illustrate the +road network model, and followed by the definition of states, +actions and reward. +A. Road network +The simulation road network is a 5 × 5 as shown in Fig. 2. +The total size of the grid is 600 × 600 meters. The vehicle +information of the road network, including input vehicle +number (IVN)(vehicle number/episode) and traffic flow rate +(TFR)(vehicle number/seconds) are summarized in Table I. +Direction +Entrance +IVN +TFR +North +Entrance 1 +1000 veh/epi +1/20 veh/s +Entrance 2 +1000 +1/10 +Entrance 3 +1100 +1/15 +Entrance 4 +1050 +1/20 +Entrance 5 +900 +1/10 +South +Entrance 1 +1100 +1/15 +Entrance 2 +900 +1/15 +Entrance 3 +900 +1/20 +Entrance 4 +1000 +1/10 +Entrance 5 +950 +1/20 +East +Entrance 1 +1050 +1/20 +Entrance 2 +1000 +1/15 +Entrance 3 +950 +1/15 +Entrance 4 +1000 +1/20 +Entrance 5 +850 +1/10 +West +Entrance 1 +950 +1/15 +Entrance 2 +1000 +1/20 +Entrance 3 +1050 +1/10 +Entrance 4 +1000 +1/10 +Entrance 5 +1000 +1/20 +TABLE I: The vehicle input information of the road network. +B. States +Fig. 3 and Fig. 4 show how to set up the state values. Fig. +3 shows a snapshot of the traffic status at a simple network +(four intersections for example), which is divided into square- +shape grids. The position matrix has the same size of the grids, +which is shown in Fig. 4. The vehicle’s position is set to 1, +and the blank cells mean no vehicle in the corresponding grid, +which set to 0. It is worth noting that the length of each grid +is equal to the length of the car, and a grid only have a 1 or +0. + +Equation Iteration (Fictitious Part) +a, change +=argmaxQ: (s: ,db ,al,db;e,)sep +Agent i +Agent i, +Step 1: +argmaxQ(st,at,ah,de;e, )st +dr +argmaxQ(s,at,at,ab;e, ), +a' change +argmaxQi (s: ,h ,dl,dt;e, )ste +Step 2: +di +a's +Nash Game +Agenti, +Agent i. +Step j: +argmaxQ(st,at, b ,a ;e,)s += argmaxQ: (st ,at,al,at ;0. )ste +a. +1 +Nash Q' Value +Agent i Training Model +Evaluation Network +s' +nashQi(st,at,at,a ;0) +(Environment) +Memory M +Update 0, +(si,di,f,,s2,ri) +Minibatch +(2,2,,,s,2) +Update ' +MSE +(st,at,a ab,$+r) +: +($,a,at,ae,$+++') +Target Network +: +nashQ'($++1,a++1, df+1,d+1;0,) +St+1 +SamplingEntrance1 +Entrance2 +Entrance3 +Entrance4 +Entrance55 +Fig. 3: The snapshot of traffic at one time instant. +Fig. 4: The corresponding position matrix of Fig. 3. +C. Actions +In our model, the action space is defined by the green light +phase duration of each stage change. We formulate a fixed +period of 40 seconds. The green light duration of each traffic +light can be changed to 10 seconds, 15 seconds, 20 seconds, 25 +seconds, or 30 seconds. The remaining period is the duration +of the red light (Each 40 seconds is an epoch, and after all +vehicles of a route file run, the episode is incremented by 1 +and the route file is reset). Each agent has the same action +space. +D. Reward +The role of the reward is to provide feedback to the RL +model about the performance of the previous action, so the +formulation of the reward is particularly important. We define +reward as the change in cumulative waiting time between two +adjacent epochs. We define the cumulative waiting time of all +vehicles in the T epoch as WT , then the reward in the T epoch +is defined as: +rT = WT −1 − WT +(11) +The meaning of this reward is an increment in the cumulative +waiting time of all vehicles after the action. If the reward in the +current epoch is larger than before, it means that the waiting +time of vehicles is decreasing, which can achieve the purpose +of optimization. +IV. EVALUATION +In this section, we will show the simulation results as +well as the OPNDQN performance analysis for this research. +We firstly illustrate the parameters used by the OPNDQN +algorithm. +A. Evaluation Parameters +TABLE II: PARAMETERS IN OFF POLICY NASH DEEP +Q-NETWORK +Parameter +Value +Replay memory size M +20000 +Minibatch size B +64 +Starting ϵ +1 +Ending ϵ +0.01 +Pre-training steps tp +2000 +Target network update iteration +100 +Discount factor γ +0.99 +Learning rate ϵr +0.0001 +The model is trained in iterations. One iteration is an +episode. The entire development environment is written in +Tensorflow [20], and the parameters of the model are shown +in Table II. +B. Baseline Methods +We compare the performance of the OPNDQN with the +following baseline methods: +(1) Multi-agent Q-learning (MAQL) [7]: We use the Q- +learning method separately at each intersection, the Q-table +is applied to find the optimal traffic light control policy. Each +intersection is independent, and no information sharing. +(2) Multi-agent Advantage Actor-Critic (MA2C) [14]: We +use A2C method separately at each intersection. Each agent +uses critic network evaluates the policy of each actor and +guides them to optimize their policies. +(3) Fully Decentralized DQN Approach [8]: We use the +traditional DQN method separately at each intersection, each +agent is essentially influenced by the latest actions of its +neighbours, while searching the optimal strategy to control +an intersection by a DQN. +(4) Cooperative Multi-agent Deep RL Approach (Co- +MARL): Haddad et al., proposed a Co-MARL approach in +2022 [19]. The Co-MARL method applies a DQN, while trans- +ferring state, action and reward received from their neighbour +agents to its own loss function during the learning process. +Unlike our OPNDQN approach, Co-MARL does not let the +agent play a game but uses the information transmitted by the +neighbor agent as the state of learning to achieve the purpose +of agent cooperation. +C. Performance Analysis +The simulation results are shown in Fig. 5, 6, 7. Fig. 5 +shows the performance comparisons of cumulative reward in +every episode under the same traffic flow rate. As we can see, + +N +个 ++ ++ +T +1 +1 +1 +一 +1 ++ +1 +00 +1 +1 +- +1 +一 +一 +1 ++ +W +E +1 +1 +- +一 +1 +1 +1 +1 +00 +个 +0 +1 +- +-个 +1 +1 +1 +1 +F +T +T +一 +L ++ +一 +1 +S0 +0 +0 +1 +1 +1 +11 +1 +0 +0 +06 +Fig. 5: Performance comparisons for reward. +when episodes approximately equal to 80, our method has con- +verged. It takes 250 episodes for the fully decentralized DQN +method, MA2C method and Co-MARL method to converge. +The MAQL method is the least effective, and the model is +almost impossible to converge. Meanwhile, in terms of training +effect, the value of reward after convergence of our approach +method is higher than the other four methods. Fig. 6 shows +five methods of average vehicle waiting time per episode. It +can be seen that after the training of OPNDQN method, the +average waiting time of vehicles is shortened to about 6.4 +seconds, which is 0.2 seconds better than MA2C method, and +0.4 seconds lower than fully decentralized DQN method and +MA2C method (MAQL was not taken into account, because +it does not converge). Although in the average waiting time of +vehicles, our method is not much more than the other methods, +but in the convergence speed, our method is far better than the +other four methods. To evaluate the performance of various +signal control methods, we also compared the average queue +length of vehicles at each intersection per episode in Fig. +7. We can see that OPNDQN is also performing very well, +converging before 100 episodes and keeping the queue length +at around 25. It can also be noted that the OPNDQN method +has stronger stability after the completion of convergence. The +three standard curves of fully decentralized method, MA2C +method and Co-MARL method have large oscillation after +convergence due to the mutual influence between intersec- +tions, which proves that our OPNDQN approach has better +robustness and stability. +V. CONCLUSION +In this paper, we study the ATSC problem for a large +traffic network, and then propose an OPNDQN approach. +Firstly, we find a nash equilibrium between agents by using +a fictitious game method. Secondly, we use the nash Q-value +between agents rather than the maximum Q-value, it allows the +agents to achieve equilibrium through learning. At the same +time, OPNDQN method overcomes the problem that the fully +centralized method is challenging to train large-scale neural +networks. From the experimental results, the convergence +Fig. 6: Performance comparisons for average waiting time. +Fig. 7: Performance comparisons for average queue length. +speed of the OPNDQN algorithm is much higher than such +approaches. Furthermore, our method has high stability after +the algorithm converges. Experiments also demonstrate that +with a large number of agents, the advantages of our method +are prominent. +REFERENCES +[1] X. Liang, T. Yan, J. Lee, and G. Wang, “A Distributed Intersection +Management Protocol for Safety, Efficiency, and Driver’s Comfort,” IEEE +Internet of Things Journal, 2018. +[2] N. Casas, “Deep deterministic policy gradient for urban traffic light +control,” arXiv preprint arXiv:1703.09035, 2017. +[3] C. Li, W. Yue, G. Mao, and Z. Xu, “Congestion Propagation Based +Bottleneck Identification in Urban Road Networks,” IEEE Transactions +on Vehicular Technology, 2020. +[4] J. Jin and X. Ma, “A multi-objective agent-based control approach with +application in intelligent traffic signal system,” IEEE Transactions on +Intelligent Transportation Systems, vol. 20, no. 10, pp. 3900–3912, 2019. +[5] T. Chu and J. 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Chai, “Large-scale traffic signal control +using a novel multiagent reinforcement learning,” IEEE transactions on +cybernetics, vol. 51, no. 1, pp. 174–187, 2020. +[14] T. Chu, J. Wang, L. Codeca, and Z. Li, “Multi-agent deep reinforcement +learning for large-scale traffic signal control,” IEEE Transactions on +Intelligent Transportation Systems, 2020. +[15] Y. Chen, C. Li, W. Yue, H. Zhang, and G. Mao, “Engineering a +large-scale traffic signal control: A multi-agent reinforcement learning +approach,” in IEEE INFOCOM 2021 - IEEE Conference on Computer +Communications Workshops, INFOCOM WKSHPS 2021, 2021. +[16] “A course in game theory,” Computers and Mathematics with Applica- +tions, 1995. +[17] J. Hu and M. P. Wellman, “Nash Q-learning for general-sum stochastic +games,” Journal of Machine Learning Research, 2004. +[18] D. Krajzewicz, J. Erdmann, M. Behrisch, and L. Bieker, “Recent +development and applications of sumo-simulation of urban mobility,” +International journal on advances in systems and measurements, vol. 5, +no. 3&4, 2012. +[19] T. A. Haddad, D. Hedjazi, and S. Aouag, “A deep reinforcement +learning-based cooperative approach for multi-intersection traffic signal +control,” Engineering Applications of Artificial Intelligence, vol. 114, +p. 105019, 2022. [Online]. Available: https://www.sciencedirect.com/ +science/article/pii/S0952197622001993 +[20] F. Marcham, “TensorFlow: Large-Scale Machine Learning on Hetero- +geneous Distributed Systems (Preliminary White Paper, November 9, +2015),” The Library, 1929. + diff --git a/ZdAyT4oBgHgl3EQfvvmr/content/tmp_files/load_file.txt b/ZdAyT4oBgHgl3EQfvvmr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..327f5dfffda4ff3a0fbe56c63ef132fbd4f69eb7 --- /dev/null +++ b/ZdAyT4oBgHgl3EQfvvmr/content/tmp_files/load_file.txt @@ -0,0 +1,441 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf,len=440 +page_content='1 Large-Scale Traffic Signal Control by a Nash Deep Q-network Approach Yuli Zhang, Shangbo Wang, Ruiyuan Jiang Abstract—Reinforcement Learning (RL) is currently one of the most commonly used techniques for traffic signal control (TSC), which can adaptively adjusted traffic signal phase and duration according to real-time traffic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' However, a fully centralized RL approach is beset with difficulties in a multi- network scenario because of exponential growth in state-action space with increasing intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Multi-agent reinforcement learning (MARL) can overcome the high-dimension problem by employing the global control of each local RL agent, but it also brings new challenges, such as the failure of convergence caused by the non-stationary Markov Decision Process (MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In this paper, we introduce an off-policy nash deep Q-Network (OPNDQN) algorithm, which mitigates the weakness of both fully centralized and MARL approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The OPNDQN algorithm solves the problem that traditional algorithms cannot be used in large state-action space traffic models by utilizing a fictitious game approach at each iteration to find the nash equilibrium among neighboring intersections, from which no intersection has incentive to unilaterally deviate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' One of main advantages of OPNDQN is to mitigate the non-stationarity of multi-agent Markov process because it considers the mutual influence among neighboring intersections by sharing their actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' On the other hand, for training a large traffic network, the convergence rate of OPNDQN is higher than that of existing MARL approaches because it does not incorporate all state information of each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We conduct an extensive experiments by using Simulation of Urban MObility simulator (SUMO), and show the dominant superiority of OPNDQN over several existing MARL approaches in terms of average queue length, episode training reward and average waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Index Terms—Multi-agent, reinforcement learning, deep Q- network, nash equilibrium, traffic light control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' INTRODUCTION W ITH the growth of vehicle ownership, the current traffic demand is rising rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Traffic jams and vehicular accidents are becoming more seriously with the increasing travel demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' One of the main reasons is the unreasonable traffic signal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' It has been reported in [1] that the current traffic signal network cannot adaptively respond to the real time traffic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Existing traffic light control either deploys fixed-time mode without considering real-time traffic or considering the traffic to a very limited degree [2], , which cause the vehicle accumulation at intersections and degrade traffic efficiency [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Therefore, developing an effective traffic signal control strategy is of great significance in urban net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In recent years, reinforcement learning (RL) technique had many significant achievements in traffic signal control (TSC) because RL can better solve the decision optimization prob- lem of sequential actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In the literature, TSC using RL methods could be categorized into model-based and model- free approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In a model-based approach, agent trying to understand environment and creating a model based on interact with environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' A model-free RL approach learns the optimal policy base without thoroughly understanding the dynamics of the traffic system [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' A model-free method is straightforward and convenient for implementation, therefore, model-free RL methods are widely used in TSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' There are two variants for model-free RL: Value-based and Policy- based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Q-learning is a value-based RL method which able to compare the expected utility of the available actions without requiring a model of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' However, the Q-table will be huge if state space is vast, and searching and storing it involves a lot of time and memory, so Q-learning approach performance depends on the quality of traffic feature design [5], [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Deep Q-Network (DQN), first proposed in 2015 by Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=', successfully combines RL with deep learning by using a multi-layer convolutional neural network to approximate the Q-function [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' It can be observed from [9], [10] that DQN can achieve good performance for small state-action space in single-agent control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In 2019, Liang and Du proposed a Double-Dueling-Deep Q-Network method, including dueling network, double network and deep network [11] to control the traffic light cycle in a single intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Even though RL technique has made good achievements in TSC for single intersection scenario, it is infeasible for large- scale urban road network with multiple intersections because of exponential growth of state-action space with increase of number of intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' To overcome the extremely high di- mension of joint action space issue, some researchers proposed multi-agent reinforcement learning (MARL) approach, which distributes global control to each local agent [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In 2020, Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=', [14] proposed an Advantage Actor-Critic (A2C) algorithm in an adaptive TSC environment, which can improve the observability of each agent by considering the control policies of other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=', improved A2C by using difference reward method [15] in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' However, the disadvantage of the distributed A2C approach is that when the state dimension or the number of surrounding intersections increases, intersections will have a severe mutual influence, resulting in a non-stationary environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Since Nash presented the concept of nash equilibrium in game theory in [16], there are some research about nash Q- learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In 2004, Hu and Wellman designed an on-policy nash Q-learning algorithm which extended Q-learning to non- cooperative multi-agent environments [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' However, this on- policy nash Q-learning algorithm has an apparent defect in that it needs to know or be able to calculate the control arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='00637v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='GT] 2 Jan 2023 2 policies of each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We conclude that the on-policy ap- proach is impractical in complex traffic networks, because it is cumbersome to obtain the accurate control policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' To tackle the problem, this paper proposes an off-policy nash deep Q-Network (OPNDQN) algorithm that find the optimal strategy by searching the nash equilibrium among neighboring intersections iteratively without need of knowing or calculating the control policy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' the conditional probability of selecting an action based on current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The main contributions are as follows: We propose the OPNDQN algorithm that can achieve the nash equilibrium of the expected cumulative reward for non-cooperative agents without knowing or calculating the control policies, and apply it in a large-scale traffic signal control scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We evaluate the performance of the proposed approach, by conducting extensive experiments using SUMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' (Sim- ulation of Urban MObility) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We compare the OPNDQN algorithm to Fully Decentralized DQN ap- proach, Multi-agent Q-Learning (MAQL), Cooperative Multi-agent Deep Reinforcement Learning approach (Co- MARL) [19] and Multi-agent Advantage Actor-Critic (MA2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The results show that the proposed OPNDQN approach outperforms these methods in terms of multiple traffic metrics that are: average queue length, episode training reward, and average waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' METHODOLOGY To let reader better understand the OPNDQN approach, we firstly briefly review the Markov Decision Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Markov Decision Process Markov Decision Process (MDP) is the basic framework for illustrating RL problem, which is described by a sequential decision process that state space, action space, reward, state transition function, and discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' RL is a sequential de- cision process that interacts with the environment to maximize a target reward function through long-term trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' A MDP can be defined with a quin-tuple ⟨S, A, R, T, γ⟩: State space S: The set of all possible states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' State is the description and generalization of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' st (st ∈ S) is a state observed by an agent at t-th time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Action space A: The set of all possible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' at (at ∈ A) is an action of an agent at t-th time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Reward R: The reward space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' After the agent performs an action, the environment returns a value (reward) to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' rt (rt ∈ R) is a reward of an agent at t-th time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' State transition function T: A function used by the environment to generate new states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' It represents the probability of transitioning from one state to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The state transition function T can be expressed as: p (st+1 | st, at) = P � S′ = st+1 | S = st, A = at � , where p is the probability, at is the action of an agent at t-th time instant, st is the state of an agent, and st+1 represents the state that all agents transfer to at the next moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Discount factor γ: Discounts on future reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' (γ ∈ [0, 1]) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' OPNDQN To facilitate the reader’s understanding of the OPNDQN method, we will start with a single-agent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Let us define st, at, rt as the state matrix, action vector and reward vector at t-th time instant, respectively, in a MDP, we define all rewards from start to end as: r1, · · · , rt, · · · , rtEnd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Combine discount rates, future discounted return at time t as Ut, it can be represented as: Ut = rt + γ · rt+1 + γ2 · rt+2 + · · · + γtEnd−t · rtEnd (1) The randomness of Ut comes from all sequences of actions and observations after time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' To eliminate the effect of randomness, we use the expectation of Equation (1) and define the action-value function Qπ (st, at) as Qπ (st, at) = E [Ut | St = st, At = at, π] (2) where π is a policy mapping sequences to actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Traditional DQN method uses the maximization method to determine the optimal policy π∗ and derive the optimal action-value function: Q∗(st, at) = max π E [Ut | St = s, At = a, π∗] (3) The optimal action-value function obeys an important identity known as the Bellman equation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Assuming that the agent knows the optimal Q-value for the subsequent state, then the optimal policy is to select action at+1 maximising the expected value: rt + γQ∗ (st+1, at+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In practice, it is common to use a neural network to approximate the Q-value: Q(st, at;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θ) ≈ Q∗(st, at) (4) where θ is the parameter of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We define a neural network function approximator as a Q-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' However, in MARL, multi-agent system interacts with the same environment, which make the neural network unable to converge because of non-stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' To fill the gap, we try to investigate how to mitigate the non-stationarity of the MDP by considering the influence caused by neighboring agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Some of literature tends to let each agent collect the information of all other agents to increase the observability in the process of model training to increase the stationarity, however, it will negatively affect the convergence rate because of extremely large dimension of state-action space [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In this paper, we propose an OPNDQN approach, which finds the nash equilirium strategies among neighboring agents at each iteration, by only sharing their actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The advantage of this approach is not simply to increase the Q-value of each agent, but to find a balance between the agent and its neighbors so that each agent increases its own rewards without lowering the rewards of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We consider a non-cooperative multi- agent system with N agents, each of which has n neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' For all s1 ∈ S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' , sN ∈ SN, a1 ∈ A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' , aN ∈ AN, we use a fictitious game method to find the nash Q-value for complex model-free learning of multi-agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We define that the action selection by the agent i (i ∈ N) depends not only on its own 3 state, action and reward values but also on the action of their neighbours ˆi (ˆi ∈ n): ai t = NashQt � si t, ai t, a ˆi t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θi � (5) The fictitious game process can be described as follows: initialize the actions ai t and aˆi t at each episode, then update the group actions to increase each agent’s Q-value to the Nash Equilibrium, where no agent can further increase its Q-value by unilaterally change its action (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' NashQi � si, ai, a ˆi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θi � Step j ≤ NashQi � si, ai, a ˆi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θi � Step j−1 (6) where j is fictitious game loop counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The action (real action to do) with the largest Q-value among all the previously selected actions is the nash equilibrium strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' It is worth noting that in the above fictitious game process, the process of a searching for nash equilibrium by changing actions aˆi t is fictitious, while the agent i and its neighbor ˆi do not make real action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The agents does not make real actions until the final nash strategy ai Step j is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The flow of the fictitious game can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Fictitious game in pseudocode of OPNDQN is as follows: Algorithm 1 Fictitious Game Input: random actions ai Output: nash joint actions ai Step j or random actions ai Notation: The parameters in the neural network θ, agents number n, nash parameter ϵ 1: Initialize ai to the random action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 2: if probability ϵ select random action then 3: Output ai 4: else 5: while Step j = 1 do 6: ai Step j = argmax Qi � si , ai Step j−1, aˆi Step j−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θi � 7: if eq (6) then 8: break 9: end if 10: end while 11: Output ai Step j−1 12: end if We define the target yi k and train a Q-network by minimising a sequence of loss functions Li k (θk) that changes at each iteration k: yi k = ri t + γ · NashQi � si t+1, ai,ˆi t+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θk � (7) Li k (θk) = E �� yi k − Qi t � si t, ai,ˆi t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θk ��2� (8) Compute the gradient of loss L with respect to θ: ∇θkLk(θk) = E[ri t + γ · NashQi(si t+1, ai,ˆi t+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θk) −Qi t(si t, ai,ˆi t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θk))∇θkQi t(si t, ai,ˆi t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θk)] (9) The stochastic gradient descent method is used to optimize the loss function Li k (θk), the parameters θk of the neural network converge to a fixed value with the increase of the number of training step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Target Network To update the parameters θ in the neural network, a target value θ∗ is defined to help guide the update process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Using the target network to update the parameters, Equation (10) can be written as: yi k = ri t + γ · NashQi � si t+1, ai,ˆi t+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θ∗ k � (10) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Overall Architecture Algorithm 2 OPNDQN Algorithm Input: replay memory size M, minibatch size B, nash parameter ϵ, pre-train steps tp, target network update rate α, discount factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Notation: The parameters in the primary neural network θ, the parameters in the target neural network θ∗, the replay memory m, the agents number n, training step number k, time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 1: Initialize parameters θ, θ∗ with random values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 2: Initialize m to be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 3: Initialize si with the starting scenario at the intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 4: Initialize ai to the random action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 5: for Episode=1 do 6: Initialize state si 7: while the number of vehicles N > 0, k=1 do 8: Algorithm 1→ ai t 9: Take action ai t 10: observe reward ri t and new state si t, 11: Add episode to buffer 12: Assign si t+1 to si t : si t ← si t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 13: k ← k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 14: if |M| > B and k > tp then 15: Select B samples from replay memory m 16: Algorithm 1 → ai t+1 17: nash Qi t+1 = Qi t+1 � si t+1, ai t+1, aˆi t+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θ∗ i � 18: Calculate the loss L: 19: L = �� r + γnashQi t+1 � − Q(si t, ai t, a ˆi t+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' θi)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 20: Update θi with ∇L using RMS optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 21: Assign θi to θ∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 22: if the number of vehicles N = 0 then 23: break 24: end if 25: end if 26: end while 27: end for As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 1, we took four intersections as an example to show the training process of an agent i (red box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We can see that the intersection with red box has three neighbors (blue box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The intersections can transmit action information to each other, and the agent can receive the action that the adjacent intersections want to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' After train step number k greater than or equal to the pre-train steps value tp is completed, the parameter of agent i is trained in the training model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We use evaluation network and the target network with the same fully connected network for the DQN used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Each network contains three hidden layers with sizes 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 1: Overall architecture of the OPNDQN approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 2: A road network with 25 traffic signal control agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' of 32, 64, 64, and each layer selects the relu function as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The output layer outputs the Q-value of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Root Mean Square (RMS) with learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='0001 was selected as the gradient optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' In training, we utilize a mini-bacth method with a batchsize of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' SIMULATION ENVIRONMENT In this section, we will describe the applied simulation environment for this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We will firstly illustrate the road network model, and followed by the definition of states, actions and reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Road network The simulation road network is a 5 × 5 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The total size of the grid is 600 × 600 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The vehicle information of the road network, including input vehicle number (IVN)(vehicle number/episode) and traffic flow rate (TFR)(vehicle number/seconds) are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Direction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='IVN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='TFR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='North ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1000 veh/epi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1/20 veh/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1/10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1/20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='850 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1/10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='West ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='950 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1/15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1/20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1050 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1/10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1/10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='Entrance 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='1/20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='TABLE I: The vehicle input information of the road network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' States Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 4 show how to set up the state values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 3 shows a snapshot of the traffic status at a simple network (four intersections for example), which is divided into square- shape grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The position matrix has the same size of the grids, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The vehicle’s position is set to 1, and the blank cells mean no vehicle in the corresponding grid, which set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' It is worth noting that the length of each grid is equal to the length of the car, and a grid only have a 1 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Equation Iteration (Fictitious Part) a, change =argmaxQ: (s: ,db ,al,db;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='e,)sep Agent i Agent i, Step 1: argmaxQ(st,at,ah,de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='e, )st dr argmaxQ(s,at,at,ab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content="e, ), a' change argmaxQi (s: ,h ,dl,dt;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content="e, )ste Step 2: di a's Nash Game Agenti, Agent i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Step j: argmaxQ(st,at, b ,a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='e,)s = argmaxQ: (st ,at,al,at ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' )ste a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=" 1 Nash Q' Value Agent i Training Model Evaluation Network s' nashQi(st,at,at,a ;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content="0) (Environment) Memory M Update 0, (si,di,f,,s2,ri) Minibatch (2,2,,,s,2) Update ' MSE (st,at,a ab,$+r) : ($,a,at,ae,$+++') Target Network : nashQ'($++1,a++1, df+1,d+1;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='0,) St+1 SamplingEntrance1 Entrance2 Entrance3 Entrance4 Entrance55 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 3: The snapshot of traffic at one time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 4: The corresponding position matrix of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Actions In our model, the action space is defined by the green light phase duration of each stage change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We formulate a fixed period of 40 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The green light duration of each traffic light can be changed to 10 seconds, 15 seconds, 20 seconds, 25 seconds, or 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The remaining period is the duration of the red light (Each 40 seconds is an epoch, and after all vehicles of a route file run, the episode is incremented by 1 and the route file is reset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Each agent has the same action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Reward The role of the reward is to provide feedback to the RL model about the performance of the previous action, so the formulation of the reward is particularly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We define reward as the change in cumulative waiting time between two adjacent epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We define the cumulative waiting time of all vehicles in the T epoch as WT , then the reward in the T epoch is defined as: rT = WT −1 − WT (11) The meaning of this reward is an increment in the cumulative waiting time of all vehicles after the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' If the reward in the current epoch is larger than before, it means that the waiting time of vehicles is decreasing, which can achieve the purpose of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' EVALUATION In this section, we will show the simulation results as well as the OPNDQN performance analysis for this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We firstly illustrate the parameters used by the OPNDQN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Evaluation Parameters TABLE II: PARAMETERS IN OFF POLICY NASH DEEP Q-NETWORK Parameter Value Replay memory size M 20000 Minibatch size B 64 Starting ϵ 1 Ending ϵ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='01 Pre-training steps tp 2000 Target network update iteration 100 Discount factor γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='99 Learning rate ϵr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='0001 The model is trained in iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' One iteration is an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The entire development environment is written in Tensorflow [20], and the parameters of the model are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Baseline Methods We compare the performance of the OPNDQN with the following baseline methods: (1) Multi-agent Q-learning (MAQL) [7]: We use the Q- learning method separately at each intersection, the Q-table is applied to find the optimal traffic light control policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Each intersection is independent, and no information sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' (2) Multi-agent Advantage Actor-Critic (MA2C) [14]: We use A2C method separately at each intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Each agent uses critic network evaluates the policy of each actor and guides them to optimize their policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' (3) Fully Decentralized DQN Approach [8]: We use the traditional DQN method separately at each intersection, each agent is essentially influenced by the latest actions of its neighbours, while searching the optimal strategy to control an intersection by a DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' (4) Cooperative Multi-agent Deep RL Approach (Co- MARL): Haddad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=', proposed a Co-MARL approach in 2022 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The Co-MARL method applies a DQN, while trans- ferring state, action and reward received from their neighbour agents to its own loss function during the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Unlike our OPNDQN approach, Co-MARL does not let the agent play a game but uses the information transmitted by the neighbor agent as the state of learning to achieve the purpose of agent cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Performance Analysis The simulation results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 5, 6, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 5 shows the performance comparisons of cumulative reward in every episode under the same traffic flow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' As we can see, N 个 + + T 1 1 1 一 1 + 1 00 1 1 1 一 一 1 + W E 1 1 一 1 1 1 1 00 个 0 1 个 1 1 1 1 F T T 一 L + 一 1 S0 0 0 1 1 1 11 1 0 0 06 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 5: Performance comparisons for reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' when episodes approximately equal to 80, our method has con- verged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' It takes 250 episodes for the fully decentralized DQN method, MA2C method and Co-MARL method to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The MAQL method is the least effective, and the model is almost impossible to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Meanwhile, in terms of training effect, the value of reward after convergence of our approach method is higher than the other four methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 6 shows five methods of average vehicle waiting time per episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' It can be seen that after the training of OPNDQN method, the average waiting time of vehicles is shortened to about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='4 seconds, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='2 seconds better than MA2C method, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content='4 seconds lower than fully decentralized DQN method and MA2C method (MAQL was not taken into account, because it does not converge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Although in the average waiting time of vehicles, our method is not much more than the other methods, but in the convergence speed, our method is far better than the other four methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' To evaluate the performance of various signal control methods, we also compared the average queue length of vehicles at each intersection per episode in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' We can see that OPNDQN is also performing very well, converging before 100 episodes and keeping the queue length at around 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' It can also be noted that the OPNDQN method has stronger stability after the completion of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' The three standard curves of fully decentralized method, MA2C method and Co-MARL method have large oscillation after convergence due to the mutual influence between intersec- tions, which proves that our OPNDQN approach has better robustness and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' CONCLUSION In this paper, we study the ATSC problem for a large traffic network, and then propose an OPNDQN approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Firstly, we find a nash equilibrium between agents by using a fictitious game method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Secondly, we use the nash Q-value between agents rather than the maximum Q-value, it allows the agents to achieve equilibrium through learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' At the same time, OPNDQN method overcomes the problem that the fully centralized method is challenging to train large-scale neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' From the experimental results, the convergence Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 6: Performance comparisons for average waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' 7: Performance comparisons for average queue length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' speed of the OPNDQN algorithm is much higher than such approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Furthermore, our method has high stability after the algorithm converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfvvmr/content/2301.00637v1.pdf'} +page_content=' Experiments also demonstrate that with a large number of agents, the advantages of our method are prominent.' 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Wen-Tsun Mathematics +University of Science and Technology of China, Hefei, Anhui 230026, China. +Abstract +For a graph T and a set of graphs H, let ex(n, T, H) denote the maximum number of +copies of T in an n-vertex H-free graph. Recently, Alon and Frankl (arXiv2210.15076) +determined the exact value of ex(n, K2, {Kk+1, Ms+1}), where Kk+1 and Ms+1 are +complete graph on k + 1 vertices and matching of size s + 1, respectively. Soon after, +Gerbner (arXiv2211.03272) continued the study by extending Kk+1 to general fixed +graph H. +In this paper, we continue the study of the function ex(n, T, {H, Ms+1}) +when T = Kr for r ≥ 3. We determine the exact value of ex(n, Kr, {Kk+1, Ms+1}) and +give the value of ex(n, Kr, {H, Ms+1}) for general H with an error term O(1). +1 +Introduction +Let G = (V, E) be a graph with vertex set V = V (G) and edge set E = E(G) ⊂ +�V +2 +� +. We +may write G instead of E(G). +Let T be a fixed graph and H be a set of given graphs. A graph G is called H-free if +G contains no copy of any member in H as its subgraph. Write N(G, T) for the number of +copies of T in a graph G. Define the generalized Tur´an number as +ex(n, T, H) = max{N(G, T) : G is an n-vertex H-free graph}. +We call an n-vertex graph G with N(G, T) attaining the maximum an extremal graph of H. +This function has been systematically studied by Alon and Shikhelman [2] and has received +∗The work was supported by the National Natural Science Foundation of China (No. 12071453), the +National Key R and D Program of China(2020YFA0713100), the Anhui Initiative in Quantum Informa- +tion Technologies (AHY150200) and the Innovation Program for Quantum Science and Technology, China +(2021ZD0302904). +1 + +much attention, for example, in [7, 8, 9, 10, 11, 12, 13, 14, 15, 19]. When T = K2, it is the +classical Tur´an number ex(n, H). +Let Kr denote a complete graph on r vertices for some integer r. For a set U, write +K[U] for a complete graph on vertex set U. Let U1, U2, . . . , Ur be disjoint sets and U = +{U1, . . . , Ur}, write K[U] = K[U1, U2, . . . , Ur] for a complete r-partite graph with partition +sets U1, . . . , Ur. Let G = (V, E) be a graph. For a set U ⊆ V , write G[U] for the subgraph +induced by U. For disjoint sets U1, U2, · · · , Ur ⊆ V , write G[U1, · · · , Ur] for the induced +r-partite subgraph of G, i.e. G[U1, · · · , Ur] = K[U1, · · · , Ur]∩G. Let Ka1,a2,··· ,ar denote the +complete r-partite graph with partition sets of size a1, a2, · · · , ar. For graphs G1, · · · , Gr, +let �r +i=1 G1 + · · · + Gr be the union of vertex-disjoint copies of G1, · · · , Gr. +A Tur´an graph Tk(n) is a complete k-partite graph on n vertices whose partition sets +have sizes as equal as possible. Let tk(n) = |Tk(n)| = N(Tk(n), K2) be the Tur´an number. +The famous Tur´an Theorem [4] states that ex(n, K2, Kk+1) = tk(n). Erd˝os [5] gave the +generalized version of Tur´an Theorem as follows. +Theorem 1.1 ([5]). For all n ≥ k ≥ r ≥ 2, +ex(n, Kr, Kk+1) = N(Tk(n), Kr), +and Tk(n) is the unique extremal graph. +Write χ(G) for the chromatic number of graph G. We say a graph is edge-critical if there +exists some edge whose deletion reduces its chromatic number. Simonovits [18] proved that +for any edge-critical graph H with χ(H) = k + 1 ≥ 3, ex(n, K2, H) = tk(n) for sufficiently +large n, and Tk(n) is the unique extremal graph. This result was extended by Ma and +Qiu [17] as follows: For sufficiently large n, ex(n, Kr, H) = N(Tk(n), Kr), and Tk(n) is the +unique extremal graph, where H is an edge-critical graph with χ(H) = k + 1 > r ≥ 2. +Recently, Alon and Frankl [1] studied the function ex(n, K2, H) when H = {Kk+1, Ms+1}. +Let Gk(n, s) = Kn−s ∨ Tk−1(s), the join of an empty graph Kn−s and Tur´an graph Tk−1(s), +i.e. a complete k-partite graph on n vertices with one partition set of size n−s and the oth- +ers having sizes as equal as possible. Given a graph H with χ(H) ≥ 3, define H(H) to be the +family of graphs obtained by deleting a color class from H. Define DH(n, s) = D ∨ Kn−s, +where D is a copy of extremal graph of H(H) on s vertices. Write Mk for a matching +consisting of k edges. +Theorem 1.2 ([1]). (1) For n ≥ 2s + 1 and k ≥ 2, +ex(n, K2, {Kk+1, Ms+1}) = max {|Tk(2s + 1)|, |Gk(n, s)|} . +(2) Let H be an edge-critical graph with χ(H) = k + 1 > 2. Then, for sufficiently large s +and n ≫ s, +ex(n, K2, {H, Ms+1}) = |Gk(n, s)|. +2 + +Theorem 1.2 (2) was soon strengthened by Gerbner [6] in the following theorem. For a +bipartite graph H, let p = p(H) denote the smallest possible number of a color class in a +proper 2-coloring of H. +Theorem 1.3 ([6]). Let H be a fixed graph. +(1) Suppose χ(H) ≥ 3 and n is large enough. Then +ex(n, K2, {H, Ms+1}) = s(n − s) + ex(s, K2, H(H)) = sn + O(1), +and the graph DH(n, s) is an extremal graph of {H, Ms+1}. +(2) Suppose χ(H) = 2. The following holds. +(i) If p > s, then +ex(n, K2, {H, Ms+1}) = ex(n, K2, Ms+1), +and K2s+1 and Gs+1(n, s) are extremal graphs. +(ii) If p ≤ s, then +ex(n, K2, {H, Ms+1}) = (p − 1)n + O(1), +and Gp(n, p − 1) is an asymptotically optimal graph up to additive error of O(1). +In this article, we first extend the result of Alon and Frankl [1] as shown in the following. +Theorem 1.4. For n ≥ 2s + 1 and k ≥ r ≥ 3, +ex(n, Kr, {Kk+1, Ms+1}) = max{N(Tk(2s + 1), Kr), N(Gk(n, s), Kr)}. +Second, we extend Gebner’s result [6] with a similar way. +Theorem 1.5. (I) For every graph H with χ(H) ≥ 3, r ≥ 3 and sufficiently large n, +ex(n, Kr, {H, Ms+1}) = ex(s, Kr−1, H(H))n + O(1) = N(DH(n, s), Kr) + O(1). +(II) For every graph H with χ(H) = 2 and r ≥ 3, the following holds. +(i) If p > s, then +ex(n, Kr, {H, Ms+1}) = ex(n, Kr, Ms+1), +and K2s+1 and Gs+1(n, s) are extremal graphs. +(ii) If p ≤ s, then +ex(n, Kr, {H, Ms+1}) = N(Gp(n, p − 1), Kr) + O(1) = +�p − 1 +r − 1 +� +n + O(1), +and Gp(n, p − 1) is an asymptotically optimal graph up to additive error of O(1). +The rest of the article is arranged as follows. We give the proofs of Theorems 1.4 and +1.5 in Sections 2 and 3. We give some discussions in the last section. +3 + +2 +Proof of Theorem 1.4 +We need the following fundamental theorem in graph theory. +Theorem 2.1 (Tutte-Berge Theorem [3], see also [16]). A graph G is Ms+1-free if and only +if there is a set B ⊂ V (G) such that all the components G1, . . . , Gm of G − B are odd (i.e. +|V (Gi)| ≡ 1 (mod 2) for i ∈ [m]), and +|B| + +m +� +i=1 +|V (Gi)| − 1 +2 += s. +Let graph G = (V, E) and integer r ≥ 1. For a vertex v ∈ V (G), define N (r) +G (v) = +{U ∈ +�V +r +� +: G[U ∪ {v}] ∼= Kr+1} be the r-clique neighborhood of v and d(r) +G (v) = |N (r) +G (v)| +be the r-clique-degree of v. For a set U ⊂ V (G), write d(r) +G (U) = � +u∈U d(r) +G (u). As usual, +write neighborhood NG(v) and degree dG(v) instead of 1-clique neighborhood N (1) +G (v) and +1-clique-degree d(1) +G (v) for short. +For two non-adjacent vertices u, v in a graph G, we define the switching operation u → v +as deleting the edges joining u to its neighbors and adding new edges connecting u to vertices +in NG(v). Let Gu→v to be the graph obtained from G by the switching operation u → v, +that is V (Gu→v) = V (G) and +E(Gu→v) = (E(G) \ EG(u, NG(u))) ∪ EG(u, NG(v)), +where EG(S, T) = E(G[S, T]) for disjoint subsets S, T ⊂ V (G). Note that the edges between +u and the common neighbors of u and v remain unchanged by the definition of Gu→v. For +two disjoint independent sets S and T in a graph G, if all of vertices in S (resp. +T) +have the same neighborhood NG(S) (resp. NG(T)), we similarly define GS→T to be the +graph obtained from G by deleting the edges between S and NG(S) and adding new edges +connecting S and NG(T). +Proposition 2.2. For r ≥ 2 and two disjoint independent sets S and T in a graph G, +if all of vertices in S (resp. T) have the same neighborhood NG(S) (resp. NG(T)) and +EG(S, T) = ∅, then either G′ = GS→T or G′ = GT→S has the property that N(G′, Kr) ≥ +N(G, Kr), the equality holds if and only if d(r−1) +G +(S) = d(r−1) +G +(T). +Proof. Let S and T be two such independent sets of G. Without loss of generality, suppose +d(r−1) +G +(T) ≥ d(r−1) +G +(S). Let G′ = GS→T. Then +N(G′, Kr) = N(G, Kr) − d(r−1) +G +(S) + d(r−1) +G +(T) ≥ N(G, Kr), +the equality holds if and only if d(r−1) +G +(T) = d(r−1) +G +(S). +4 + +Let ∆r +t,k = N(Tk(t), Kr) for some integers t ≥ k ≥ r. +Observation 2.3. (1) For positive integers t ≥ k ≥ r ≥ 2, +∆r +t+1,k − ∆r +t,k = ∆r−1 +t−⌊ t +k ⌋,k−1 and ∆r +t,k = ∆r +t−⌊ t +k ⌋,k−1 + +� t +k +� +∆r−1 +t−⌊ t +k⌋,k−1. +(2) For n ≥ 2s + 1 and s ≥ t ≥ k ≥ r ≥ 3, define +gn,k,r(t) := (n − t)∆r−1 +t,k−1 + ∆r +t,k−1. +Then gn,k,r(t) is a strictly increasing function of t. In particular, gn,k,r(s) = N(Gk(n, s), Kr). +Proof. (1) can be checked directly by the definitions of ∆r +t,k and the Tur´an graph. +(2) By (1), for t ≤ s − 1, +gn,k,r(t + 1) − gn,k,r(t) += +(n − t − 1)∆r−2 +t−⌊ +t +k−1 ⌋,k−2 − ∆r−1 +t,k−1 + ∆r−1 +t−⌊ +t +k−1⌋,k−2 += +(n − t − 1)∆r−2 +t−⌊ +t +k−1 ⌋,k−2 − +� +t +k − 1 +� +∆r−2 +t−⌊ +t +k−1⌋,k−2 += +� +n − 1 − t − +� +t +k − 1 +�� +∆r−2 +t−⌊ +t +k−1⌋,k−2 > 0. +This completes the proof. +Now we are ready to give the proof of Theorem 1.4. +Proof of Theorem 1.4: Let G be an extremal graph of {Kk+1, Ms+1} on n ≥ 2s+1 vertices. +By Theorem 2.1, there is a vertex set B ⊂ V (G) such that G−B consists of odd components +G1, . . . , Gm, and +|B| + +m +� +i=1 +|V (Gi)| − 1 +2 += s. +Let Ai = V (Gi) and |Ai| = ai for i ∈ [m]. Denote A = ∪m +i=1Ai. Let IG(A) = {i ∈ [m] : ai = +1}. We may choose G maximizing |IG(A)| (assumption (*)). Let |B| = b. +Define two vertices u and v in B are equivalent if and only if NG(u) = NG(v). Clearly, +it is an equivalent relation. Therefore, the vertices of B can be partitioned into equiva- +lent classes according to the equivalent relation defined above. We may choose G (among +graphs G satisfying assumption (*)) with the minimum number of equivalent classes of B +(assumption (**)). Note that each equivalent class of B is an independent set of G by the +definition of the equivalent relation. We first claim that every two non-adjacent vertices of +B have the same neighborhood (a clique version of Lemma 2.1 of [1]), which is also a simple +consequence of the Zykov symmetrization method introduced in [20], for completeness we +include the proof. +5 + +Claim 2.4. Every two non-adjacent vertices of B have the same neighborhood. +Proof. Suppose there are two non-adjacent vertices u, w ∈ B with NG(u) ̸= NG(w). Then +u and w must be in distinct equivalent classes U and W by the definition of the equivalence. +Since uw /∈ E(G), we have EG(U, W) = ∅. Without loss of generality, suppose d(r−1) +G +(w) ≥ +d(r−1) +G +(u). Let G′ = GU→W . By Proposition 2.2, N(G′, Kr) ≥ N(G, Kr). Now we show that +G′ is {Kk+1, Ms+1}-free too. Clearly, G′ − B still consists of odd components G1, . . . , Gm. +Hence G′ is Ms+1-free by Theorem 2.1. If G′ contains a copy T of Kk+1, we must have a +vertex u ∈ V (T) ∩ U. Choose a vertex w ∈ W. Since NG′(u) = NG′(w) = NG(w), (V (T) \ +{u}) ∪ {w} induces a copy of Kk+1 in G, a contradiction. Hence, GU→W is {Kk+1, Ms+1}- +free. By the extremality of G, we have N(G′, Kr) = N(G, Kr). But the number of equivalent +classes of G′ (U and W merge into one class in G′) is less than the one in G, a contradiction +to assumption (**). +By Claim 2.4 and G is Kk+1-free, G[B] is a complete ℓ-partite graph with ℓ ≤ k. Let its +partition sets be B1, . . . , Bℓ and let Bℓ+1 = · · · = Bk = ∅ if ℓ < k. Let bi = |Bi| for i ∈ [k]. +Without loss of generality, assume b1 ≥ b2 ≥ . . . ≥ bk ≥ 0. Write B = {B1, . . . , Bk−1}. Let +∆r−1 +B += N(K[B], Kr−1). Since �k−1 +i=1 bi = b − bk, by Theorem 1.1, ∆r−1 +B +≤ ∆r−1 +b−bk,k−1. +Recall that A = ∪m +i=1Ai. For those isolated vertices in G[A], we have the following claim. +Claim 2.5. For v ∈ A with dG[A](v) = 0, we have d(r−1) +G +(v) ≤ ∆r−1 +B +≤ ∆r−1 +b−bk,k−1. +Proof. Let T be the vertex set of a copy of Kr covering v. Then T ∩ A = {v} and |T ∩ B| = +r − 1 because dG[A](v) = 0. Therefore, G[T ∩ B] ∼= Kr−1. Note that NG(v) ⊂ B. We have +d(r−1) +G +(v) ≤ N(G[NG(v)], Kr−1). +Let I(v) = {i ∈ [k] : NG(v)∩Bi ̸= ∅}. Apparently, |I(v)| ≤ k −1. Otherwise, I(v) = [k]. +Note that G[B] = K[B1, . . . , Bk] in this case. Thus G[B∪{v}] = K[{v}, B1, . . . , Bk] contains +a copy of Kk+1, a contradiction. Now let B′ = ∪i∈I(v){Bi} ⊆ B. Then +d(r−1) +G +(v) ≤ N(G[NG(v)], Kr−1) ≤ N(K[B′], Kr−1) ≤ N(K[B], Kr−1) = ∆r−1 +B +. +When A is an independent set of G, we have the following claim. +Claim 2.6. If A is an independent set of G, then G ∼= Gk(n, s). +Proof. Let B0 = {B1, · · · , Bk}. Recall that B = {B1, . . . , Bk−1}. Then +N(K[B0], Kr) = ∆r +B0 = ∆r−1 +B +|Bk| + ∆r +B. +6 + +Note that b + �m +i=1 +ai−1 +2 += s. Hence when A = ∪m +i=1Ai is an independent set, we have +a1 = . . . = am = 1 and b = s. By Claim 2.5, +N(G, Kr) +≤ +∆r +B0 + +� +v∈A +dr−1 +G +(v) +≤ +∆r +B0 + (n − s)∆r−1 +B += +(n − s + |Bk|)∆r−1 +B ++ ∆r +B +≤ +[n − (s − bk)]∆r−1 +s−bk,k−1 + ∆r +s−bk,k−1 += +gn,k,r(s − bk). +By Observation 2.3 (2), we have +N(G, Kr) ≤ gn,k,r(s − bk) ≤ gn,k,r(s) = N(Gk(n, s), s). +When the equality holds, we must have bk = 0, G[B] ∼= Tk−1(s) by Theorem 1.1, and +G[B, A] = K[B, A]. This implies that G ∼= Gk(n, s). +Claim 2.7. a2 = a3 = · · · = am = 1. +Proof. If A is an independent set of G, then we are done. Now suppose |G[A]| > 0. Then +b < s. Let v0 be a vertex in A with d(r−1) +G +(v0) = max +v∈A d(r−1) +G +(v). Without loss of generality, +suppose v0 ∈ A1. +If dG[A](v0) = 0, let G′ be the resulting graph by applying the switching operations u → +v0 for all vertices u ∈ A \ {v0} one by one. Then we have |G′[A]| = 0. By Proposition 2.2, +N(G′, Kr) ≥ N(G, Kr). With the same discussion as in the proof of Claim 2.4, we have +that G′ is still {Kk+1, Ms+1}-free. But |IG′(A)| = m > |IG(A)|, a contradiction to the +assumption (*). +If dG[A](v0) > 0, then a1 ≥ 3. If a2 = . . . = am = 1, we are done. Now, without +loss of generality, assume a2 ≥ 3. Since G[A2] is connected, we can pick two vertices, say +u1, u2 in A such that G[A2\{u1, u2}] is still connected (u1, u2 exist, for example, we can +take two leaves of a spanning tree of G[A2]). Let G1 be the resulting graph by applying the +switching operations u1 → v0 and u2 → v0 one by one. With similar discussion as in the +above case, we have N(G1, Kr) ≥ N(G, Kr) and G1 is {Kk+1, Ms+1}-free. Continue the +process after t = a2−1 +2 +steps, we obtain a graph Gt with N(Gt, Kr) ≥ N(G, Kr) and Gt is +{Kk+1, Ms+1}-free. But |IGt(A)| = |IG(A)| + 1, a contradiction to the assumption (*). +Now by Claim 2.7 and Theorem 2.1, we have b + a1−1 +2 += s. Thus, a1 = 2s − 2b + 1 +and then |B ∪ A1| = a1 + b = 2s − b + 1. +Note that 0 ≤ b ≤ s. +By Claim 2.5, for +vertex v ̸∈ B ∪ A1, d(r−1) +G +(v) ≤ ∆r−1 +b−bk,k−1 ≤ ∆r−1 +b,k−1. Also, since G[B ∪ A1] is Kk+1-free, by +Theorem 1.1, N(G[B ∪ A1], Kr) ≤ ∆r +2s−b+1,k. Therefore, +N(G, Kr) ≤ ∆r +2s−b+1,k + (n − 2s + b − 1)∆r−1 +b,k−1. +7 + +Define fn,k,r,s(b) := ∆r +2s−b+1,k +(n−2s+b−1)∆r−1 +b,k−1. If b = 0, then fn,k,r,s(0) = N(Tk(2s+ +1), Kr), and the proof is done. If b = s, then a1 = 1 and thus A is an independent set of G, +the proof is done by Claim 2.6. In particular, fn,k,r,s(s) ≤ N(Gk(n, s), Kr). +By Observation 2.3 (1), for 0 ≤ b ≤ s − 1, +fn,k,r,s(b + 1) − fn,k,r,s(b) = −∆r−1 +(2s−b)−⌊ 2s−b +k +⌋,k−1 + (n − 2s + b)∆r−2 +b−⌊ +b +k−1⌋,k−2 + ∆r−1 +b,k−1. +For fixed k and r, ∆r +t,k is an increasing function of t, and t−⌊ t +k⌋ is a non-decreasing function +of t. It is easy to check that g(b) = fn,k,r,s(b + 1) − fn,k,r,s(b) is an increasing function on +0 ≤ b ≤ s − 1. This implies that fn,k,r,s(b) is convex on [0, s − 1]. Therefore, +fn,k,r,s(b) ≤ max{fn,k,r,s(0), fn,k,r,s(s)} ≤ max{N(Tk(2s + 1), Kr), N(Gk(n, s), Kr)}. +3 +Proof of Theorem 1.5 +Proof of Theorem 1.5 (I). Let H be a graph with χ(H) ≥ 3 and let G be an extremal graph +of {H, Ms+1} on sufficiently large n vertices. Let H = H(H). Let N = N(G, Kr), we will +prove N ≤ ex(n, Kr−1, H)n + O(1). +Since G is Ms+1-free, by Theorem 2.1, there is a vertex set B ⊂ V (G) such that G−B = +G[A1] + G[A2] + · · · + G[Am] for some Ai ⊂ V (G) (i ∈ [m]) of odd sizes, and |B| + +�m +i=1 +|Ai|−1 +2 += s. Let A = ∪m +i=1Ai. +For integer 0 ≤ j ≤ r, let Nj = |{T ⊂ G : T ∼= Kr, |V (T) ∩ A| = j}|, the number of +copies of Kr with exactly j vertices in A. Apparently, N = �r +j=0 Nj. +Since |B| ≤ s, we have N0 ≤ +�s +r +� += O(1). Since �m +i=1(|Ai| − 1) ≤ 2s, we have +|G[A]| +≤ +m +� +i=1 +|K[Ai]| = 1 +2 +m +� +i=1 +� +(|Ai| − 1)2 + |Ai| − 1 +� +≤ +1 +2 + + +� m +� +i=1 +(|Ai| − 1) +�2 ++ +m +� +i=1 +(|Ai| − 1) + + +≤ +2s2 + s. +Since every copy of Kr with exactly j vertices in A must induce a copy of Kr−j in B and +a copy of Kj in A for 2 ≤ j ≤ r, we have +Nj ≤ N(G[B], Kr−j) · N(G[A], Kj) ≤ +� +s +r − j +��2s2 + s +j(j−1) +2 +� += O(1). +Therefore, N = N0 + N1 + �m +j=2 Nj = N1 + O(1). +8 + +Now for any U ⊂ A, let +N1(U) = |{T ⊂ G : T ∼= Kr, |V (T) ∩ U| = |V (T) ∩ A| = 1}|. +For any W ⊂ B, let +AW = {v ∈ A : NG(v) ∩ A = W}. +Apparently, A = ∪W ⊂BAW and N1 = N1(A) = � +W ⊂B +N1(AW ). +For some W ⊂ B, if |AW | ≥ |V (H)|, then G[W] must be H-free, otherwise, suppose +G[W] contains some H′ ∈ H, then any |V (H)| − |V (H′)| vertices in AW together with +H′ would induce a copy of H in G[W ∪ AW ], a contradiction. Thus N(G[W], Kr−1) ≤ +ex(|W|, Kr−1, H). Therefore, for W ⊂ B with |AW | ≥ |V (H)|, +N1(AW ) ≤ ex(|W|, Kr−1, H)|AW | ≤ ex(s, Kr−1, H)|AW |. +Let Q = {W ⊂ B : |AW | ≥ |V (H)|}. Then +� +W ∈Q +N1(AW ) ≤ ex(s, Kr−1, H) +� +W ∈Q +|AW |. +Let R = {W ⊂ B : |AW | < |V (H)|}. Then R = 2B − Q. Thus +� +W ∈R +|AW | ≤ |R|(|V (H)| − 1) ≤ 2s(|V (H)| − 1). +Hence, for every W ∈ R, we have +N1(AW ) ≤ N(G[W], Kr−1)|AW | ≤ +� |W| +r − 1 +� +|AW | ≤ +� +s +r − 1 +� +|AW |. +Therefore, +� +W ∈R +N1(AW ) ≤ +� +s +r − 1 +� � +W ∈R +|AW | ≤ 2s +� +s +r − 1 +� +(|V (H)| − 1) = O(1). +Finally, we have +N1 = +� +W ∈2B +N1(AW ) += +� +W ∈Q +N1(AW ) + +� +W ∈R +N1(AW ) +≤ +ex(s, Kr−1, H) +� +W ∈Q +|AW | + O(1) +≤ +ex(s, Kr−1, H)|A| + O(1) +< +ex(s, Kr−1, H)n + O(1). +The proof is completed because of N(G, Kr) = N1 + O(1). +9 + +Now we give the proof of Theorem 1.5 (II). +Proof of Theorem 1.5 (II). Let H be a graph with χ(H) = 2. Recall that p = p(H) is the +smallest possible number of vertices of a color class in a proper 2-coloring of H. Suppose +r ≥ 3 and G is an extremal graph of {H, Ms+1} on n vertices. +If p > s, then the result (i) follows the fact that ex(n, Kr, {H, Ms+1}) ≤ ex(n, Kr, Ms+1), +where the equality can be obtained by max{N(K2s+1, Kr), N(Gs+1(n, s), Kr)}. +Now, assume p ≤ s and n is sufficiently large. We prove the result (ii). Pick a maximum +matching M in G. Since G is Ms+1-free, |M| ≤ s. Note that H ⊆ Kp,|V (H)|−p. This implies +that for any W ∈ +�V (M) +p +� +, we have +����� +� +u∈W +NG(u) +����� ≤ |V (H)| − p. +Let D≥p = {v ∈ V (G)\V (M) : dG(v) ≥ p}. Note that NG(v) ⊆ V (M). Hence, for v ∈ D≥p +and W ∈ +�NG(v) +p +� +⊆ +�V (M) +p +� +, we have v ∈ ∩u∈WNG(u). Therefore, +|D≥p| ≤ +������� +� +W ∈(V (M) +p ) +� � +u∈W +NG(u) +�������� +≤ +�|V (M)| +p +� +(|V (H)| − p) . +Let B = D≥p ∪ V (M) and A = V (G)\B. Let N0 = N(G[B], Kr). Then +N(G, Kr) ≤ N0 + +� +v∈A +d(r−1) +G +(v). +Since +|B| = |D≥p| + |V (M)| ≤ +�2s +p +� +(|V (H)| − p) + 2s = O(1), +we have N0 ≤ +�|B| +r +� += O(1). For any v ∈ A, dG(v) ≤ p − 1 by the definition. Hence +d(r−1) +G +(v) = N(G[NG(v)], Kr−1) ≤ +�dG(v) +r − 1 +� +≤ +�p − 1 +r − 1 +� +. +Therefore, +N(G, Kr) = N0 + +� +v∈A +d(r−1) +G +(v) ≤ O(1) + +�p − 1 +r − 1 +� +|A| = +�p − 1 +r − 1 +� +n + O(1). +This completes the proof. +10 + +4 +Concluding Remarks +In this paper, we determine the exact value of ex(n, Kr, {Kk+1, Ms+1}) (Theorem 1.4) and +give an asymptotic value of ex(n, Kr, {H, Ms+1}) up to additive error of O(1) for general +H (Theorem 1.5). There is a natural question that if the error term O(1) can be omitted? +Unfortunately, the answer is no. +For example, when H = C5 and for r ∈ {3, 4} and +sufficiently large s, ex(s, Kr−1, H) = 0, while apparently ex(n, Kr, {C5, Ms+1}) > 0. It will +be interesting to consider the following question. +Question 4.1. For every graph H with χ(H) ≥ 3 and r ≥ 3, if ex(s, Kr−1, H(H)) > 0, +then, for sufficiently large n, +ex(n, Kr, {H, Ms+1}) = N(DH(n, s), Kr). +References +[1] N. Alon, P. Frankl, Tur´an graphs with bounded matching number, arXiv2210.15076. +[2] N. Alon and C. Shikhelman. Many T copies in H-free graphs. Journal of Combinatorial +Theory, Series B, 121:146–172, 2016. +[3] C. Berge, Sur le couplage maximum d’un graphe, C.R. Acad. Sci. Paris S´er. I Math, +247(1958), 258-259. +[4] P. Tur´an, On an extremal problem in graph theory, Matematikai´es Fizikai Lapok (in +Hungarian), 48(1941), 436-452. +[5] P. Erd¨os, On the number of complete subgraphs contained in certain graphs, Magy. +Tud. Akad. Mat. Kut. Int´ez. K¨ozl, 7(1962), 459-474. +[6] D. Gerbner, On Tur´an problems with bounded matching number, arXiv2211.03272. +[7] D. Gerbner, Generalized Tur´an problems for small graphs. Discussiones Mathematicae +Graph Theory, 2021, 10.7151/dmgt.2388. +[8] D. Gerbner, On Tur´an-good graphs. Discrete Mathematics. 344 (2021), 112445. +10.1016/j.disc.2021.112445. +[9] D. Gerbner, A non-aligning variant of generalized Tur´an problems, arXiv:2109.02181v1, +2021. +[10] D. Gerbner, Generalized Tur´an problems for double stars, arXiv:2112.11144v2, 2022. +11 + +[11] D. Gerbner, E. Gy˝ori, A. Methuku and M. Vizer, Generalized Tur´an problems for even +cycles, Journal of Combinatorial Theory, Series B, 145:169-213, 2020. +[12] D. Gerbner, C. Palmer, Counting copies of a fixed subgraph in F-free graphs, European +Journal of Combinatorics, 82(2019), pp 103001 +[13] D. +Gerbner, +C. +Palmer, +Some +exact +results +for +generalized +Tur´an +prob- +lems, +European +Journal +of +Combinatorics, +Volume +103 +(2022), +103519, +https://doi.org/10.1016/j.ejc.2022.103519. +[14] D. Hei, +X. Hou, +B. Liu, +Some exact results of the generalized Tur´an num- +bers for paths, European Journal of Combinatorics, Volume 110 (2023), 103682, +https://doi.org/10.1016/j.ejc.2022.103682. +[15] J. Kritschgau, A. Methuku, M. Tait, and C. Timmons, Few H copies in F-saturated +graphs, Journal of Graph Theory, 94(3):320–348, 2020. +[16] L. Lov´asz, M.D. Plummer, Mathching theory, Ann. Discrete Math., 29(1986). +[17] J. Ma, Y. Qiu, Some sharp results on the generalized Tur´an numbers, European J. +Combin., 84(2020), 103026. +[18] M. Simonovits, A method for solving extremal problems in graph theory, stability +problems, in Theory of Graphs, Proc. Colloq., Tihany, 1966, Academic Press, New +York, (1968), pp.279-319. +[19] L. Shoham, Many H-Copies in Graphs with a Forbidden Tree, SIAM Journal on Dis- +crete Mathematics, 33(4):2360-2368, 2019. +[20] A.A. Zykov, On some properties of linear complexes, Mat. Sbornik N.S., 24(66)(1949), +163-188. +12 + diff --git a/b9E5T4oBgHgl3EQffQ9l/content/tmp_files/load_file.txt b/b9E5T4oBgHgl3EQffQ9l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..59dec78fa07374a249b174bafe58ddc17400c68c --- /dev/null +++ b/b9E5T4oBgHgl3EQffQ9l/content/tmp_files/load_file.txt @@ -0,0 +1,464 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf,len=463 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='05625v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='CO] 13 Jan 2023 Generalized Tur´an problem with bounded matching number ∗ Yue Maa, Xinmin Houa,b a School of Mathematical Sciences University of Science and Technology of China, Hefei, Anhui 230026, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' b CAS Key Laboratory of Wu Wen-Tsun Mathematics University of Science and Technology of China, Hefei, Anhui 230026, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Abstract For a graph T and a set of graphs H, let ex(n, T, H) denote the maximum number of copies of T in an n-vertex H-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Recently, Alon and Frankl (arXiv2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='15076) determined the exact value of ex(n, K2, {Kk+1, Ms+1}), where Kk+1 and Ms+1 are complete graph on k + 1 vertices and matching of size s + 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Soon after, Gerbner (arXiv2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='03272) continued the study by extending Kk+1 to general fixed graph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' In this paper, we continue the study of the function ex(n, T, {H, Ms+1}) when T = Kr for r ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We determine the exact value of ex(n, Kr, {Kk+1, Ms+1}) and give the value of ex(n, Kr, {H, Ms+1}) for general H with an error term O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 1 Introduction Let G = (V, E) be a graph with vertex set V = V (G) and edge set E = E(G) ⊂ �V 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We may write G instead of E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let T be a fixed graph and H be a set of given graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' A graph G is called H-free if G contains no copy of any member in H as its subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Write N(G, T) for the number of copies of T in a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Define the generalized Tur´an number as ex(n, T, H) = max{N(G, T) : G is an n-vertex H-free graph}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We call an n-vertex graph G with N(G, T) attaining the maximum an extremal graph of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' This function has been systematically studied by Alon and Shikhelman [2] and has received ∗The work was supported by the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 12071453), the National Key R and D Program of China(2020YFA0713100), the Anhui Initiative in Quantum Informa- tion Technologies (AHY150200) and the Innovation Program for Quantum Science and Technology, China (2021ZD0302904).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 1 much attention, for example, in [7, 8, 9, 10, 11, 12, 13, 14, 15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' When T = K2, it is the classical Tur´an number ex(n, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let Kr denote a complete graph on r vertices for some integer r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For a set U, write K[U] for a complete graph on vertex set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Ur be disjoint sets and U = {U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Ur}, write K[U] = K[U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Ur] for a complete r-partite graph with partition sets U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let G = (V, E) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For a set U ⊆ V , write G[U] for the subgraph induced by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For disjoint sets U1, U2, · · · , Ur ⊆ V , write G[U1, · · · , Ur] for the induced r-partite subgraph of G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' G[U1, · · · , Ur] = K[U1, · · · , Ur]∩G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let Ka1,a2,··· ,ar denote the complete r-partite graph with partition sets of size a1, a2, · · · , ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For graphs G1, · · · , Gr, let �r i=1 G1 + · · · + Gr be the union of vertex-disjoint copies of G1, · · · , Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' A Tur´an graph Tk(n) is a complete k-partite graph on n vertices whose partition sets have sizes as equal as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let tk(n) = |Tk(n)| = N(Tk(n), K2) be the Tur´an number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' The famous Tur´an Theorem [4] states that ex(n, K2, Kk+1) = tk(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Erd˝os [5] gave the generalized version of Tur´an Theorem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1 ([5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For all n ≥ k ≥ r ≥ 2, ex(n, Kr, Kk+1) = N(Tk(n), Kr), and Tk(n) is the unique extremal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Write χ(G) for the chromatic number of graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We say a graph is edge-critical if there exists some edge whose deletion reduces its chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Simonovits [18] proved that for any edge-critical graph H with χ(H) = k + 1 ≥ 3, ex(n, K2, H) = tk(n) for sufficiently large n, and Tk(n) is the unique extremal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' This result was extended by Ma and Qiu [17] as follows: For sufficiently large n, ex(n, Kr, H) = N(Tk(n), Kr), and Tk(n) is the unique extremal graph, where H is an edge-critical graph with χ(H) = k + 1 > r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Recently, Alon and Frankl [1] studied the function ex(n, K2, H) when H = {Kk+1, Ms+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let Gk(n, s) = Kn−s ∨ Tk−1(s), the join of an empty graph Kn−s and Tur´an graph Tk−1(s), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' a complete k-partite graph on n vertices with one partition set of size n−s and the oth- ers having sizes as equal as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Given a graph H with χ(H) ≥ 3, define H(H) to be the family of graphs obtained by deleting a color class from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Define DH(n, s) = D ∨ Kn−s, where D is a copy of extremal graph of H(H) on s vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Write Mk for a matching consisting of k edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='2 ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (1) For n ≥ 2s + 1 and k ≥ 2, ex(n, K2, {Kk+1, Ms+1}) = max {|Tk(2s + 1)|, |Gk(n, s)|} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (2) Let H be an edge-critical graph with χ(H) = k + 1 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then, for sufficiently large s and n ≫ s, ex(n, K2, {H, Ms+1}) = |Gk(n, s)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='2 (2) was soon strengthened by Gerbner [6] in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For a bipartite graph H, let p = p(H) denote the smallest possible number of a color class in a proper 2-coloring of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='3 ([6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let H be a fixed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (1) Suppose χ(H) ≥ 3 and n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then ex(n, K2, {H, Ms+1}) = s(n − s) + ex(s, K2, H(H)) = sn + O(1), and the graph DH(n, s) is an extremal graph of {H, Ms+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (2) Suppose χ(H) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' The following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (i) If p > s, then ex(n, K2, {H, Ms+1}) = ex(n, K2, Ms+1), and K2s+1 and Gs+1(n, s) are extremal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (ii) If p ≤ s, then ex(n, K2, {H, Ms+1}) = (p − 1)n + O(1), and Gp(n, p − 1) is an asymptotically optimal graph up to additive error of O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' In this article, we first extend the result of Alon and Frankl [1] as shown in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For n ≥ 2s + 1 and k ≥ r ≥ 3, ex(n, Kr, {Kk+1, Ms+1}) = max{N(Tk(2s + 1), Kr), N(Gk(n, s), Kr)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Second, we extend Gebner’s result [6] with a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (I) For every graph H with χ(H) ≥ 3, r ≥ 3 and sufficiently large n, ex(n, Kr, {H, Ms+1}) = ex(s, Kr−1, H(H))n + O(1) = N(DH(n, s), Kr) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (II) For every graph H with χ(H) = 2 and r ≥ 3, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (i) If p > s, then ex(n, Kr, {H, Ms+1}) = ex(n, Kr, Ms+1), and K2s+1 and Gs+1(n, s) are extremal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (ii) If p ≤ s, then ex(n, Kr, {H, Ms+1}) = N(Gp(n, p − 1), Kr) + O(1) = �p − 1 r − 1 � n + O(1), and Gp(n, p − 1) is an asymptotically optimal graph up to additive error of O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' The rest of the article is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We give the proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5 in Sections 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We give some discussions in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 3 2 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='4 We need the following fundamental theorem in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1 (Tutte-Berge Theorem [3], see also [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' A graph G is Ms+1-free if and only if there is a set B ⊂ V (G) such that all the components G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Gm of G − B are odd (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' |V (Gi)| ≡ 1 (mod 2) for i ∈ [m]), and |B| + m � i=1 |V (Gi)| − 1 2 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let graph G = (V, E) and integer r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For a vertex v ∈ V (G), define N (r) G (v) = {U ∈ �V r � : G[U ∪ {v}] ∼= Kr+1} be the r-clique neighborhood of v and d(r) G (v) = |N (r) G (v)| be the r-clique-degree of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For a set U ⊂ V (G), write d(r) G (U) = � u∈U d(r) G (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' As usual, write neighborhood NG(v) and degree dG(v) instead of 1-clique neighborhood N (1) G (v) and 1-clique-degree d(1) G (v) for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For two non-adjacent vertices u, v in a graph G, we define the switching operation u → v as deleting the edges joining u to its neighbors and adding new edges connecting u to vertices in NG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let Gu→v to be the graph obtained from G by the switching operation u → v, that is V (Gu→v) = V (G) and E(Gu→v) = (E(G) \\ EG(u, NG(u))) ∪ EG(u, NG(v)), where EG(S, T) = E(G[S, T]) for disjoint subsets S, T ⊂ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Note that the edges between u and the common neighbors of u and v remain unchanged by the definition of Gu→v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For two disjoint independent sets S and T in a graph G, if all of vertices in S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' T) have the same neighborhood NG(S) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' NG(T)), we similarly define GS→T to be the graph obtained from G by deleting the edges between S and NG(S) and adding new edges connecting S and NG(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For r ≥ 2 and two disjoint independent sets S and T in a graph G, if all of vertices in S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' T) have the same neighborhood NG(S) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' NG(T)) and EG(S, T) = ∅, then either G′ = GS→T or G′ = GT→S has the property that N(G′, Kr) ≥ N(G, Kr), the equality holds if and only if d(r−1) G (S) = d(r−1) G (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let S and T be two such independent sets of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Without loss of generality, suppose d(r−1) G (T) ≥ d(r−1) G (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let G′ = GS→T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then N(G′, Kr) = N(G, Kr) − d(r−1) G (S) + d(r−1) G (T) ≥ N(G, Kr), the equality holds if and only if d(r−1) G (T) = d(r−1) G (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 4 Let ∆r t,k = N(Tk(t), Kr) for some integers t ≥ k ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (1) For positive integers t ≥ k ≥ r ≥ 2, ∆r t+1,k − ∆r t,k = ∆r−1 t−⌊ t k ⌋,k−1 and ∆r t,k = ∆r t−⌊ t k ⌋,k−1 + � t k � ∆r−1 t−⌊ t k⌋,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (2) For n ≥ 2s + 1 and s ≥ t ≥ k ≥ r ≥ 3, define gn,k,r(t) := (n − t)∆r−1 t,k−1 + ∆r t,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then gn,k,r(t) is a strictly increasing function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' In particular, gn,k,r(s) = N(Gk(n, s), Kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (1) can be checked directly by the definitions of ∆r t,k and the Tur´an graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' (2) By (1), for t ≤ s − 1, gn,k,r(t + 1) − gn,k,r(t) = (n − t − 1)∆r−2 t−⌊ t k−1 ⌋,k−2 − ∆r−1 t,k−1 + ∆r−1 t−⌊ t k−1⌋,k−2 = (n − t − 1)∆r−2 t−⌊ t k−1 ⌋,k−2 − � t k − 1 � ∆r−2 t−⌊ t k−1⌋,k−2 = � n − 1 − t − � t k − 1 �� ∆r−2 t−⌊ t k−1⌋,k−2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Now we are ready to give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='4: Let G be an extremal graph of {Kk+1, Ms+1} on n ≥ 2s+1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1, there is a vertex set B ⊂ V (G) such that G−B consists of odd components G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Gm, and |B| + m � i=1 |V (Gi)| − 1 2 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let Ai = V (Gi) and |Ai| = ai for i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Denote A = ∪m i=1Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let IG(A) = {i ∈ [m] : ai = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We may choose G maximizing |IG(A)| (assumption (*)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let |B| = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Define two vertices u and v in B are equivalent if and only if NG(u) = NG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Clearly, it is an equivalent relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Therefore, the vertices of B can be partitioned into equiva- lent classes according to the equivalent relation defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We may choose G (among graphs G satisfying assumption (*)) with the minimum number of equivalent classes of B (assumption (**)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Note that each equivalent class of B is an independent set of G by the definition of the equivalent relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We first claim that every two non-adjacent vertices of B have the same neighborhood (a clique version of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1 of [1]), which is also a simple consequence of the Zykov symmetrization method introduced in [20], for completeness we include the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 5 Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Every two non-adjacent vertices of B have the same neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Suppose there are two non-adjacent vertices u, w ∈ B with NG(u) ̸= NG(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then u and w must be in distinct equivalent classes U and W by the definition of the equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since uw /∈ E(G), we have EG(U, W) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Without loss of generality, suppose d(r−1) G (w) ≥ d(r−1) G (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let G′ = GU→W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='2, N(G′, Kr) ≥ N(G, Kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Now we show that G′ is {Kk+1, Ms+1}-free too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Clearly, G′ − B still consists of odd components G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Hence G′ is Ms+1-free by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' If G′ contains a copy T of Kk+1, we must have a vertex u ∈ V (T) ∩ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Choose a vertex w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since NG′(u) = NG′(w) = NG(w), (V (T) \\ {u}) ∪ {w} induces a copy of Kk+1 in G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Hence, GU→W is {Kk+1, Ms+1}- free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' By the extremality of G, we have N(G′, Kr) = N(G, Kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' But the number of equivalent classes of G′ (U and W merge into one class in G′) is less than the one in G, a contradiction to assumption (**).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='4 and G is Kk+1-free, G[B] is a complete ℓ-partite graph with ℓ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let its partition sets be B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Bℓ and let Bℓ+1 = · · · = Bk = ∅ if ℓ < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let bi = |Bi| for i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Without loss of generality, assume b1 ≥ b2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' ≥ bk ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Write B = {B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Bk−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let ∆r−1 B = N(K[B], Kr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since �k−1 i=1 bi = b − bk, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1, ∆r−1 B ≤ ∆r−1 b−bk,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Recall that A = ∪m i=1Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For those isolated vertices in G[A], we have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For v ∈ A with dG[A](v) = 0, we have d(r−1) G (v) ≤ ∆r−1 B ≤ ∆r−1 b−bk,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let T be the vertex set of a copy of Kr covering v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then T ∩ A = {v} and |T ∩ B| = r − 1 because dG[A](v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Therefore, G[T ∩ B] ∼= Kr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Note that NG(v) ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We have d(r−1) G (v) ≤ N(G[NG(v)], Kr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let I(v) = {i ∈ [k] : NG(v)∩Bi ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Apparently, |I(v)| ≤ k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Otherwise, I(v) = [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Note that G[B] = K[B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Bk] in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Thus G[B∪{v}] = K[{v}, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Bk] contains a copy of Kk+1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Now let B′ = ∪i∈I(v){Bi} ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then d(r−1) G (v) ≤ N(G[NG(v)], Kr−1) ≤ N(K[B′], Kr−1) ≤ N(K[B], Kr−1) = ∆r−1 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' When A is an independent set of G, we have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' If A is an independent set of G, then G ∼= Gk(n, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let B0 = {B1, · · · , Bk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Recall that B = {B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' , Bk−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then N(K[B0], Kr) = ∆r B0 = ∆r−1 B |Bk| + ∆r B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 6 Note that b + �m i=1 ai−1 2 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Hence when A = ∪m i=1Ai is an independent set, we have a1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' = am = 1 and b = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5, N(G, Kr) ≤ ∆r B0 + � v∈A dr−1 G (v) ≤ ∆r B0 + (n − s)∆r−1 B = (n − s + |Bk|)∆r−1 B + ∆r B ≤ [n − (s − bk)]∆r−1 s−bk,k−1 + ∆r s−bk,k−1 = gn,k,r(s − bk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' By Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='3 (2), we have N(G, Kr) ≤ gn,k,r(s − bk) ≤ gn,k,r(s) = N(Gk(n, s), s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' When the equality holds, we must have bk = 0, G[B] ∼= Tk−1(s) by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1, and G[B, A] = K[B, A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' This implies that G ∼= Gk(n, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' a2 = a3 = · · · = am = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' If A is an independent set of G, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Now suppose |G[A]| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then b < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let v0 be a vertex in A with d(r−1) G (v0) = max v∈A d(r−1) G (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Without loss of generality, suppose v0 ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' If dG[A](v0) = 0, let G′ be the resulting graph by applying the switching operations u → v0 for all vertices u ∈ A \\ {v0} one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then we have |G′[A]| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='2, N(G′, Kr) ≥ N(G, Kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' With the same discussion as in the proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='4, we have that G′ is still {Kk+1, Ms+1}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' But |IG′(A)| = m > |IG(A)|, a contradiction to the assumption (*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' If dG[A](v0) > 0, then a1 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' If a2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' = am = 1, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Now, without loss of generality, assume a2 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since G[A2] is connected, we can pick two vertices, say u1, u2 in A such that G[A2\\{u1, u2}] is still connected (u1, u2 exist, for example, we can take two leaves of a spanning tree of G[A2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let G1 be the resulting graph by applying the switching operations u1 → v0 and u2 → v0 one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' With similar discussion as in the above case, we have N(G1, Kr) ≥ N(G, Kr) and G1 is {Kk+1, Ms+1}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Continue the process after t = a2−1 2 steps, we obtain a graph Gt with N(Gt, Kr) ≥ N(G, Kr) and Gt is {Kk+1, Ms+1}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' But |IGt(A)| = |IG(A)| + 1, a contradiction to the assumption (*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Now by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='7 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1, we have b + a1−1 2 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Thus, a1 = 2s − 2b + 1 and then |B ∪ A1| = a1 + b = 2s − b + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Note that 0 ≤ b ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' By Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5, for vertex v ̸∈ B ∪ A1, d(r−1) G (v) ≤ ∆r−1 b−bk,k−1 ≤ ∆r−1 b,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Also, since G[B ∪ A1] is Kk+1-free, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1, N(G[B ∪ A1], Kr) ≤ ∆r 2s−b+1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Therefore, N(G, Kr) ≤ ∆r 2s−b+1,k + (n − 2s + b − 1)∆r−1 b,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 7 Define fn,k,r,s(b) := ∆r 2s−b+1,k +(n−2s+b−1)∆r−1 b,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' If b = 0, then fn,k,r,s(0) = N(Tk(2s+ 1), Kr), and the proof is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' If b = s, then a1 = 1 and thus A is an independent set of G, the proof is done by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' In particular, fn,k,r,s(s) ≤ N(Gk(n, s), Kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' By Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='3 (1), for 0 ≤ b ≤ s − 1, fn,k,r,s(b + 1) − fn,k,r,s(b) = −∆r−1 (2s−b)−⌊ 2s−b k ⌋,k−1 + (n − 2s + b)∆r−2 b−⌊ b k−1⌋,k−2 + ∆r−1 b,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For fixed k and r, ∆r t,k is an increasing function of t, and t−⌊ t k⌋ is a non-decreasing function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' It is easy to check that g(b) = fn,k,r,s(b + 1) − fn,k,r,s(b) is an increasing function on 0 ≤ b ≤ s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' This implies that fn,k,r,s(b) is convex on [0, s − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Therefore, fn,k,r,s(b) ≤ max{fn,k,r,s(0), fn,k,r,s(s)} ≤ max{N(Tk(2s + 1), Kr), N(Gk(n, s), Kr)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 3 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5 (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let H be a graph with χ(H) ≥ 3 and let G be an extremal graph of {H, Ms+1} on sufficiently large n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let H = H(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let N = N(G, Kr), we will prove N ≤ ex(n, Kr−1, H)n + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since G is Ms+1-free, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1, there is a vertex set B ⊂ V (G) such that G−B = G[A1] + G[A2] + · · · + G[Am] for some Ai ⊂ V (G) (i ∈ [m]) of odd sizes, and |B| + �m i=1 |Ai|−1 2 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let A = ∪m i=1Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For integer 0 ≤ j ≤ r, let Nj = |{T ⊂ G : T ∼= Kr, |V (T) ∩ A| = j}|, the number of copies of Kr with exactly j vertices in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Apparently, N = �r j=0 Nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since |B| ≤ s, we have N0 ≤ �s r � = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since �m i=1(|Ai| − 1) ≤ 2s, we have |G[A]| ≤ m � i=1 |K[Ai]| = 1 2 m � i=1 � (|Ai| − 1)2 + |Ai| − 1 � ≤ 1 2 \uf8ee \uf8f0 � m � i=1 (|Ai| − 1) �2 + m � i=1 (|Ai| − 1) \uf8f9 \uf8fb ≤ 2s2 + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since every copy of Kr with exactly j vertices in A must induce a copy of Kr−j in B and a copy of Kj in A for 2 ≤ j ≤ r, we have Nj ≤ N(G[B], Kr−j) · N(G[A], Kj) ≤ � s r − j ��2s2 + s j(j−1) 2 � = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Therefore, N = N0 + N1 + �m j=2 Nj = N1 + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 8 Now for any U ⊂ A, let N1(U) = |{T ⊂ G : T ∼= Kr, |V (T) ∩ U| = |V (T) ∩ A| = 1}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For any W ⊂ B, let AW = {v ∈ A : NG(v) ∩ A = W}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Apparently, A = ∪W ⊂BAW and N1 = N1(A) = � W ⊂B N1(AW ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For some W ⊂ B, if |AW | ≥ |V (H)|, then G[W] must be H-free, otherwise, suppose G[W] contains some H′ ∈ H, then any |V (H)| − |V (H′)| vertices in AW together with H′ would induce a copy of H in G[W ∪ AW ], a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Thus N(G[W], Kr−1) ≤ ex(|W|, Kr−1, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Therefore, for W ⊂ B with |AW | ≥ |V (H)|, N1(AW ) ≤ ex(|W|, Kr−1, H)|AW | ≤ ex(s, Kr−1, H)|AW |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let Q = {W ⊂ B : |AW | ≥ |V (H)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then � W ∈Q N1(AW ) ≤ ex(s, Kr−1, H) � W ∈Q |AW |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let R = {W ⊂ B : |AW | < |V (H)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then R = 2B − Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Thus � W ∈R |AW | ≤ |R|(|V (H)| − 1) ≤ 2s(|V (H)| − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Hence, for every W ∈ R, we have N1(AW ) ≤ N(G[W], Kr−1)|AW | ≤ � |W| r − 1 � |AW | ≤ � s r − 1 � |AW |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Therefore, � W ∈R N1(AW ) ≤ � s r − 1 � � W ∈R |AW | ≤ 2s � s r − 1 � (|V (H)| − 1) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Finally, we have N1 = � W ∈2B N1(AW ) = � W ∈Q N1(AW ) + � W ∈R N1(AW ) ≤ ex(s, Kr−1, H) � W ∈Q |AW | + O(1) ≤ ex(s, Kr−1, H)|A| + O(1) < ex(s, Kr−1, H)n + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' The proof is completed because of N(G, Kr) = N1 + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 9 Now we give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5 (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5 (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let H be a graph with χ(H) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Recall that p = p(H) is the smallest possible number of vertices of a color class in a proper 2-coloring of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Suppose r ≥ 3 and G is an extremal graph of {H, Ms+1} on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' If p > s, then the result (i) follows the fact that ex(n, Kr, {H, Ms+1}) ≤ ex(n, Kr, Ms+1), where the equality can be obtained by max{N(K2s+1, Kr), N(Gs+1(n, s), Kr)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Now, assume p ≤ s and n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' We prove the result (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Pick a maximum matching M in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since G is Ms+1-free, |M| ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Note that H ⊆ Kp,|V (H)|−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' This implies that for any W ∈ �V (M) p � , we have ����� � u∈W NG(u) ����� ≤ |V (H)| − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let D≥p = {v ∈ V (G)\\V (M) : dG(v) ≥ p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Note that NG(v) ⊆ V (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Hence, for v ∈ D≥p and W ∈ �NG(v) p � ⊆ �V (M) p � , we have v ∈ ∩u∈WNG(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Therefore, |D≥p| ≤ ������� � W ∈(V (M) p ) � � u∈W NG(u) �������� ≤ �|V (M)| p � (|V (H)| − p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let B = D≥p ∪ V (M) and A = V (G)\\B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Let N0 = N(G[B], Kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Then N(G, Kr) ≤ N0 + � v∈A d(r−1) G (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Since |B| = |D≥p| + |V (M)| ≤ �2s p � (|V (H)| − p) + 2s = O(1), we have N0 ≤ �|B| r � = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For any v ∈ A, dG(v) ≤ p − 1 by the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Hence d(r−1) G (v) = N(G[NG(v)], Kr−1) ≤ �dG(v) r − 1 � ≤ �p − 1 r − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Therefore, N(G, Kr) = N0 + � v∈A d(r−1) G (v) ≤ O(1) + �p − 1 r − 1 � |A| = �p − 1 r − 1 � n + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 10 4 Concluding Remarks In this paper, we determine the exact value of ex(n, Kr, {Kk+1, Ms+1}) (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='4) and give an asymptotic value of ex(n, Kr, {H, Ms+1}) up to additive error of O(1) for general H (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' There is a natural question that if the error term O(1) can be omitted?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Unfortunately, the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For example, when H = C5 and for r ∈ {3, 4} and sufficiently large s, ex(s, Kr−1, H) = 0, while apparently ex(n, Kr, {C5, Ms+1}) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' It will be interesting to consider the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' For every graph H with χ(H) ≥ 3 and r ≥ 3, if ex(s, Kr−1, H(H)) > 0, then, for sufficiently large n, ex(n, Kr, {H, Ms+1}) = N(DH(n, s), Kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Alon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Frankl, Tur´an graphs with bounded matching number, arXiv2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content='15076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Alon and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Shikhelman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Many T copies in H-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' Journal of Combinatorial Theory, Series B, 121:146–172, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' [3] C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=', 24(66)(1949), 163-188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E5T4oBgHgl3EQffQ9l/content/2301.05625v1.pdf'} diff --git a/btE1T4oBgHgl3EQfxQWA/content/tmp_files/2301.03420v1.pdf.txt b/btE1T4oBgHgl3EQfxQWA/content/tmp_files/2301.03420v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb286faff3e6366d1ce215c32ecea219e49d5147 --- /dev/null +++ b/btE1T4oBgHgl3EQfxQWA/content/tmp_files/2301.03420v1.pdf.txt @@ -0,0 +1,493 @@ +arXiv:2301.03420v1 [math.CO] 9 Jan 2023 +Criticality in Sperner’s Lemma +Tom´aˇs Kaiser 1 +Matˇej Stehl´ık 2 +Riste ˇSkrekovski 3 +Abstract +We answer a question posed by Tibor Gallai in 1969 concerning +criticality in Sperner’s lemma, listed as Problem 9.14 in the collection +of Jensen and Toft [Graph coloring problems, John Wiley & Sons, +Inc., New York, 1995]. +Sperner’s lemma states that if a labelling of the vertices of a trian- +gulation of the d-simplex ∆d with labels 1, 2, . . . , d+1 has the property +that (i) each vertex of ∆d receives a distinct label, and (ii) any vertex +lying in a face of ∆d has the same label as one of the vertices of that +face, then there exists a rainbow facet (a facet whose vertices have +pairwise distinct labels). For d ≤ 2, it is not difficult to show that for +every facet σ, there exists a labelling with the above properties where +σ is the unique rainbow facet. For every d ≥ 3, however, we construct +an infinite family of examples where this is not the case, which im- +plies the answer to Gallai’s question as a corollary. The construction +is based on the properties of a 4-polytope which had been used earlier +to disprove a claim of Theodore Motzkin on neighbourly polytopes. +1 +Introduction +A central result of combinatorial topology, Sperner’s lemma [20] has found +wide-ranging applications in diverse areas of mathematics and beyond; for +1Department of Mathematics and European Centre of Excellence NTIS (New Technolo- +gies for the Information Society), University of West Bohemia, Pilsen, Czech Republic. +E-mail: kaisert@kma.zcu.cz. Supported by project GA20-09525S of the Czech Science +Foundation. +2Universit´e Paris Cit´e, CNRS, IRIF, F-75006, Paris, France. E-mail: matej@irif.fr. +Partially supported by ANR project DISTANCIA (ANR-17-CE40-0015). +3Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana & Fac- +ulty of Information Studies, 8000 Novo Mesto, Slovenia. E-mail: skrekovski@gmail.com. +Partially supported by the Slovenian Research Agency Program P1-0383, Project J1-3002, +and BI-FR/22-23-Proteus-011. +1 + +example, it can be used to give an elementary proof of Brouwer’s fixed point +theorem [18, p. 117] from algebraic topology, and to solve various fair-division +problems, such as rental harmony [22, 23]. +Let ∆d be a d-simplex with vertices v1, . . . , vd+1, and let K be a finite +triangulation of ∆d (see Section 2 for a precise definition). A Sperner labelling +of K is a map λ : V (K) → {1, . . . , d+1} such that λ(vi) = i, and every vertex +lying in a face of ∆d has the same label as one of the vertices of that face. A +simplex is said to be rainbow if its vertices receive pairwise distinct labels. +The following classical result, proved by Sperner [20], is usually referred to +as Sperner’s lemma (see e.g. [5, p. 423], [6, p. 3]). +Theorem 1.1 (Sperner’s lemma). If K is a finite triangulation of the d- +simplex ∆d and λ : V (K) → {1, . . . , d + 1} is any Sperner labelling of K, +then there is a rainbow facet. +A natural question, asked by T. Gallai in 1969 (in a different but equiv- +alent form; see Problem 9.14 in Jensen and Toft [12] and Proposition 3.3 +below) is whether Sperner’s lemma is ‘critical’, in the following sense: +Problem 1.2. Given any triangulation K of ∆d, and any facet σ ∈ K, is +there a Sperner labelling of K where σ is the unique rainbow facet? +In dimension 1, the answer to Problem 1.2 is is easily seen to be positive. +With a little more work, we can show that Problem 1.2 also has a positive +answer in dimension 2. +Theorem 1.3. For every triangulation K of ∆2 and every facet σ ∈ K, +there exists a Sperner labelling of K where σ is the unique rainbow facet. +Proof. Let K be a triangulation of the 2-simplex ∆2; add edges and faces +as shown in Figure 1 to make it into a triangulation K′ of S2. Let τ be +the 2-simplex corresponding to the outer face, and assume that σ ∈ K is +a given facet. +Every planar triangulation is 3-connected, so by Steinitz’s +theorem [16, p. 63], it is the graph of a convex polyhedron, and the dual +graph is also 3-connected. Hence, by Menger’s theorem [16, p. 14], there exist +three internally vertex-disjoint paths between σ and τ in the dual graph. The +union of these paths (in the given embedding) has three faces; let us assign +the vertices of K′ embedded in the i-th face (1 ≤ i ≤ 3) label i. It is not +hard to see that this determines a Sperner labelling of K. Furthermore, if +a 3-cycle of K′ contains vertices of all three labels and differs from the face +boundaries of σ and τ, then it separates a vertex of σ from a vertex of τ — +in particular, it is not a face boundary. Thus, σ is the unique rainbow facet +in the Sperner labelling of K as claimed. +2 + +σ +τ +Figure 1: A triangulation K of ∆2 (in gray) extended to a triangulation K′ of +S2 (thick edges and white faces; the outer face is labelled τ). Three internally +vertex-disjoint paths (dashed) in the dual graph partition the vertices into +three subsets (red, green and blue). +In this paper, we will show that the answer to Problem 1.2 is negative +in dimension 3 and higher. Namely, we will prove the following theorem in +Section 5. +Theorem 1.4. For every integer d ≥ 3, there exist infinitely many triangu- +lations K of ∆d such that there exists a facet σ ∈ K with the property that +every Sperner labelling of K has a rainbow facet distinct from σ. +2 +Terminology and preliminary results +For topological and graph-theoretic concepts not defined here, the reader +is referred to Munkres [18] and to Bondy and Murty [3], respectively. An +introduction to Sperner’s lemma — and combinatorial topology in general — +can be found in de Longueville [6]. The book by Mohar and Thomassen [16] +provides an introduction to topological graph theory. For polytope theory, +we refer the reader to Gr¨unbaum [10] and to Ziegler [25]. +A k-colouring of a graph G is a function c : V (G) → {1, . . . , k} such that +c(u) ̸= c(v) whenever uv ∈ E(G). The minimum k for which a k-colouring +of G exists is known as the chromatic number of G, denoted by χ(G). If +χ(G) ≤ k or χ(G) = k, then G is said to be k-colourable or k-chromatic, +respectively. Furthermore, a graph G is said to be k-critical (resp. k-vertex- +critical) if G is k-chromatic and deleting any edge (resp. vertex) results in +3 + +a (k − 1)-colourable graph. A graph is said to be k-vertex-connected if it +has at least k + 1 vertices, and removing any set of k − 1 vertices does not +disconnect the graph. +A (geometric) simplicial complex K is a collection of geometric simplices +of various dimensions satisfying two conditions: firstly, if σ ∈ K and τ is a +face of σ, then τ ∈ K; secondly, if σ, τ ∈ K and σ∩τ ̸= ∅, then σ∩τ is a face +of both σ and τ. We denote the regular d-dimensional simplex, unit d-ball +and unit d-sphere by ∆d, Bd and Sd, respectively. Simplices of K that are +not strictly contained in other simplices are the facets of K. The union of +all simplices of K, known as the polyhedron of K, is denoted by |K|. The k- +skeleton of a simplicial complex K, denoted K(k), consists of all the simplices +of K of dimension at most k; note that K(1) consists of vertices and edges, +i.e., it is a graph. If two topological spaces X and Y are homeomorphic, we +write X ∼= Y . A simplicial complex K such that X ∼= |K|, if one exists, is +called a triangulation of X. The join of two simplicial complexes K and L +is denoted by K ⋆ L. +A quadrangulation of a surface (i.e., a 2-dimensional manifold) X is a +graph embedded in X so that every face of the embedding is bounded by a 4- +cycle. In [13], the definition of quadrangulations was extended to triangulable +topological spaces in the following way. A graph G is a quadrangulation of +a topological space X if G is a spanning subgraph of the 1-skeleton K(1) +of some triangulation K of X, with the property that the vertices of every +facet of K induce a complete bipartite subgraph of G with at least one edge. +This definition coincides with the usual definition in the case when X is a +2-dimensional manifold. In [13, Lemma 3.2], the following characterisation +was given of non-bipartite quadrangulations of the d-dimensional projective +space RPd: +Lemma 2.1. A graph G is a non-bipartite quadrangulation of RPd if and only +if there exists a centrally symmetric triangulation K of Sd and a 2-colouring +of the vertices with colours black and white, such that no pair of antipodal +vertices is adjacent, antipodal vertices receive different colours, every facet of +K contains a black and a white vertex, and G is obtained from the 1-skeleton +K(1) by first deleting all monochromatic edges, and then identifying antipodal +vertices and edges. +If P is a (convex) d-polytope, its (d − 1)-dimensional faces are its facets. +If every facet is a simplex, then the boundary of P, denoted ∂P, is a sim- +plicial complex such that |∂P| ∼= Sd−1, i.e., ∂P is a triangulation of Sd−1. +The convex hull of the vectors of the standard orthonormal basis and their +negatives in Rd is known as the the d-dimensional cross polytope. +4 + +Figure 2: A triangulation K (dashed edges) of a 2-simplex (in gray), and the +graph GK (solid edges). +3 +Relation to chromatic number +Gallai used Sperner’s lemma to construct a family of 4-chromatic planar +graphs; see [12, Problem 9.14] or [19]. Let K be a triangulation of ∆d, and +define the graph GK as follows (see Figure 2 for an illustration). There are +three types of vertices of GK: +(V1) The vertices u1, . . . , un of K; +(V2) A vertex vρ for every d-simplex ρ ∈ K; +(V3) A vertex wσ for every (d − 1)-simplex σ ∈ ∆d. +There are also three types of edges of GK: +(E1) uivρ ∈ E(GK) if and only if ui ∈ ρ, where ui is a vertex of K and ρ is +a d-simplex of K; +(E2) uiwσ ∈ E(GK) if and only if ui belongs to the facet σ of ∆d; +(E3) wσwτ ∈ E(GK) for all (d − 1)-simplices σ, τ ∈ ∆d (i.e., the vertices wσ +induce a (d + 1)-clique of GK). +Gallai observed that the statement that GK is not (d + 1)-colourable is +equivalent to Sperner’s lemma; we formalise it as the following proposition. +Proposition 3.1. GK is not (d + 1)-colourable if and only if every Sperner +labelling of K has a rainbow facet. +5 + +Proof. Suppose for a contradiction that there exists a Sperner labelling λ of +K with no rainbow facet. Then we can define a (k − 1)-colouring c of GK as +follows. For each vertex ui of type (V1), let c(ui) = λ(ui). For each vertex +wσ of type (V3), let uj be the unique vertex of ∆d not belonging to σ, and +set c(vσ) = λ(uj). Finally, each vertex vρ of type (V2) is adjacent to the d+1 +vertices of a facet ρ ∈ K, which is not rainbow by assumption. Hence, the +colouring c can be extended to a (d+1)-colouring of GK, which is impossible. +Conversely, suppose for a contradiction that there is a (d + 1)-colouring +of GK. The labelling of K induced by the colouring is a Sperner labelling, +so by Sperner’s lemma, there must be a rainbow d-simplex ρ ∈ K. But then +the vertex vρ of GK is adjacent to vertices of colours 1, . . . , d + 1, which is +impossible. +Observe that if K is a triangulation of the 2-dimensional simplex, then +GK is a 4-chromatic planar graph with exactly 4 triangles. This is best possi- +ble in view of the sharpening of Gr¨otzsch’s theorem due to Gr¨unbaum [9] and +Aksenov [1], which states that every planar graph with at most 3 triangles is +3-colourable. Indeed, Borodin et al. [4] characterised the 4-chromatic planar +graphs with exactly 4 triangles; in this characterisation, the graphs GK be- +long among those graphs that can be obtained from the complete graph on +4 vertices by replacing a single 3-vertex by a ‘critical’ patch (in a sense made +precise in [4]). +It is easy to see that for any triangulation K of the 1-simplex, the graph +GK is an odd cycle and thus 3-critical. Gallai [12, 19] proved that GK is 4- +critical whenever K is a triangulation of the 2-simplex; note that this follows +easily from Theorem 1.3 and Proposition 3.3, as it does from Theorem 4.1 in +conjunction with [8, Theorem 5.4]. +In 1969 Gallai asked the following question, presented as Problem 9.14 +by Jensen and Toft [12]. +Problem 3.2. If d ≥ 3, and K is a triangulation of ∆d, is the graph GK +(d + 2)-critical? +Problems 1.2 and 3.2 are easily seen to be equivalent. +Proposition 3.3. Let K be a triangulation of ∆d. Then GK is (d + 2)- +critical if and only if for any facet σ ∈ K, there exists a Sperner labelling of +K where σ is the unique rainbow facet. +Proof. Suppose GK is (d + 2)-critical, for some triangulation K of ∆d. Then +for any facet σ ∈ K, the graph GK − vσ is (d + 1)-colourable. Hence, the +labelling of K induced by the (d+1)-colouring of G−vσ is a Sperner labelling +with no rainbow facet distinct from σ. +6 + +Conversely, suppose that for every facet σ ∈ K, there exists a Sperner +labelling of K where σ is the unique rainbow facet. Fix an arbitrary edge e +of GK. We shall define a proper colouring c : V (GK − e) → {1, . . . , d + 1}. +First, suppose that e is of type (E1), say e = uivρ, and let λ be a Sperner +labelling of K in which ρ is the unique rainbow facet. Define a colouring c +of GK − e as follows. Every vertex uj of type (V1) gets colour c(uj) = λ(ui). +If vσ is a vertex of type (V2) and σ ̸= ρ, then σ is not rainbow, and c(vσ) +can be any label not present in σ. If, on the other hand, ρ = σ, then put +c(vρ) = λ(ui). Finally, a vertex wσ of type (V3) is coloured by the label that +is not used by the Sperner labelling λ in σ. +Next, suppose e is of type (E2), say e = uiwσ, and let λ be a Sperner +labelling of K in which the unique rainbow facet ρ contains the vertex ui, +and has a (d−1)-face contained in σ. Let ℓ be the unique label not appearing +on the face σ. We can define a (d + 1)-colouring of G − e as follows. If uj is +a vertex of type (V1), let +c(uj) = +� +λ(uj) +if j ̸= i +ℓ +otherwise. +Note that vertices of type (V1) corresponding to any facet of K get at most +d colours. This means that any vertex of type (V2) is adjacent to at most +d colours, so the colouring can be extended to the vertices of type (V2). +Finally, if wτ is a vertex of type (V3), we let c(wτ) be the unique label not +appearing on τ. +Finally, suppose e is of type (E3); say e = wσwτ. Then we can colour the +subgraph induced by the vertices of type (V3) with colours {1, . . . , d}, colour +all the vertices of type (V1) with colour d + 1, and all vertices of type (V2) +with colour 1. +We have thus shown that G − e is (d + 1)-colourable, for any edge e +of G, and G is not (d + 1)-colourable by Proposition 3.1 and Theorem 1.1. +Therefore, G is (d + 2)-critical. +Theorem 1.4, in conjunction with Proposition 3.3, immediately implies a +negative answer to Gallai’s question. +Theorem 3.4. For every d ≥ 3, there exist infinitely many triangulations +K of ∆d such that GK is not (d + 2)-critical. +4 +GK as projective quadrangulations +Jensen and Toft [12, Problem 9.14] also raised the question of whether the +graphs GK belong to a ‘larger class of (d + 2)-chromatic graphs defined in +7 + +Figure 3: The complexes P − ω (left), P − ω − β (centre) and P − β (right) +in the proof of Theorem 4.1 (case d = 2). +purely graph-theoretic terms’. +While we cannot do away with the topol- +ogy, we can show that the graphs GK belong to a larger class of graphs +whose chromatic number is large for ‘topological reasons’, namely the class +of quadrangulations of projective spaces. Indeed, a result by the first two +authors [13, Theorem 1.1], generalising an earlier result of Youngs [24], shows +that all non-bipartite quadrangulations of the d-dimensional projective space +RPd are at least (d + 2)-chromatic. +Theorem 4.1. If K is a triangulation of the d-simplex, then GK is a non- +bipartite quadrangulation of the d-dimensional projective space RPd. +Proof. Let P be the 2-coloured complex obtained from the boundary complex +of the (d + 1)-dimensional cross polytope by colouring all the vertices of one +facet (say ω) white, and all the vertices of the complementary facet (say β) +black. The complex P − ω − β obtained by deleting the facets ω and β from +P contains ∂ω and ∂β as subcomplexes. Similarly, P − ω and P − β contain +∂ω and ∂β, respectively (see Figure 3 for an illustration). Glue P − ω − β to +P − ω along ∂ω, and then glue the result to P − β along ∂β. The resulting +complex C is a centrally symmetric triangulation of Sd. +Since K is a subdivision of a d-simplex ∆d, there is a sequence of stellar +moves (see e.g. [14, Theorem 4.5]) that transforms ∆d into K. Identify the +white and black facets of C with the vertices of ∆d in an antisymmetric +fashion, and perform the sequence of stellar moves on the white and black +facets of C, colouring all new vertices with the colour of the facet (so all new +vertices in the black facet will be black, and all new vertices in the white +facet will be white). Finally, perform a stellar subdivision of each white facet +with a black vertex, and of each black facet with a white vertex. Call the +resulting 2-coloured complex ˜K. +Clearly, ˜K is a centrally symmetric triangulation of Sd, with an anti- +symmetric 2-colouring such that no facet is monochromatic. Therefore, by +Lemma 2.1, the graph G obtained from the 1-skeleton of the complex by +8 + +deleting monochromatic edges, and by identifying antipodal vertices, is a +non-bipartite quadrangulation of RPd. We claim that G ∼= GK. To see this, +note that the vertices of P −ω−β in the above construction correspond to the +vertices of type (V3), the vertices subdividing ω and β are type (V1), while +the ones added at the end, via stellar subdivisions, are of type (V2). +Non-bipartite quadrangulations of RP1 are odd cycles, so they are 3- +critical. Youngs [24] showed that non-bipartite quadrangulations of RP2 are +4-chromatic, and conjectured that they are 4-critical as long as they contain +no non-bipartite quadrangulation of RP2 as a proper subgraph; this was +subsequently proved by Gimbel and Thomassen [8]. For d ≥ 3, non-bipartite +quadrangulations of RPd are not (d + 2)-critical in general; indeed, their +chromatic number is unbounded [13]. +5 +Proof of the main result +Our construction uses a specific convex 4-polytope known as H8 and de- +scribed in [10, Section 7.2]. This polytope is 2-neighbourly (recall that a poly- +tope is t-neighbourly if every set of t vertices constitutes a face). Motzkin [17] +claimed that for even d, every d/2-neighbourly d-polytope is combinatorially +equivalent to a cyclic polytope (see [10] for the necessary definitions). This +assertion is true for d-polytopes with at most d + 3 vertices [7] but fails in +general; H8 is one of the two counterexamples for d = 4 constructed in [10]. +Our example is in line with Sturmfels’ remark [21] that d + 4 vertices seem +to be a ‘threshold of counterexamples’. +The convex 4-polytope H8 has 8 vertices A, B, C, D, E, F, G, Z and the +following are the 20 facets of its boundary complex. (We underline the facets +relevant for the proof of Lemma 5.1 below.) +ABCD +ABCG +ABDE +ABEF +ABFZ +ABGZ +ACDZ +ACGZ +ADEZ +AEFZ +BCDE +BCEF +BCFG +BFGZ +CDEF +CDFG +CDGZ +DEFG +DEGZ +EFGZ +Lemma 5.1. Let the vertices of H8 be labelled by 1, 2, 3, 4 in such a way that +the facets ABGZ and CDEF of ∂H8 are rainbow. Then there exists another +rainbow facet of ∂H8. +Proof. We proceed by contradiction. Let ϕ be the given labelling. Suppose +that σ0 := ABGZ and σ1 := CDEF are the only rainbow facets of ∂H8, +and assume (without loss of generality) that the vertices A, B, G, Z are la- +belled 1, 2, 3, 4 in order. +Observe the following direct consequence of the +assumptions. +9 + +Claim 1. If a facet τ of ∂H8 shares a 2-dimensional face with σi (for some +i ∈ {0, 1}), then the two vertices in the symmetric difference of τ and σi have +different labels. +Applying Claim 1 to the facet ABCG (and σ0) we find ϕ(C) ̸= ϕ(Z) = 4. +Similarly, ϕ(C) ̸= 2 (considering ACGZ and σ0) and ϕ(C) ̸= 3 (considering +DEFG and σ1). Consequently, ϕ(C) = 1. +Neither E nor F have label 3, as seen by applying Claim 1 to the facets +CDFG and ABFZ, respectively. Since σ1 is rainbow, these vertices do not +have label 1 either, since it is already taken by C. Thus, {ϕ(E), ϕ(F)} = +{2, 4}. This implies that ϕ(D) = 3. +The facet ABDE shares two vertices A, B with σ0. Since ABDE is not +rainbow, we deduce that {ϕ(D), ϕ(E)} ̸= {ϕ(G), ϕ(Z)} = {3, 4}. Hence, +ϕ(E) ̸= 4 as we know that ϕ(D) = 3. This leaves only the option ϕ(E) = 2 +for the label of E. But then the facet ADEZ is rainbow, a contradiction. +We remark that H8 is far from the only polytope satisfying an analogue +of Lemma 5.1. While it is the only such polytope among 4-polytopes on 8 +vertices which are simplicial (all their faces are simplices; these polytopes are +listed in [11]), the situation is much different when the number of vertices +is increased. +For instance, a complete search of the 23 2-neighbourly 4- +polytopes on 9 vertices listed in [2] showed that 18 of them have the same +property as H8. +Proof of Theorem 1.4. Let K be the simplicial complex obtained from the +boundary complex of H8 by deleting the simplex σ0. Since ∂H8 ∼= S3, K is +a triangulation of B3, or, equivalently, a triangulation of the 3-simplex with +vertices A, B, G, Z. +For an integer d ≥ 3, let ρ be a (d−4)-simplex, and define Kd = K ⋆ρ (we +take the convention that the empty simplex has dimension −1; in particular, +K3 = K). We can view Kd as a triangulation of the d-simplex with vertices +{A, B, G, Z}∪V (ρ). Let σ = σ1⋆ρ, and fix any Sperner labelling λ : V (Kd) → +{1, . . . , d + 1} of Kd (observe that ρ is rainbow with respect to λ). If σ is not +rainbow, then by Sperner’s lemma, there must be a rainbow facet distinct +from σ and we are done. We may thus assume that σ is rainbow. This means +that the simplices σ0 and σ1 receive the same set of labels. In particular, the +labelling λ′ of V (K) induced by λ uses 4 labels. By Lemma 5.1, some 3- +simplex τ ∈ K distinct from σ1 is rainbow with respect to λ′. Hence, τ ⋆ ρ is +a rainbow d-simplex with respect to λ. We have thus shown that any Sperner +labelling of Kd has a rainbow d-simplex distinct from σ. +Now consider any facet π ∈ Kd distinct from σ, and let L be a triangu- +lation of π such that ∂L = ∂π. Let K′ +d be the complex obtained from Kd by +10 + +replacing π by L. Fix any Sperner labelling λ : V (K′ +d) → {1, . . . , d + 1}, and +suppose that no facet of K′ +d distinct from σ is rainbow. The d + 1 vertices +in ∂L must receive labels 1, . . . , d + 1. Therefore, the labelling induced by L +is a Sperner labelling of L, so by Sperner’s lemma, there is at least one rain- +bow facet in L. This shows that any Sperner labelling of K′ +d has a rainbow +d-simplex distinct from σ. +By iterating this construction, we can obtain infinitely many triangula- +tions K of ∆d with the property that every Sperner labelling of K has a +rainbow d-simplex distinct from σ. +It may well be that even in higher dimensions, the property in Prob- +lem 1.2 holds for triangulations of a special type. 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SIAM J. Discrete Math., 1(1):121–133, 1988. +[22] F. E. Su. Rental harmony: Sperner’s lemma in fair division. Amer. +Math. Monthly, 106(10):930–942, 1999. +[23] A. Sun. To divide the rent, start with a triangle. New York Times, April +29, 2014. +[24] D. A. Youngs. +4-chromatic projective graphs. +J. Graph Theory, +21(2):219–227, 1996. +[25] G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in +Mathematics. Springer-Verlag, New York, 1995. +13 + diff --git a/btE1T4oBgHgl3EQfxQWA/content/tmp_files/load_file.txt b/btE1T4oBgHgl3EQfxQWA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ef8144f82870457b9c4491f38b70b22c55dbbdd4 --- /dev/null +++ b/btE1T4oBgHgl3EQfxQWA/content/tmp_files/load_file.txt @@ -0,0 +1,530 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf,len=529 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='03420v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='CO] 9 Jan 2023 Criticality in Sperner’s Lemma Tom´aˇs Kaiser 1 Matˇej Stehl´ık 2 Riste ˇSkrekovski 3 Abstract We answer a question posed by Tibor Gallai in 1969 concerning criticality in Sperner’s lemma, listed as Problem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='14 in the collection of Jensen and Toft [Graph coloring problems, John Wiley & Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=', New York, 1995].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Sperner’s lemma states that if a labelling of the vertices of a trian- gulation of the d-simplex ∆d with labels 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , d+1 has the property that (i) each vertex of ∆d receives a distinct label, and (ii) any vertex lying in a face of ∆d has the same label as one of the vertices of that face, then there exists a rainbow facet (a facet whose vertices have pairwise distinct labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For d ≤ 2, it is not difficult to show that for every facet σ, there exists a labelling with the above properties where σ is the unique rainbow facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For every d ≥ 3, however, we construct an infinite family of examples where this is not the case, which im- plies the answer to Gallai’s question as a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The construction is based on the properties of a 4-polytope which had been used earlier to disprove a claim of Theodore Motzkin on neighbourly polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 1 Introduction A central result of combinatorial topology, Sperner’s lemma [20] has found wide-ranging applications in diverse areas of mathematics and beyond;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' for 1Department of Mathematics and European Centre of Excellence NTIS (New Technolo- gies for the Information Society), University of West Bohemia, Pilsen, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' E-mail: kaisert@kma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='cz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Supported by project GA20-09525S of the Czech Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 2Universit´e Paris Cit´e, CNRS, IRIF, F-75006, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' E-mail: matej@irif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Partially supported by ANR project DISTANCIA (ANR-17-CE40-0015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 3Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana & Fac- ulty of Information Studies, 8000 Novo Mesto, Slovenia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' E-mail: skrekovski@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Partially supported by the Slovenian Research Agency Program P1-0383, Project J1-3002, and BI-FR/22-23-Proteus-011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 1 example, it can be used to give an elementary proof of Brouwer’s fixed point theorem [18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 117] from algebraic topology, and to solve various fair-division problems, such as rental harmony [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let ∆d be a d-simplex with vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , vd+1, and let K be a finite triangulation of ∆d (see Section 2 for a precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A Sperner labelling of K is a map λ : V (K) → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , d+1} such that λ(vi) = i, and every vertex lying in a face of ∆d has the same label as one of the vertices of that face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A simplex is said to be rainbow if its vertices receive pairwise distinct labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The following classical result, proved by Sperner [20], is usually referred to as Sperner’s lemma (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 423], [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1 (Sperner’s lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If K is a finite triangulation of the d- simplex ∆d and λ : V (K) → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , d + 1} is any Sperner labelling of K, then there is a rainbow facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A natural question, asked by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Gallai in 1969 (in a different but equiv- alent form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' see Problem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='14 in Jensen and Toft [12] and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='3 below) is whether Sperner’s lemma is ‘critical’, in the following sense: Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Given any triangulation K of ∆d, and any facet σ ∈ K, is there a Sperner labelling of K where σ is the unique rainbow facet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' In dimension 1, the answer to Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2 is is easily seen to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' With a little more work, we can show that Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2 also has a positive answer in dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For every triangulation K of ∆2 and every facet σ ∈ K, there exists a Sperner labelling of K where σ is the unique rainbow facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let K be a triangulation of the 2-simplex ∆2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' add edges and faces as shown in Figure 1 to make it into a triangulation K′ of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let τ be the 2-simplex corresponding to the outer face, and assume that σ ∈ K is a given facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Every planar triangulation is 3-connected, so by Steinitz’s theorem [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 63], it is the graph of a convex polyhedron, and the dual graph is also 3-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Hence, by Menger’s theorem [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 14], there exist three internally vertex-disjoint paths between σ and τ in the dual graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The union of these paths (in the given embedding) has three faces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' let us assign the vertices of K′ embedded in the i-th face (1 ≤ i ≤ 3) label i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' It is not hard to see that this determines a Sperner labelling of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Furthermore, if a 3-cycle of K′ contains vertices of all three labels and differs from the face boundaries of σ and τ, then it separates a vertex of σ from a vertex of τ — in particular, it is not a face boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Thus, σ is the unique rainbow facet in the Sperner labelling of K as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 2 σ τ Figure 1: A triangulation K of ∆2 (in gray) extended to a triangulation K′ of S2 (thick edges and white faces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' the outer face is labelled τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Three internally vertex-disjoint paths (dashed) in the dual graph partition the vertices into three subsets (red, green and blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' In this paper, we will show that the answer to Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2 is negative in dimension 3 and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Namely, we will prove the following theorem in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For every integer d ≥ 3, there exist infinitely many triangu- lations K of ∆d such that there exists a facet σ ∈ K with the property that every Sperner labelling of K has a rainbow facet distinct from σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 2 Terminology and preliminary results For topological and graph-theoretic concepts not defined here, the reader is referred to Munkres [18] and to Bondy and Murty [3], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' An introduction to Sperner’s lemma — and combinatorial topology in general — can be found in de Longueville [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The book by Mohar and Thomassen [16] provides an introduction to topological graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For polytope theory, we refer the reader to Gr¨unbaum [10] and to Ziegler [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A k-colouring of a graph G is a function c : V (G) → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , k} such that c(u) ̸= c(v) whenever uv ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The minimum k for which a k-colouring of G exists is known as the chromatic number of G, denoted by χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If χ(G) ≤ k or χ(G) = k, then G is said to be k-colourable or k-chromatic, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Furthermore, a graph G is said to be k-critical (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' k-vertex- critical) if G is k-chromatic and deleting any edge (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' vertex) results in 3 a (k − 1)-colourable graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A graph is said to be k-vertex-connected if it has at least k + 1 vertices, and removing any set of k − 1 vertices does not disconnect the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A (geometric) simplicial complex K is a collection of geometric simplices of various dimensions satisfying two conditions: firstly, if σ ∈ K and τ is a face of σ, then τ ∈ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' secondly, if σ, τ ∈ K and σ∩τ ̸= ∅, then σ∩τ is a face of both σ and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We denote the regular d-dimensional simplex, unit d-ball and unit d-sphere by ∆d, Bd and Sd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Simplices of K that are not strictly contained in other simplices are the facets of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The union of all simplices of K, known as the polyhedron of K, is denoted by |K|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The k- skeleton of a simplicial complex K, denoted K(k), consists of all the simplices of K of dimension at most k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' note that K(1) consists of vertices and edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=', it is a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If two topological spaces X and Y are homeomorphic, we write X ∼= Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A simplicial complex K such that X ∼= |K|, if one exists, is called a triangulation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The join of two simplicial complexes K and L is denoted by K ⋆ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A quadrangulation of a surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=', a 2-dimensional manifold) X is a graph embedded in X so that every face of the embedding is bounded by a 4- cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' In [13], the definition of quadrangulations was extended to triangulable topological spaces in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A graph G is a quadrangulation of a topological space X if G is a spanning subgraph of the 1-skeleton K(1) of some triangulation K of X, with the property that the vertices of every facet of K induce a complete bipartite subgraph of G with at least one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' This definition coincides with the usual definition in the case when X is a 2-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' In [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2], the following characterisation was given of non-bipartite quadrangulations of the d-dimensional projective space RPd: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A graph G is a non-bipartite quadrangulation of RPd if and only if there exists a centrally symmetric triangulation K of Sd and a 2-colouring of the vertices with colours black and white, such that no pair of antipodal vertices is adjacent, antipodal vertices receive different colours, every facet of K contains a black and a white vertex, and G is obtained from the 1-skeleton K(1) by first deleting all monochromatic edges, and then identifying antipodal vertices and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If P is a (convex) d-polytope, its (d − 1)-dimensional faces are its facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If every facet is a simplex, then the boundary of P, denoted ∂P, is a sim- plicial complex such that |∂P| ∼= Sd−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=', ∂P is a triangulation of Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The convex hull of the vectors of the standard orthonormal basis and their negatives in Rd is known as the the d-dimensional cross polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 4 Figure 2: A triangulation K (dashed edges) of a 2-simplex (in gray), and the graph GK (solid edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 3 Relation to chromatic number Gallai used Sperner’s lemma to construct a family of 4-chromatic planar graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' see [12, Problem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='14] or [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let K be a triangulation of ∆d, and define the graph GK as follows (see Figure 2 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' There are three types of vertices of GK: (V1) The vertices u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , un of K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' (V2) A vertex vρ for every d-simplex ρ ∈ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' (V3) A vertex wσ for every (d − 1)-simplex σ ∈ ∆d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' There are also three types of edges of GK: (E1) uivρ ∈ E(GK) if and only if ui ∈ ρ, where ui is a vertex of K and ρ is a d-simplex of K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' (E2) uiwσ ∈ E(GK) if and only if ui belongs to the facet σ of ∆d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' (E3) wσwτ ∈ E(GK) for all (d − 1)-simplices σ, τ ∈ ∆d (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=', the vertices wσ induce a (d + 1)-clique of GK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Gallai observed that the statement that GK is not (d + 1)-colourable is equivalent to Sperner’s lemma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' we formalise it as the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' GK is not (d + 1)-colourable if and only if every Sperner labelling of K has a rainbow facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Suppose for a contradiction that there exists a Sperner labelling λ of K with no rainbow facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Then we can define a (k − 1)-colouring c of GK as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For each vertex ui of type (V1), let c(ui) = λ(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For each vertex wσ of type (V3), let uj be the unique vertex of ∆d not belonging to σ, and set c(vσ) = λ(uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Finally, each vertex vρ of type (V2) is adjacent to the d+1 vertices of a facet ρ ∈ K, which is not rainbow by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Hence, the colouring c can be extended to a (d+1)-colouring of GK, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Conversely, suppose for a contradiction that there is a (d + 1)-colouring of GK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The labelling of K induced by the colouring is a Sperner labelling, so by Sperner’s lemma, there must be a rainbow d-simplex ρ ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' But then the vertex vρ of GK is adjacent to vertices of colours 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , d + 1, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Observe that if K is a triangulation of the 2-dimensional simplex, then GK is a 4-chromatic planar graph with exactly 4 triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' This is best possi- ble in view of the sharpening of Gr¨otzsch’s theorem due to Gr¨unbaum [9] and Aksenov [1], which states that every planar graph with at most 3 triangles is 3-colourable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Indeed, Borodin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' [4] characterised the 4-chromatic planar graphs with exactly 4 triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' in this characterisation, the graphs GK be- long among those graphs that can be obtained from the complete graph on 4 vertices by replacing a single 3-vertex by a ‘critical’ patch (in a sense made precise in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' It is easy to see that for any triangulation K of the 1-simplex, the graph GK is an odd cycle and thus 3-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Gallai [12, 19] proved that GK is 4- critical whenever K is a triangulation of the 2-simplex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' note that this follows easily from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='3 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='3, as it does from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1 in conjunction with [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' In 1969 Gallai asked the following question, presented as Problem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='14 by Jensen and Toft [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If d ≥ 3, and K is a triangulation of ∆d, is the graph GK (d + 2)-critical?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Problems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2 are easily seen to be equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let K be a triangulation of ∆d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Then GK is (d + 2)- critical if and only if for any facet σ ∈ K, there exists a Sperner labelling of K where σ is the unique rainbow facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Suppose GK is (d + 2)-critical, for some triangulation K of ∆d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Then for any facet σ ∈ K, the graph GK − vσ is (d + 1)-colourable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Hence, the labelling of K induced by the (d+1)-colouring of G−vσ is a Sperner labelling with no rainbow facet distinct from σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 6 Conversely, suppose that for every facet σ ∈ K, there exists a Sperner labelling of K where σ is the unique rainbow facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Fix an arbitrary edge e of GK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We shall define a proper colouring c : V (GK − e) → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , d + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' First, suppose that e is of type (E1), say e = uivρ, and let λ be a Sperner labelling of K in which ρ is the unique rainbow facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Define a colouring c of GK − e as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Every vertex uj of type (V1) gets colour c(uj) = λ(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If vσ is a vertex of type (V2) and σ ̸= ρ, then σ is not rainbow, and c(vσ) can be any label not present in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If, on the other hand, ρ = σ, then put c(vρ) = λ(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Finally, a vertex wσ of type (V3) is coloured by the label that is not used by the Sperner labelling λ in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Next, suppose e is of type (E2), say e = uiwσ, and let λ be a Sperner labelling of K in which the unique rainbow facet ρ contains the vertex ui, and has a (d−1)-face contained in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let ℓ be the unique label not appearing on the face σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We can define a (d + 1)-colouring of G − e as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If uj is a vertex of type (V1), let c(uj) = � λ(uj) if j ̸= i ℓ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Note that vertices of type (V1) corresponding to any facet of K get at most d colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' This means that any vertex of type (V2) is adjacent to at most d colours, so the colouring can be extended to the vertices of type (V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Finally, if wτ is a vertex of type (V3), we let c(wτ) be the unique label not appearing on τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Finally, suppose e is of type (E3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' say e = wσwτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Then we can colour the subgraph induced by the vertices of type (V3) with colours {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , d}, colour all the vertices of type (V1) with colour d + 1, and all vertices of type (V2) with colour 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We have thus shown that G − e is (d + 1)-colourable, for any edge e of G, and G is not (d + 1)-colourable by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Therefore, G is (d + 2)-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='4, in conjunction with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='3, immediately implies a negative answer to Gallai’s question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For every d ≥ 3, there exist infinitely many triangulations K of ∆d such that GK is not (d + 2)-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 4 GK as projective quadrangulations Jensen and Toft [12, Problem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='14] also raised the question of whether the graphs GK belong to a ‘larger class of (d + 2)-chromatic graphs defined in 7 Figure 3: The complexes P − ω (left), P − ω − β (centre) and P − β (right) in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1 (case d = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' purely graph-theoretic terms’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' While we cannot do away with the topol- ogy, we can show that the graphs GK belong to a larger class of graphs whose chromatic number is large for ‘topological reasons’, namely the class of quadrangulations of projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Indeed, a result by the first two authors [13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1], generalising an earlier result of Youngs [24], shows that all non-bipartite quadrangulations of the d-dimensional projective space RPd are at least (d + 2)-chromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If K is a triangulation of the d-simplex, then GK is a non- bipartite quadrangulation of the d-dimensional projective space RPd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let P be the 2-coloured complex obtained from the boundary complex of the (d + 1)-dimensional cross polytope by colouring all the vertices of one facet (say ω) white, and all the vertices of the complementary facet (say β) black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The complex P − ω − β obtained by deleting the facets ω and β from P contains ∂ω and ∂β as subcomplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Similarly, P − ω and P − β contain ∂ω and ∂β, respectively (see Figure 3 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Glue P − ω − β to P − ω along ∂ω, and then glue the result to P − β along ∂β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The resulting complex C is a centrally symmetric triangulation of Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Since K is a subdivision of a d-simplex ∆d, there is a sequence of stellar moves (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' [14, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='5]) that transforms ∆d into K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Identify the white and black facets of C with the vertices of ∆d in an antisymmetric fashion, and perform the sequence of stellar moves on the white and black facets of C, colouring all new vertices with the colour of the facet (so all new vertices in the black facet will be black, and all new vertices in the white facet will be white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Finally, perform a stellar subdivision of each white facet with a black vertex, and of each black facet with a white vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Call the resulting 2-coloured complex ˜K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Clearly, ˜K is a centrally symmetric triangulation of Sd, with an anti- symmetric 2-colouring such that no facet is monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Therefore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1, the graph G obtained from the 1-skeleton of the complex by 8 deleting monochromatic edges, and by identifying antipodal vertices, is a non-bipartite quadrangulation of RPd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We claim that G ∼= GK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' To see this, note that the vertices of P −ω−β in the above construction correspond to the vertices of type (V3), the vertices subdividing ω and β are type (V1), while the ones added at the end, via stellar subdivisions, are of type (V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Non-bipartite quadrangulations of RP1 are odd cycles, so they are 3- critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Youngs [24] showed that non-bipartite quadrangulations of RP2 are 4-chromatic, and conjectured that they are 4-critical as long as they contain no non-bipartite quadrangulation of RP2 as a proper subgraph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' this was subsequently proved by Gimbel and Thomassen [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For d ≥ 3, non-bipartite quadrangulations of RPd are not (d + 2)-critical in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' indeed, their chromatic number is unbounded [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 5 Proof of the main result Our construction uses a specific convex 4-polytope known as H8 and de- scribed in [10, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' This polytope is 2-neighbourly (recall that a poly- tope is t-neighbourly if every set of t vertices constitutes a face).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Motzkin [17] claimed that for even d, every d/2-neighbourly d-polytope is combinatorially equivalent to a cyclic polytope (see [10] for the necessary definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' This assertion is true for d-polytopes with at most d + 3 vertices [7] but fails in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' H8 is one of the two counterexamples for d = 4 constructed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Our example is in line with Sturmfels’ remark [21] that d + 4 vertices seem to be a ‘threshold of counterexamples’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The convex 4-polytope H8 has 8 vertices A, B, C, D, E, F, G, Z and the following are the 20 facets of its boundary complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' (We underline the facets relevant for the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=') ABCD ABCG ABDE ABEF ABFZ ABGZ ACDZ ACGZ ADEZ AEFZ BCDE BCEF BCFG BFGZ CDEF CDFG CDGZ DEFG DEGZ EFGZ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let the vertices of H8 be labelled by 1, 2, 3, 4 in such a way that the facets ABGZ and CDEF of ∂H8 are rainbow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Then there exists another rainbow facet of ∂H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We proceed by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let ϕ be the given labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Suppose that σ0 := ABGZ and σ1 := CDEF are the only rainbow facets of ∂H8, and assume (without loss of generality) that the vertices A, B, G, Z are la- belled 1, 2, 3, 4 in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Observe the following direct consequence of the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' 9 Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If a facet τ of ∂H8 shares a 2-dimensional face with σi (for some i ∈ {0, 1}), then the two vertices in the symmetric difference of τ and σi have different labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Applying Claim 1 to the facet ABCG (and σ0) we find ϕ(C) ̸= ϕ(Z) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Similarly, ϕ(C) ̸= 2 (considering ACGZ and σ0) and ϕ(C) ̸= 3 (considering DEFG and σ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Consequently, ϕ(C) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Neither E nor F have label 3, as seen by applying Claim 1 to the facets CDFG and ABFZ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Since σ1 is rainbow, these vertices do not have label 1 either, since it is already taken by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Thus, {ϕ(E), ϕ(F)} = {2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' This implies that ϕ(D) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The facet ABDE shares two vertices A, B with σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Since ABDE is not rainbow, we deduce that {ϕ(D), ϕ(E)} ̸= {ϕ(G), ϕ(Z)} = {3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Hence, ϕ(E) ̸= 4 as we know that ϕ(D) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' This leaves only the option ϕ(E) = 2 for the label of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' But then the facet ADEZ is rainbow, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We remark that H8 is far from the only polytope satisfying an analogue of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' While it is the only such polytope among 4-polytopes on 8 vertices which are simplicial (all their faces are simplices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' these polytopes are listed in [11]), the situation is much different when the number of vertices is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For instance, a complete search of the 23 2-neighbourly 4- polytopes on 9 vertices listed in [2] showed that 18 of them have the same property as H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let K be the simplicial complex obtained from the boundary complex of H8 by deleting the simplex σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Since ∂H8 ∼= S3, K is a triangulation of B3, or, equivalently, a triangulation of the 3-simplex with vertices A, B, G, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For an integer d ≥ 3, let ρ be a (d−4)-simplex, and define Kd = K ⋆ρ (we take the convention that the empty simplex has dimension −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' in particular, K3 = K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We can view Kd as a triangulation of the d-simplex with vertices {A, B, G, Z}∪V (ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let σ = σ1⋆ρ, and fix any Sperner labelling λ : V (Kd) → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , d + 1} of Kd (observe that ρ is rainbow with respect to λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' If σ is not rainbow, then by Sperner’s lemma, there must be a rainbow facet distinct from σ and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We may thus assume that σ is rainbow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' This means that the simplices σ0 and σ1 receive the same set of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' In particular, the labelling λ′ of V (K) induced by λ uses 4 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='1, some 3- simplex τ ∈ K distinct from σ1 is rainbow with respect to λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Hence, τ ⋆ ρ is a rainbow d-simplex with respect to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We have thus shown that any Sperner labelling of Kd has a rainbow d-simplex distinct from σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Now consider any facet π ∈ Kd distinct from σ, and let L be a triangu- lation of π such that ∂L = ∂π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Let K′ d be the complex obtained from Kd by 10 replacing π by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Fix any Sperner labelling λ : V (K′ d) → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , d + 1}, and suppose that no facet of K′ d distinct from σ is rainbow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The d + 1 vertices in ∂L must receive labels 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' , d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Therefore, the labelling induced by L is a Sperner labelling of L, so by Sperner’s lemma, there is at least one rain- bow facet in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' This shows that any Sperner labelling of K′ d has a rainbow d-simplex distinct from σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' By iterating this construction, we can obtain infinitely many triangula- tions K of ∆d with the property that every Sperner labelling of K has a rainbow d-simplex distinct from σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' It may well be that even in higher dimensions, the property in Prob- lem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2 holds for triangulations of a special type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' For example, one could take cyclic polytopes (a simple and well-understood class) and consider the triangulations associated with them in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Given a poly- tope P, let us say that a triangulation is associated with P if it is obtained from the boundary of P by removing a facet τ and projecting the rest of the boundary into the convex hull of the vertex set of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content='2 has an affirmative answer if K is a triangu- lation of ∆d associated with a d-dimensional cyclic polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Acknowledgements We are indebted to Frank Lutz for the valuable information on small trian- gulations of manifolds collected in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' We used Sage to test these triangu- lations (as well as those obtained from [2, 11]) for specific properties, and we thank the developers of this fine piece of software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Aksenov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' On continuation of 3-colouring of planar graphs (in Russian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Diskret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Anal.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Dvoˇr´ak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Kostochka, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Lidick´y, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Yancey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Planar 4-critical graphs with four triangles.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Meunier, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Mustafa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' The discrete yet ubiquitous theorems of Carath´eodory, Helly, Sperner, Tucker, and Tverberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfxQWA/content/2301.03420v1.pdf'} +page_content=' Amer.' metadata={'source': 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a/cdFJT4oBgHgl3EQf-C0r/content/tmp_files/2301.11690v1.pdf.txt b/cdFJT4oBgHgl3EQf-C0r/content/tmp_files/2301.11690v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c7cd0e3a9b1bd98af835615e8b773b5302bde68 --- /dev/null +++ b/cdFJT4oBgHgl3EQf-C0r/content/tmp_files/2301.11690v1.pdf.txt @@ -0,0 +1,2036 @@ +A statistical framework for planning and analysing test-retest +studies for repeatability of quantitative biomarker +measurements +Moritz Fabian Danzer1,*, Maria Eveslage1, Dennis G¨orlich1, and Benjamin +Noto1, 2, 3, 4 +1Institute of Biostatistics and Clinical Research, University of M¨unster, M¨unster, +48149, Germany +2Clinic for Radiology, University Hospital M¨unster, M¨unster, 48149, Germany +3Department of Nuclear Medicine, University Hospital M¨unster, M¨unster, 48149, +Germany +4West German Cancer Centre (WTZ) Essen-M¨unster – M¨unster site, University +Hospital M¨unster, M¨unster, 48149, Germany +*moritzfabian.danzer@ukmuenster.de +Abstract +There is an increasing number of potential biomarkers that could allow for early assess- +ment of treatment response or disease progression. However, measurements of quantitative +biomarkers are subject to random variability. Hence, differences of a biomarker in longitudinal +measurements do not necessarily represent real change but might be caused by this random +measurement variability. Before utilizing a quantitative biomarker in longitudinal studies, it +is therefore essential to assess the measurement repeatability. Measurement repeatability ob- +tained from test-retest studies can be quantified by the repeatability coefficient (RC), which +is then used in the subsequent longitudinal study to determine if a measured difference rep- +resents real change or is within the range of expected random measurement variability. The +quality of the point estimate of RC therefore directly governs the assessment quality of the +longitudinal study. +RC estimation accuracy depends on the case number in the test-retest study, but despite +its pivotal role, no comprehensive framework for sample size calculation of test-retest studies +exists. To address this issue, we have established such a framework, which allows for flexible +sample size calculation of test-retest studies, based upon newly introduced criteria concerning +assessment quality in the longitudinal study. This also permits retrospective assessment of +prior test-retest studies. +1 +Introduction +A biomarker is a characteristic objectively measured and evaluated as an indicator of normal +biological processes, pathogenic processes, or response to a therapeutic intervention [5]. Biomarkers +used as indicators of response to a therapeutic intervention, or disease progression, are called +treatment response biomarkers. One prime, established treatment response biomarker is lesion size +change in cross-sectional imaging. For clinical trials concerning solid tumors, the measurement of +lesion size is formalized in the so-called Response Evaluation Criteria in Solid Tumors (RECIST) +[11], that categorize treatment response. With the rapid advancement in medical sciences, there +is an increasing number of new potential treatment response biomarkers that could possibly allow +for early and objective assessment of treatment response or disease progression in clinical trials +and clinical practice [17]. +arXiv:2301.11690v1 [stat.ME] 27 Jan 2023 + +However, using a biomarker in practice requires some basic research into the reliability of its +measurement. In addition to a fixed systematic measurement error (bias), which can be investigated +by comparing measurements with a known target value (e.g. phantom studies), it is important to +take into account that measurements of quantitative biomarkers are subject to random variability. +Hence, changes in a biomarker in longitudinal measurements made under the same conditions do +not necessarily represent real change but might be caused by exactly this random measurement +variability. Before testing or even utilizing a quantitative biomarker in longitudinal studies, it is +therefore of principal importance to assess the measurement repeatability [27]. +The repeatability of measurement is determined by test-retest studies, which then are also re- +ferred to as repeatability studies. In such studies, replicate measurements are made on a sample +of subjects under conditions that are as constant as possible [2]. Measurement repeatability can +be quantified by the within-subject standard deviation (wSD). Using wSD, the repeatability coef- +ficient (RC) can be calculated [6,24,27]. RC is then used in the longitudinal study to determine +if a difference in the biomarker represents presumed real change or is within the range of random +measurement variability. It is defined in such a way that a desired specificity to detect changes – +usually 95% – is targeted. +The wSD and the RC, as determined by the test-retest study, are point estimates, and hence suffer +from random error. As we will show, the targeted specificity is therefore generally not achieved +in practice. Following standard statistical results, the more subjects and the more repeated mea- +surements are included in the test-retest study, the more reliable the estimates of wSD and RC +will be. Accordingly, the probability of a relevant deviation of the actually achieved value from +the targeted specificity will decrease. The quality of assessments in the longitudinal study and +consequently the validity of its results is directly governed by the precision of the estimates of wSD +and RC. +Of course, exact knowledge of measurement repeatability is not only crucial for biomarkers. For +example, excellent measurement repeatability of scales and other laboratory instruments is manda- +tory. The reliability of a scale can be checked using weights with a known mass and it is possible +to perform many repeated measurements. In contrast, many biomarkers are measured in-vivo, +rendering attainment of large sample sizes difficult. Also, it might be necessary from an ethical +point of view to keep sample sizes as low as possible, since the measurement in question might be +inconvenient, invasive, or even harmful for the patient or the healthy test person. For example, a +biomarker might be derived from computed tomography, which involves ionizing radiation. Yet, +if the sample size in the test-retest study is small, there is a high chance of obtaining suboptimal +estimates of RC with associated detrimental effects on sensitivity and specificity in the longitudinal +study. +In what follows, we will focus on such and related issues concerning repeatability. Before doing +so, note that, related to but different from repeatability is reproducibility. While repeatability +represents the measurement precision under constant conditions, i.e, same measurement proce- +dure, same operators, same measuring system, etc., reproducibility is, in contrast, measurement +precision under differing conditions as various operators, measuring systems, etc. [16]. +Statistical literature concerning requirements for test-retest studies is scarce. One notable study +investigating sample size requirements is by Obuchowski and Bullen [24]. In their work, Obu- +chowski and Bullen conducted a simulation study to investigate the relation between the sample +size in the test-retest study and the specificity achieved in a following longitudinal study. The +authors give a blanket recommendation for sample size of test-retest studies based on their results +from a fixed set of simulation parameters. +Our goal is to expand upon the results of Obuchowski and Bullen [24] in several areas. First, we +want to introduce new quality criteria for the planning of test-retest studies. Furthermore, we will +expand the considerations to include sensitivity, which has not been investigated in the literature +so far. Finally, we aim to provide analytical solutions. In contrast to simulation studies, this +allows for flexible calculation of sample size requirements and also the retrospective assessment of +test-retest studies, as we will show. In doing so, we establish a comprehensive framework in which +the notions introduced above are precisely defined. +In what follows, we will introduce the model used for our framework and study the aspects of +specificity and sensitivity in separate sections. Afterwards, we demonstrate the application of our +2 + +concepts in a practical example and discuss our results. +2 +Definitions +One possible approach to distinguish true change from random variation in the longitudinal study +is to estimate measurement variability in a test-retest study. +For this purpose, n patients are +measured m times within a short period of time, in which their true value presumably does not +change. For our considerations we assume independent subjects, e.g. measurement of one target +per patient. In addition, independent replicate measurements are necessary, i.e. measurements on +a subject need to be made independent of the knowledge of its previous value(s) [8]. Consequently, +we establish the following model for the j-th measurement of the i-th patient Yij of the test-retest +study: +Yij = µi + εij +(1) +where µi is the true value for the i-th patient and εij is the random error. +We assume the +random errors to be independent and normally distributed with mean 0 and variance w2 +SD [24]. +In particular, it follows that Yij ∼ N(µi, w2 +SD) for any i ∈ {1, . . . , n} and j ∈ {1, . . . , m} . This +model is appropriate when true replicates are studied and a learning effect can be ruled out. As we +are only addressing measurement repeatability, a fixed bias does not need to be considered since +it cancels out. We also assume that measurement error is independent from the magnitude of µi. +From this data, we can estimate the within-patient standard deviation wSD [6] by +�wSD := +� +� +� +� 1 +n +n +� +i=1 +1 +m − 1 +m +� +j=1 +(Yij − ¯Yi·)2, +(2) +where ¯Yi· := 1/m �m +j=1 Yij denotes the mean value of the measurements of patient i. Following +Cochran’s theorem [10], the distribution of this entity is given by +n(m − 1) �w2 +SD +w2 +SD +∼ χ2 +n(m−1). +(3) +According to standard asymptotic theory, the following central limit theorem holds for �wSD: +�wSD − wSD +wSD +√ +2n(m−1) +D +→ +n,m→∞ Z, +(4) +where Z is a standard normally distributed random variable. If the number of repeated measure- +ments differs between subjects, i.e. the i-th subject is measured mi times, the value n(m−1) needs +to be replaced by �n +i=1(mi − 1) in all formulas. For the sake of simplicity, we restrict ourselves to +the simple case of an equal number of repetitions m per subject. +In order to assess changes in the measurements of a single patient in the subsequent longitudinal +study, the repeatability coefficient (RC) is computed [6]. It indicates the range in which two re- +peated measurements are expected to fall with a certain probability. In what follows, we restrict +ourselves to the assessment of changes in both directions. We want to keep our decision rules +flexible, i.e. we establish a target specificity psp ∈ (0, 1) which shall be reached for patients with +no change in their true biomarker value. Hence RC is a function of psp and is given by +RC(psp) := Φ−1(1 − (1 − psp)/2) · +√ +2 · wSD. +(5) +In most literature the RC is only considered for a fixed targeted specificity of 95%, i.e. RC(0.95) +[26,27]. +In practice, wSD is unknown and hence replaced by its consistent estimator �wSD to obtain the +estimated repeatability coefficient +ˆ +RC(psp) := Φ−1(1 − (1 − psp)/2) · +√ +2 · �wSD. +(6) +3 + +This quantity can then be applied as cutpoint in the longitudinal study to determine whether there +has been change between two consecutive measurements Ypre and Ypost. Here, we also assume, that +the measured values have independent errors, but the true levels µpre and µpost might actually be +different, i.e. +we have Ypre = µpre + εpre and Ypost = µpost + εpost with εpre and εpost being +independent and normally distributed with mean 0 and variance w2 +SD. +In case the true values have not changed, i.e. µpre = µpost, the difference Ypost − Ypre is normally +distributed with mean 0 and variance 2w2 +SD. Hence, with a probability of psp, we have Ypost−Ypre ∈ +[−RC(psp), RC(psp)]. +The rule to decide whether there is a change for a patient with the two measured values Ypre and +Ypost should thus be whether their difference lies outside or inside the interval [−RC(psp), RC(psp)]. +As the bounds are unknown in practice, this decision rule is replaced by the decision rule based on +the estimated interval [− ˆ +RC(psp), ˆ +RC(psp)]. Consequently, the targeted specificity psp will never +be exactly met. This applies analogously to considerations for the sensitivity of this procedure. +3 +Effective specificity as a criterion for sample size estima- +tion +Our goal is to quantify the uncertainty introduced by the replacement of wSD by its estimator +�wSD. As mentioned, the targeted specificity (psp) is not met in practice. To assess this problem, +we introduce the effective specificity Pesp which is the specificity actually achieved if a realisation +of the estimate �wSD is plugged in. Hence, Pesp is a random quantity as it depends on the value of +�wSD. We use a capital letter to emphasise that it is indeed a random variable. It can be implicitly +defined via +RC(Pesp) = +ˆ +RC(psp). +(7) +Although this quantity is unknown in practice, we can nevertheless analyse its distribution. Firstly, +we can compute the expected value E[Pesp] and the bias, i.e. the difference E[Pesp] − psp. This +is also the quantity targeted by Obuchowski and Bullen [24]. +Their quality criterion requires +|E[Pesp] − psp| to be smaller than 0.01, i.e. they want the mean effective specificity to deviate less +than 1 percentage point from the target specificity, which they set to 95%. But what is even more +important, from our point of view, is that we can compute quantiles of the distribution of Pesp +which will enable us to establish quality guarantees on the effective specificity of the longitudinal +studies based on the design parameters n and m of the test-retest study. +Expected value and bias +According to (7), Pesp is given by +Pesp = RC−1( ˆ +RC(psp)) +(8) += 1 − 2 · +� +1 − Φ +� +Φ−1 +� +1 − 1 − psp +2 +� +· �wSD +wSD +�� +. +(9) +The function RC can be inverted as it is a continuous, monotonically increasing function on (0, 1). +The expectation of this random quantity can be computed exactly using (3) or approximately using +the central limit theorem (4), according to which the distribution of �wSD/wSD can be approximated +with a normal distribution with expectation 1 and variance 1/(2n(m − 1)). Hence, we get +E[Pesp] +=1 − 2 · +� +1 − +� ∞ +0 +Φ +� +Φ−1 +� +1 − 1 − psp +2 +� +· w +� +fχ2 +n(m−1)(n(m − 1)w2) 2wn(m − 1) dw +� +(10) +≈1 − 2 · +� +1 − +� ∞ +−∞ +Φ +� +Φ−1 +� +1 − 1 − psp +2 +� +· w +� � +n(m − 1) +π +exp +� +−n(m − 1)(w − 1)2� +dw +� +(11) +4 + +A +B +0.900 +0.922 +0.941 +0.945 +0.950 +20 +40 +60 +n in test−retest study +Expected value of effective specificity +Sample size of +test−retest study +10 +30 +60 +0.6 +0.8 +1.0 +Effective specificity +Figure 1: A) Expected value of the effective specificity (E[Pesp]) as a function of n in the test- +retest study for a target specificity psp of 95% and m = 2. Already for a small case number of +10 (blue dot) the expected value is comparably high. For a case number of 30 (green dot) the +bias (E[Pesp] − psp) is below 1 percentage point and does not change substantially with an increase +of the case number to 60 (red dot). B) However, while E[Pesp] is already relatively high for a +case number of 10, the tails of the corresponding PDF (blue area) are prominent, resulting in a +high chance of obtaining a low Pesp in practice. The green and red area represent the PDF of the +effective specificity for n = 30 and n = 60, respectively. +where fχ2 +n(m−1) denotes the probability density function (PDF) of a χ2-distributed random variable +with n(m − 1) degrees of freedom. By numerical evaluation of the terms in (10) and (11), the bias +can be computed. +Quantiles of the distribution of Pesp +We need to be aware that even if E[Pesp] is close to psp, i.e. the bias is low, the probability for +a substantial deviation of the actually realized specificity from the targeted specificity might be +large (Figure 1). +Therefore, we want to know with which confidence pconf we can say that the effective specificity +is larger than some lower bound pesp,lb. This is expressed by the formula +P [Pesp ≥ pesp,lb] = pconf +⇔P +� +ˆ +RC(psp) ≥ RC(pesp,lb) +� += pconf. +(12) +We want to introduce a new quality criterion based on this concept. +The quantity pconf is a function of pesp,lb and of course also depends on psp, n and m. For notational +5 + +convenience, however, we omit those arguments. After some calculations, one obtains +pconf =1 − Fχ2 +n(m−1) +� +� +� +Φ−1 � +1 − 1−pesp,lb +2 +�2 +Φ−1 +� +1 − 1−psp +2 +�2 n(m − 1) +� +� +� +(13) +≈1 − Φ +� +� +� +� +Φ−1 � +1 − 1−pesp,lb +2 +� +Φ−1 +� +1 − 1−psp +2 +� +− 1 +� +� � +2n(m − 1) +� +� , +(14) +where Fχ2 +n(m−1) denotes the cumulative distribution function of a χ2-distributed random variable +with n(m − 1) degrees of freedom. This formulas can now be used in different ways. In the above +form, one can determine the confidence with which the effective specificity exceeds a fixed bound +pesp,lb with given design parameters n and m of the test-retest study. Analogous considerations +can be made for upper bounds by computing the probability of the complementary event. +In the planning stage of the test-retest study it could be beneficial to choose the sample size n in +such a way that a desired lower bound pesp,lb is achieved with a prespecified confidence pconf. To +this end, the asymptotic formula (14) can be solved explicitly for n: +n ≥ +1 +2(m − 1) +� +� +Φ−1(1 − pconf)Φ−1 � +1 − 1−psp +2 +� +Φ−1 +� +1 − 1−pesp,lb +2 +� +− Φ−1 +� +1 − 1−psp +2 +� +� +� +2 +. +(15) +The exact formula (13) cannot be explicitly solved for n. However one can numerically solve +min +� +� +� +� +� +n ∈ N: 1 − Fχ2 +n(m−1) +� +� +� +Φ−1 � +1 − 1−pesp,lb +2 +�2 +Φ−1 +� +1 − 1−psp +2 +�2 n(m − 1) +� +� +� ≥ pconf +� +� +� +� +� +. +(16) +In our application example we will apply these formulas in the planning stage of a hypothetical +test-retest study. +If one wants to identify the worst possible cases for given n and m, one could compute the lower +bound of the effective specificity which is reached with confidence pconf: +pesp,lb = 1 − 2 +� +� +�1 − Φ +� +� +� +� +� +� +�F −1 +χ2 +n(m−1)(1 − pconf) +n(m − 1) +Φ−1 +� +1 − 1 − psp +2 +� +� +� +� +� +� +� +(17) +≈ 1 − 2 +� +�1 − Φ +� +� +Φ−1(1 − pconf)Φ−1 � +1 − 1−psp +2 +� +� +2n(m − 1) ++ Φ−1 +� +1 − 1 − psp +2 +�� +� +� +� . +(18) +Accordingly, in (1 − pconf) · 100% of all cases, the effective specificity will be even lower than the +obtained pesp,lb. +From our point of view, the probability of exceeding a lower bound pesp,lb is a valid criterion for +evaluating the quality of assessment in a longitudinal study. Different from the expected value of +Pesp which has been previously proposed as a quality criterion [24], our criterion considers the tails +of the distribution of Pesp. This allows to bound the probability of strongly deviating from the +desired specificity. +4 +Consideration of effective sensitivity +Concerning the sensitivity, i.e. the ability to detect real change between two measurements of +one patient in the longitudinal study, we can make similar considerations. Before coming back to +6 + +the problem of the uncertainty caused from the estimation of wSD, we first assume, that wSD and +hence also RC(psp) is known. Of course, the sensitivity strongly depends on the difference between +µpre and µpost. Also, such differences are more difficult to detect if wSD is large and a large target +specificity is chosen. To be more precise, the sensitivity pse to detect a difference can be written +as a function of µ∆ := µpost − µpre, wSD and the chosen specificity psp. It is given by +pse(µ∆, wSD) +:=P[Ypost − Ypre /∈ [−RC(psp), RC(psp)]] +=1 − +� +Φ +� +Φ−1 +� +1 − 1 − psp +2 +� +− +µ∆ +√ +2wSD +� +− Φ +� +Φ−1 +�1 − psp +2 +� +− +µ∆ +√ +2wSD +�� +. +(19) +In this form, the function can also be seen as a function of the effect size δ := µ∆/wSD, i.e. +pse(δ) +:=1 − +� +Φ +� +Φ−1 +� +1 − 1 − psp +2 +� +− δ +√ +2 +� +− Φ +� +Φ−1 +�1 − psp +2 +� +− δ +√ +2 +�� +. +(20) +This dependence of the sensitivity from the effect size δ is visualized in Figure 2. +0.25 +0.50 +0.75 +1.00 +−6 +−4 +−2 +0 +2 +4 +6 +δ +Sensitivity +Figure 2: Sensitivity as a function of effect size δ := µ∆/wSD for psp=0.95. Note that we assume +wSD to be known here. +As wSD is unknown and needs to be estimated by �wSD which will then be plugged in to compute +ˆ +RC(psp), the sensitivity computed in (20) will not be reached. Analogously to our considerations +for the specificity, we introduce the effective sensitivity Pese which is the sensitivity which is actually +achieved if a realisation of the estimate �wSD is plugged in. Of course, it is also a random variable +and does depend again on µ∆, wSD and psp. It can be defined by the equation +Pese(µ∆, wSD) := P[Ypost − Ypre /∈ [− ˆ +RC(psp), ˆ +RC(psp)]| �wSD] += 1 − +� +Φ +� +Φ−1 +� +1 − 1 − psp +2 +� �wSD +wSD +− +µ∆ +√ +2wSD +� +−Φ +� +Φ−1 +�1 − psp +2 +� �wSD +wSD +− +µ∆ +√ +2wSD +�� +. +(21) +With this expression and the exact distribution of �wSD given as in (3) resp. the approximation +of the distribution of +� +wSD +wSD by a normal distribution from (4) we can now quantify the bias caused +by the replacement of wSD by �wSD and compute quantiles of the distribution of Pese which will +enable us to also give quality guarantees on the effective sensitivity. Unlike our considerations +for the specificity, these values will also depend from the actual wSD and the difference µ∆ of the +longitudinal study and hence will be regarded as functions of those. +7 + +Bias +To compute the bias in dependence from psp, µ∆ and wSD, we can take the expectation of the right +hand side of (21) and use the exact distribution (3) and the central limit theorem (4) to obtain +the result +E[Pese(psp, µ∆, wSD)] +=1 − +� ∞ +0 +Φ +� +Φ−1 +� +1 − 1 − psp +2 +� +· w − +µ∆ +√ +2wSD +� +fχ2 +n(m−1)(n(m − 1)w2)2wn(m − 1) dw ++ +� ∞ +0 +Φ +� +Φ−1 +�1 − psp +2 +� +· w − +µ∆ +√ +2wSD +� +fχ2 +n(m−1)(n(m − 1)w2)2wn(m − 1) dw +≈1 − +� ∞ +−∞ +Φ +� +Φ−1 +� +1 − 1 − psp +2 +� +· w − +µ∆ +√ +2wSD +� � +n(m − 1) +π +exp +� +−n(m − 1)(w − 1)2� +dw ++ +� ∞ +−∞ +Φ +� +Φ−1 +�1 − psp +2 +� +· w − +µ∆ +√ +2wSD +� � +n(m − 1) +π +exp +� +−n(m − 1)(w − 1)2� +dw. +(22) +Please note that this can essentially be seen as a function of δ. Following (21) the bias of the effective +sensitivity can be considered as a function of δ for any given psp, i.e. E[Pese(psp, δ)] − pse(psp, δ). +Quantiles of the distribution of Pese +For the most accurate examination of the distribution of Pese we would need to consider both +events +Ypost − Ypre > +ˆ +RC(psp) and +(23) +Ypost − Ypre < − ˆ +RC(psp). +(24) +However, this leads to expressions that are difficult to handle analytically. +Actually, the two +probabilities +P[Ypost − Ypre > +ˆ +RC(psp)| �wSD] and +(25) +P[Ypost − Ypre < − ˆ +RC(psp)| �wSD] +(26) +sum up to the effective sensitivity. However, in the presence of an effect, one of them will be much +larger than the other. In the case δ > 0, the probability in (25) is larger than that from (26) which +is bounded from above by 0.025 and quickly converges to 0 as δ increases. To enable the derivation +of analytical formulas, we will therefore restrict ourselves to the consideration of δ > 0 and the +event (23). It is nevertheless possible to circumvent this simplification by numerical inversion of +the relationship given in (21). But here, we will approximate +Pese(psp, µ∆, wSD) ≈ P[Ypost − Ypre > +ˆ +RC(psp)| �wSD] +(27) += 1 − Φ +� +Φ−1 +� +1 − 1 − psp +2 +� �wSD +wSD +− +µ∆ +√ +2wSD +� +. +(28) +In analogy to the previous section we can provide confidence levels pconf which indicate the prob- +ability that the effective sensitivity for some effect δ exceeds the lower bound pese,lb: +pconf = P[Pese(psp, δ) ≥ pese,lb] +≈ Fχ2 +n(m−1) +� +� +� +� +�Φ−1(1 − pese,lb) + δ/ +√ +2 +Φ−1 +� +1 − 1−psp +2 +� +� +� +2 +n(m − 1) +� +� +� +≈ Φ +� +� +� +�Φ−1(1 − pese,lb) + δ/ +√ +2 +Φ−1 +� +1 − 1−psp +2 +� +− 1 +� +� � +2n(m − 1) +� +� +(29) +8 + +Of course, such considerations only make sense if pse > pese,lb for the chosen effect size δ. As +above, analogous considerations can be made for upper bounds by computing the probability of +the complementary event. +While (29) allows to compute the confidence of reaching a certain lower bound of the sensitivity for +an effect δ, this formula may also be transformed to be used in the planning stage of the test-retest +study. If one wants to achieve a fixed confidence with which the effective sensitivity for an effect +size δ exceeds some lower bound, one can use the exact results from above or the approximations +made thereafter to determine the sample size n of the test-retest study in which each patient is +measured m times. It shall be chosen such that +n = min +� +n ∈ N: P +� +1 − +� +Φ +� +Φ−1 +� +1 − 1 − psp +2 +� �wSD +wSD +− +µ∆ +√ +2wSD +� +(30) +−Φ +� +Φ−1 +�1 − psp +2 +� �wSD +wSD +− +µ∆ +√ +2wSD +�� +≥ pese,lb +� +≥ pconf +� +(31) +≈ min +� +� +� +� +� +n ∈ N: Fχ2 +n(m−1) +� +� +� +� +�Φ−1(1 − pese,lb) + δ/ +√ +2 +Φ−1 +� +1 − 1−psp +2 +� +� +� +2 +n(m − 1) +� +� +� ≥ pconf +� +� +� +� +� +(32) +≈ +1 +2(m − 1) +� +� +Φ−1(pconf)Φ−1 � +1 − 1−psp +2 +� +Φ−1 (1 − pese,lb) + δ/ +√ +2 − Φ−1 +� +1 − 1−psp +2 +� +� +� +2 +. +(33) +Analogous to the preceding section, we can use these results in the planning stage of a test-retest +study, as we will demonstrate in the following application example. Even if a study is planned based +on considerations of the specificity, the formulas allow to assess the distribution of the effective +sensitivity for any given effect size of interest. +Calculations for δ < 0 follow analogously to the considerations for δ > 0. +5 +Application example +To illustrate our considerations, we will discuss a hypothetical application for early treatment +response assessment in recurrent or metastatic nasopharyngeal carcinoma. While some patients +with recurrent nasopharyngeal carcinoma show response or stable disease to systemic treatment, +many patients will have progressive disease, which is invariably lethal [13,19]. Nevertheless, futile +treatments should be avoided due to associated toxicity [9, 13]. To suspend futile treatment as +soon as possible an imaging biomarker is desirable which accurately classifies treatment response +earlier than change in morphologic lesion size, the current standard. A promising biomarker in +this context is diffusion weighted magnetic resonance imaging (DWI) [18]. DWI depends on the +differences in the movement of water molecules based on Brownian motion, which can be quantified +by the apparent diffusion coefficient (ADC). An exemplary measurement of ADC is shown in Figure +3. Change in ADC has shown promise as an early treatment response marker in various tumors, +including nasopharyngeal carcinoma [18,28–30]. +Prospective planning of test-retest studies +As laid out above, before conducting a longitudinal study in which a biomarker is applied to assess +treatment response, a test-retest study should be conducted to assess repeatability. In our example, +we will set psp to 95% and m = 2, as these are the usual values in the literature. We imagine the +researcher would want to obtain a specificity of at least 90% (pesp,lb) with 95% certainty (pconf) in +the longitudinal study. What sample size (n) is necessary in the test-retest study? This question +9 + +Figure 3: Example case of a tumor in the right nose showing restricted diffusion (A) with a mean +ADC of 610 · 10−6 mm2/s. (B) Region of interest outlined in yellow. +can be answered using the asymptotic formula (15): +n ≥ +1 +2(m − 1) +� +� +Φ−1(1 − pconf)Φ−1 � +1 − 1−psp +2 +� +Φ−1 +� +1 − 1−pesp,lb +2 +� +− Φ−1 +� +1 − 1−psp +2 +� +� +� +2 +⇔n ≥ +1 +2(2 − 1) +� +Φ−1(0.05)Φ−1 (0.975) +Φ−1 (0.95) − Φ−1 (0.975) +�2 +⇔n ≥ 52.3 +(34) +This can also be concluded from Figure 4 A. Numerical solution of the exact formula (16) yields +a sample size of 54. Resulting sample sizes for other values of pesp,lb can be taken from Figure 4 +B. The resulting scenario in terms of the distribution of the relative error in the estimation of �wSD +and its effect on Pesp is displayed in Figure 5 A. +Analogous considerations can be made for the effective sensitivity. We consider the sensitivity +for an underlying true effect size of δ = 4 in a study with psp = 0.95 and m = 2. According to +formula (20), a sensitivity of 80.74% was achieved if wSD was a known quantity. However, this will +not be met in practice. What is the minimum sample size (n) of the test-retest study such that we +can be 95% (pconf) sure to achieve at least a sensitivity of 75% (pese,lb) for that effect size? This +question can be answered using the approximate formula (33). +n ≈ +1 +2(m − 1) +� +� +Φ−1(pconf)Φ−1 � +1 − 1−psp +2 +� +Φ−1 (1 − pse,lb) + δ/ +√ +2 − Φ−1 +� +1 − 1−psp +2 +� +� +� +2 +(35) +⇒n ≈ +� +Φ−1(0.95)Φ−1 (0.975) +Φ−1 (0.25) + 4/ +√ +2 − Φ−1 (0.975) +�2 +(36) +⇔n ≈ 138.1 +(37) +Accordingly, a sample size of 139 patients in the test-retest study would be recommended to achieve +the set targets. Using the exact formula (17) or the asymptotic formula (18), an effective specificity +of at least 92.25% resp. 92.27% is reached with a certainty of 95%, in this scenario. This is also +depicted by Figure 5 B. +10 + +A +BA +B +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +0 +10 +20 +30 +40 +50 +60 +70 +n +pesp,lb +m +2 +3 +pesp,lb +m +n +E[Pesp] +0.7 +2 +7 +0.9092 +0.8 +2 +12 +0.9264 +0.9 +2 +54 +0.9448 +0.925 +2 +164 +0.9483 +0.7 +3 +4 +0.9143 +0.8 +3 +6 +0.9264 +0.9 +3 +27 +0.9448 +0.925 +3 +82 +0.9483 +Figure 4: A) Lower bound of effective specificity reached with confidence of 95% as a function of +sample size (n) and number of repeated measurements (m) of the test-retest study. +B) Sample size resulting from (16) for different values of the desired lower bound pesp,lb that shall +be exceeded with a fixed confidence of 95%. +Retrospective assessment of test-retest studies +It is not always necessary to conduct a preceding test-retest study when planning a longitudinal +study. The RC used in the longitudinal study might be adopted from already published test- +retest studies. If one intends to use the point estimate of the RC obtained in a previous study, +it is advisable to retrospectively assess the resulting distribution of the effective specificity and +sensitivity. This allows to evaluate the impact of the sample size of the used test-retest study on +quality criteria of the longitudinal study, especially the probability of exceeding a given pesp,lb. +Common sample sizes in test-retest studies are around 10 and 20 [3,4,12,15,21,22]. If the point +estimator of RC(0.95) resulting from a test-retest study with a sample size of 10 and two repeated +measurements is used, the distribution of the effective specificity will have prominent tails as +illustrated in Figure 1. According to (17), the lower bound of the effective specificity obtained +with 95% confidence is 0.7814 and 0.8512 for a sample size of 10 and 20, respectively, which might +be insufficient (Figure 4). Note that for the recommendation by Obuchowski and Bullen [24] of +a sample size of 35 for test-retest studies with m = 2 the probability of achieving an effective +specificity below 94% is 39.74%. +Such considerations are also possible for the effective sensitivity. +6 +Discussion +We have established a comprehensive framework for planning of test-retest studies concerning re- +peatability. It enables flexible calculation of sample size requirements and retrospective assessment +of such studies with regard to different quality criteria. +To better discuss planning of test-retest studies we have introduced the notions of effective speci- +ficity (Pesp) and effective sensitivity (Pese), allowing for clearer differentiation of the targeted +specificity psp and sensitivity pse from the values actually achieved in the longitudinal study. Both +Pesp and Pese are random quantities and their actual values are unknown in practical application. +11 + +A +B +0.3 +0.6 +0.9 +−1.0 +−0.5 +0.0 +0.5 +1.0 +0.75 +0.92 +−1.0 +−0.5 +0.0 +0.5 +1.0 +Figure 5: Visualization of the application example. The x-axis denotes the relative error between +�wSD and wSD. The solid black line represents the asymptotic PDF of the relative error. Note the +near identity to the dashed red curve, representing the PDF of the exact χ2 distribution. The +violet line shows the effective specificity. Analogously, the blue line shows the effective sensitivity +for an underlying effect size of δ = 4. +A) For n = 53 and m = 2: The area shaded in light gray represents 95% of the area under the +normal curve. I.e. there is a 95% chance of obtaining a �wSD from the test-retest study that will +result in an effective specificity of greater than 90%. +B) For n = 139 and m = 2: The area shaded in light gray represents 95% of the area under the +normal curve. I.e. there is a 95% chance of obtaining a �wSD from the test-retest study that will +result in a specificity of greater than 75%. In this case, an effective specificity of 92.27% will be +reached with a certainty of 95%. +However, we can determine their distribution and thus can compute different characteristics which +properly reflect the uncertainty caused by the estimation process. +Expanding on the work of Obuchowski and Bullen [24], we have introduced a new quality criterion +for sample size calculation of test-retest studies. In their work, Obuchowski and Bullen [24] de- +mand that the mean effective specificity (E[Pesp]) deviates at most by 0.01 from the fixed targeted +specificity (psp) of 0.95. However, using the mean effective specificity as sole quality criterion has +limitations, since the whole distribution of the effective specificity is not properly taken into ac- +count. As illustrated in Figure 1, there is a high probability that the actually achieved effective +specificity deviates strongly from its target even if the mean effective specificity may be close to +the targeted specificity. Therefore, we propose a quality criterion for sample size calculations based +on the probability that the effective specificity exceeds a chosen lower bound, taking into account +the tails of the distribution of Pesp. +In contrast to previous works we expand our consideration also to issues of sensitivity. Here, of +course, it must also be taken into account that the sensitivity depends on the underlying effect +size. Nevertheless, we can determine the distribution of the effective sensitivity for any effect size +and provide analogous sample size formulas as for the specificity. +Finally, our study is the first to provide analytical rather than simulation results. This provides +greater flexibility as the targeted specificity psp and number of repeated measurements m may be +chosen freely. Hence, it allows the readers to avoid conducting time-consuming simulation studies +themselves. While our formulas enable flexible calculations for all scenarios, for convenience of +the reader we also provide a table with sample sizes for some exemplary scenarios in Figure 4. +12 + +Sample sizes resulting from other choices of the parameters pconf, pesp,lb, psp and m can be found +in Supplementary Tables S1-S4. +Our study has some limitations. The field of application is restricted to test-retest studies in which +true replicates of measurements are possible, for example in quantitative imaging markers. Our +considerations are not valid if the measurement process itself results in a change of the measurand +(learning / practice effect) as has been described for some psychological assessments [14,20]. +Our standard model (1) assumes independent and identically normally distributed errors. It is +therefore advisable to examine whether there is a relationship between the within-subject varia- +tion and the level of the measured value before applying our approach [23]. If the variability of +the measurement error increases with the magnitude of the measured value, a log transformation +might resolve the issue [1, 7, 23]. Beyond that, non-normally distributed error terms are not cov- +ered so far. We also have not specifically considered the scenario of clustered data, e.g. measuring +multiple lesions per subject. However, if a hierarchical model structure with independent errors +can be assumed, this does not pose a restriction to application of our approach. +It should be noted that exact solutions based on the χ2 distribution for all our considerations are +available. In some cases, when an analytic solution is not possible, these exact solutions require +the application of numerical methods. +In order to give completely analytic solutions, some of +our formulas rely on asymptotic results and approximations. The differences between exact and +approximate results are most severe for small sample sizes and small effect sizes. Applying both +exact and approximate formulas in our application example, it can be seen that these differences +are negligible in practically relevant scenarios. Implementations of exact and approximate solu- +tions can be found in our supplementary R code [25]. +So far, our considerations are limited to repeatability, i.e. assuming same measurement conditions +for the repeated measurements. However, for real world application of biomarkers, consideration +of reproducibility is also important since longitudinal measurements are often performed under +different measuring conditions, e.g. varying readers or scanners. Therefore, our model should be +perspectively enhanced to include aspects of reproducibility such as a fixed bias as e.g. in some +models considered by Obuchowski and Bullen [24]. Nevertheless, since repeatability limits repro- +ducibility, a good knowledge of the former is useful in order to interpret reproducibility studies +properly [8]. +Test-retest studies of repeatability should be well planned to guarantee for a sufficient quality of +dependent longitudinal studies. Our framework allows the derivation of analytical solutions for +quality criteria that can be used to assess implications of the test-retest study design on subsequent +longitudinal studies. +Acknowledgements +B.N. was funded as a clinician scientist by the Medical Faculty, University of M¨unster, Germany. +There was no dedicated funding for this study. +Additional information +Data availability No datasets were generated or analysed during the current study. Implementa- +tions of exact and approximate formulas can be found in the Supplementary R Code. Additionally, +we provide sample sizes based on formula (16) in Supplementary Tables S1-S4 generated using our +R code. +Competing interests The authors declare that they have no conflict of interest. +References +[1] Douglas G Altman and J Martin Bland. Measurement in medicine: the analysis of method +comparison studies. J. R. Stat. Soc. Series B, 32(3):307–317, 1983. +13 + +[2] Huiman X Barnhart, Michael J Haber, and Lawrence I Lin. 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Radiology, 293(2):374–383, 2019. +15 + +Supplementary Material +In the following tables, we want to give sample sizes based on formula (16) in our main manuscript +for different choices of parameters pconf, pesp,lb, psp and m. For further constellations that cannot +be found in the following tables, we would like to refer to the sample size function provided in our +Supplementary R Code. +Supplementary Table S1: Sample sizes for different constellations of pconf, pesp,lb and psp for m = 2 +m=2 +pconf +pesp,lb +psp +0.800 +0.900 +0.925 +0.950 +0.975 +0.990 +0.800 +0.700 +13 +4 +4 +3 +2 +2 +0.800 +10 +7 +5 +3 +3 +0.900 +68 +17 +7 +4 +0.925 +48 +11 +6 +0.950 +27 +9 +0.975 +25 +0.900 +0.700 +25 +7 +6 +5 +4 +3 +0.800 +19 +12 +8 +6 +4 +0.900 +147 +34 +13 +8 +0.925 +102 +22 +10 +0.950 +55 +16 +0.975 +52 +0.925 +0.700 +30 +9 +7 +6 +4 +4 +0.800 +23 +15 +10 +7 +5 +0.900 +183 +42 +16 +9 +0.925 +127 +26 +12 +0.950 +69 +20 +0.975 +64 +0.950 +0.700 +38 +11 +8 +7 +5 +4 +0.800 +29 +18 +12 +8 +6 +0.900 +236 +54 +20 +11 +0.925 +164 +33 +15 +0.950 +88 +25 +0.975 +82 +0.975 +0.700 +53 +14 +11 +9 +7 +5 +0.800 +40 +25 +16 +11 +8 +0.900 +332 +75 +27 +15 +0.925 +229 +46 +20 +0.950 +122 +34 +0.975 +114 +0.990 +0.700 +73 +19 +15 +12 +9 +7 +0.800 +54 +34 +22 +14 +10 +0.900 +463 +103 +37 +20 +0.925 +320 +63 +28 +0.950 +170 +46 +0.975 +159 +16 + +Supplementary Table S2: Sample sizes for different constellations of pconf, pesp,lb and psp for m = 3 +m=3 +pconf +pesp,lb +psp +0.800 +0.900 +0.925 +0.950 +0.975 +0.990 +0.800 +0.700 +7 +2 +2 +2 +1 +1 +0.800 +5 +4 +3 +2 +2 +0.900 +34 +9 +4 +2 +0.925 +24 +6 +3 +0.950 +14 +5 +0.975 +13 +0.900 +0.700 +13 +4 +3 +3 +2 +2 +0.800 +10 +6 +4 +3 +2 +0.900 +74 +17 +7 +4 +0.925 +51 +11 +5 +0.950 +28 +8 +0.975 +26 +0.925 +0.700 +15 +5 +4 +3 +2 +2 +0.800 +12 +8 +5 +4 +3 +0.900 +92 +21 +8 +5 +0.925 +64 +13 +6 +0.950 +35 +10 +0.975 +32 +0.950 +0.700 +19 +6 +4 +4 +3 +2 +0.800 +15 +9 +6 +4 +3 +0.900 +118 +27 +10 +6 +0.925 +82 +17 +8 +0.950 +44 +13 +0.975 +41 +0.975 +0.700 +27 +7 +6 +5 +4 +3 +0.800 +20 +13 +8 +6 +4 +0.900 +166 +38 +14 +8 +0.925 +115 +23 +10 +0.950 +61 +17 +0.975 +57 +0.990 +0.700 +37 +10 +8 +6 +5 +4 +0.800 +27 +17 +11 +7 +5 +0.900 +232 +52 +19 +10 +0.925 +160 +32 +14 +0.950 +85 +23 +0.975 +80 +17 + +Supplementary Table S3: Sample sizes for different constellations of pconf, pesp,lb and psp for m = 4 +m=4 +pconf +pesp,lb +psp +0.800 +0.900 +0.925 +0.950 +0.975 +0.990 +0.800 +0.700 +5 +2 +2 +1 +1 +1 +0.800 +4 +3 +2 +1 +1 +0.900 +23 +6 +3 +2 +0.925 +16 +4 +2 +0.950 +9 +3 +0.975 +9 +0.900 +0.700 +9 +3 +2 +2 +2 +1 +0.800 +7 +4 +3 +2 +2 +0.900 +49 +12 +5 +3 +0.925 +34 +8 +4 +0.950 +19 +6 +0.975 +18 +0.925 +0.700 +10 +3 +3 +2 +2 +2 +0.800 +8 +5 +4 +3 +2 +0.900 +61 +14 +6 +3 +0.925 +43 +9 +4 +0.950 +23 +7 +0.975 +22 +0.950 +0.700 +13 +4 +3 +3 +2 +2 +0.800 +10 +6 +4 +3 +2 +0.900 +79 +18 +7 +4 +0.925 +55 +11 +5 +0.950 +30 +9 +0.975 +28 +0.975 +0.700 +18 +5 +4 +3 +3 +2 +0.800 +14 +9 +6 +4 +3 +0.900 +111 +25 +9 +5 +0.925 +77 +16 +7 +0.950 +41 +12 +0.975 +38 +0.990 +0.700 +25 +7 +5 +4 +3 +3 +0.800 +18 +12 +8 +5 +4 +0.900 +155 +35 +13 +7 +0.925 +107 +21 +10 +0.950 +57 +16 +0.975 +53 +18 + +Supplementary Table S4: Sample sizes for different constellations of pconf, pesp,lb and psp for m = 5 +m=5 +pconf +pesp,lb +psp +0.800 +0.900 +0.925 +0.950 +0.975 +0.990 +0.800 +0.700 +4 +1 +1 +1 +1 +1 +0.800 +3 +2 +2 +1 +1 +0.900 +17 +5 +2 +1 +0.925 +12 +3 +2 +0.950 +7 +3 +0.975 +7 +0.900 +0.700 +7 +2 +2 +2 +1 +1 +0.800 +5 +3 +2 +2 +1 +0.900 +37 +9 +4 +2 +0.925 +26 +6 +3 +0.950 +14 +4 +0.975 +13 +0.925 +0.700 +8 +3 +2 +2 +1 +1 +0.800 +6 +4 +3 +2 +2 +0.900 +46 +11 +4 +3 +0.925 +32 +7 +3 +0.950 +18 +5 +0.975 +16 +0.950 +0.700 +10 +3 +2 +2 +2 +1 +0.800 +8 +5 +3 +2 +2 +0.900 +59 +14 +5 +3 +0.925 +41 +9 +4 +0.950 +22 +7 +0.975 +21 +0.975 +0.700 +14 +4 +3 +3 +2 +2 +0.800 +10 +7 +4 +3 +2 +0.900 +83 +19 +7 +4 +0.925 +58 +12 +5 +0.950 +31 +9 +0.975 +29 +0.990 +0.700 +19 +5 +4 +3 +3 +2 +0.800 +14 +9 +6 +4 +3 +0.900 +116 +26 +10 +5 +0.925 +80 +16 +7 +0.950 +43 +12 +0.975 +40 +19 + diff --git a/cdFJT4oBgHgl3EQf-C0r/content/tmp_files/load_file.txt b/cdFJT4oBgHgl3EQf-C0r/content/tmp_files/load_file.txt new file mode 100644 index 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' and Benjamin Noto1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 4 1Institute of Biostatistics and Clinical Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' University of M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 48149,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Germany 2Clinic for Radiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' University Hospital M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 48149,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Germany 3Department of Nuclear Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' University Hospital M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 48149,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Germany 4West German Cancer Centre (WTZ) Essen-M¨unster – M¨unster site,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' University Hospital M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 48149,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Germany moritzfabian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='danzer@ukmuenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='de Abstract There is an increasing number of potential biomarkers that could allow for early assess- ment of treatment response or disease progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' However, measurements of quantitative biomarkers are subject to random variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Hence, differences of a biomarker in longitudinal measurements do not necessarily represent real change but might be caused by this random measurement variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Before utilizing a quantitative biomarker in longitudinal studies, it is therefore essential to assess the measurement repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Measurement repeatability ob- tained from test-retest studies can be quantified by the repeatability coefficient (RC), which is then used in the subsequent longitudinal study to determine if a measured difference rep- resents real change or is within the range of expected random measurement variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The quality of the point estimate of RC therefore directly governs the assessment quality of the longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' RC estimation accuracy depends on the case number in the test-retest study, but despite its pivotal role, no comprehensive framework for sample size calculation of test-retest studies exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' To address this issue, we have established such a framework, which allows for flexible sample size calculation of test-retest studies, based upon newly introduced criteria concerning assessment quality in the longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This also permits retrospective assessment of prior test-retest studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 1 Introduction A biomarker is a characteristic objectively measured and evaluated as an indicator of normal biological processes, pathogenic processes, or response to a therapeutic intervention [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Biomarkers used as indicators of response to a therapeutic intervention, or disease progression, are called treatment response biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' One prime, established treatment response biomarker is lesion size change in cross-sectional imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' For clinical trials concerning solid tumors, the measurement of lesion size is formalized in the so-called Response Evaluation Criteria in Solid Tumors (RECIST) [11], that categorize treatment response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' With the rapid advancement in medical sciences, there is an increasing number of new potential treatment response biomarkers that could possibly allow for early and objective assessment of treatment response or disease progression in clinical trials and clinical practice [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='11690v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='ME] 27 Jan 2023 However, using a biomarker in practice requires some basic research into the reliability of its measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In addition to a fixed systematic measurement error (bias), which can be investigated by comparing measurements with a known target value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' phantom studies), it is important to take into account that measurements of quantitative biomarkers are subject to random variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Hence, changes in a biomarker in longitudinal measurements made under the same conditions do not necessarily represent real change but might be caused by exactly this random measurement variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Before testing or even utilizing a quantitative biomarker in longitudinal studies, it is therefore of principal importance to assess the measurement repeatability [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The repeatability of measurement is determined by test-retest studies, which then are also re- ferred to as repeatability studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In such studies, replicate measurements are made on a sample of subjects under conditions that are as constant as possible [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Measurement repeatability can be quantified by the within-subject standard deviation (wSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Using wSD, the repeatability coef- ficient (RC) can be calculated [6,24,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' RC is then used in the longitudinal study to determine if a difference in the biomarker represents presumed real change or is within the range of random measurement variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It is defined in such a way that a desired specificity to detect changes – usually 95% – is targeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The wSD and the RC, as determined by the test-retest study, are point estimates, and hence suffer from random error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' As we will show, the targeted specificity is therefore generally not achieved in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Following standard statistical results, the more subjects and the more repeated mea- surements are included in the test-retest study, the more reliable the estimates of wSD and RC will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Accordingly, the probability of a relevant deviation of the actually achieved value from the targeted specificity will decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The quality of assessments in the longitudinal study and consequently the validity of its results is directly governed by the precision of the estimates of wSD and RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Of course, exact knowledge of measurement repeatability is not only crucial for biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' For example, excellent measurement repeatability of scales and other laboratory instruments is manda- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The reliability of a scale can be checked using weights with a known mass and it is possible to perform many repeated measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In contrast, many biomarkers are measured in-vivo, rendering attainment of large sample sizes difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Also, it might be necessary from an ethical point of view to keep sample sizes as low as possible, since the measurement in question might be inconvenient, invasive, or even harmful for the patient or the healthy test person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' For example, a biomarker might be derived from computed tomography, which involves ionizing radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Yet, if the sample size in the test-retest study is small, there is a high chance of obtaining suboptimal estimates of RC with associated detrimental effects on sensitivity and specificity in the longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In what follows, we will focus on such and related issues concerning repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Before doing so, note that, related to but different from repeatability is reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' While repeatability represents the measurement precision under constant conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e, same measurement proce- dure, same operators, same measuring system, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=', reproducibility is, in contrast, measurement precision under differing conditions as various operators, measuring systems, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Statistical literature concerning requirements for test-retest studies is scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' One notable study investigating sample size requirements is by Obuchowski and Bullen [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In their work, Obu- chowski and Bullen conducted a simulation study to investigate the relation between the sample size in the test-retest study and the specificity achieved in a following longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The authors give a blanket recommendation for sample size of test-retest studies based on their results from a fixed set of simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Our goal is to expand upon the results of Obuchowski and Bullen [24] in several areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' First, we want to introduce new quality criteria for the planning of test-retest studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Furthermore, we will expand the considerations to include sensitivity, which has not been investigated in the literature so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Finally, we aim to provide analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In contrast to simulation studies, this allows for flexible calculation of sample size requirements and also the retrospective assessment of test-retest studies, as we will show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In doing so, we establish a comprehensive framework in which the notions introduced above are precisely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In what follows, we will introduce the model used for our framework and study the aspects of specificity and sensitivity in separate sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Afterwards, we demonstrate the application of our 2 concepts in a practical example and discuss our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 2 Definitions One possible approach to distinguish true change from random variation in the longitudinal study is to estimate measurement variability in a test-retest study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' For this purpose, n patients are measured m times within a short period of time, in which their true value presumably does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' For our considerations we assume independent subjects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' measurement of one target per patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In addition, independent replicate measurements are necessary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' measurements on a subject need to be made independent of the knowledge of its previous value(s) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Consequently, we establish the following model for the j-th measurement of the i-th patient Yij of the test-retest study: Yij = µi + εij (1) where µi is the true value for the i-th patient and εij is the random error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' We assume the random errors to be independent and normally distributed with mean 0 and variance w2 SD [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In particular, it follows that Yij ∼ N(µi, w2 SD) for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' , n} and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' , m} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This model is appropriate when true replicates are studied and a learning effect can be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' As we are only addressing measurement repeatability, a fixed bias does not need to be considered since it cancels out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' We also assume that measurement error is independent from the magnitude of µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' From this data, we can estimate the within-patient standard deviation wSD [6] by �wSD := � � � � 1 n n � i=1 1 m − 1 m � j=1 (Yij − ¯Yi·)2, (2) where ¯Yi· := 1/m �m j=1 Yij denotes the mean value of the measurements of patient i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Following Cochran’s theorem [10], the distribution of this entity is given by n(m − 1) �w2 SD w2 SD ∼ χ2 n(m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (3) According to standard asymptotic theory, the following central limit theorem holds for �wSD: �wSD − wSD wSD √ 2n(m−1) D → n,m→∞ Z, (4) where Z is a standard normally distributed random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' If the number of repeated measure- ments differs between subjects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' the i-th subject is measured mi times, the value n(m−1) needs to be replaced by �n i=1(mi − 1) in all formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' For the sake of simplicity, we restrict ourselves to the simple case of an equal number of repetitions m per subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In order to assess changes in the measurements of a single patient in the subsequent longitudinal study, the repeatability coefficient (RC) is computed [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It indicates the range in which two re- peated measurements are expected to fall with a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In what follows, we restrict ourselves to the assessment of changes in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' We want to keep our decision rules flexible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' we establish a target specificity psp ∈ (0, 1) which shall be reached for patients with no change in their true biomarker value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Hence RC is a function of psp and is given by RC(psp) := Φ−1(1 − (1 − psp)/2) · √ 2 · wSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (5) In most literature the RC is only considered for a fixed targeted specificity of 95%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' RC(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='95) [26,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In practice, wSD is unknown and hence replaced by its consistent estimator �wSD to obtain the estimated repeatability coefficient ˆ RC(psp) := Φ−1(1 − (1 − psp)/2) · √ 2 · �wSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (6) 3 This quantity can then be applied as cutpoint in the longitudinal study to determine whether there has been change between two consecutive measurements Ypre and Ypost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Here, we also assume, that the measured values have independent errors, but the true levels µpre and µpost might actually be different, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' we have Ypre = µpre + εpre and Ypost = µpost + εpost with εpre and εpost being independent and normally distributed with mean 0 and variance w2 SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In case the true values have not changed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' µpre = µpost, the difference Ypost − Ypre is normally distributed with mean 0 and variance 2w2 SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Hence, with a probability of psp, we have Ypost−Ypre ∈ [−RC(psp), RC(psp)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The rule to decide whether there is a change for a patient with the two measured values Ypre and Ypost should thus be whether their difference lies outside or inside the interval [−RC(psp), RC(psp)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' As the bounds are unknown in practice, this decision rule is replaced by the decision rule based on the estimated interval [− ˆ RC(psp), ˆ RC(psp)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Consequently, the targeted specificity psp will never be exactly met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This applies analogously to considerations for the sensitivity of this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 3 Effective specificity as a criterion for sample size estima- tion Our goal is to quantify the uncertainty introduced by the replacement of wSD by its estimator �wSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' As mentioned, the targeted specificity (psp) is not met in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' To assess this problem, we introduce the effective specificity Pesp which is the specificity actually achieved if a realisation of the estimate �wSD is plugged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Hence, Pesp is a random quantity as it depends on the value of �wSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' We use a capital letter to emphasise that it is indeed a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It can be implicitly defined via RC(Pesp) = ˆ RC(psp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (7) Although this quantity is unknown in practice, we can nevertheless analyse its distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Firstly, we can compute the expected value E[Pesp] and the bias, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' the difference E[Pesp] − psp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This is also the quantity targeted by Obuchowski and Bullen [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Their quality criterion requires |E[Pesp] − psp| to be smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='01, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' they want the mean effective specificity to deviate less than 1 percentage point from the target specificity, which they set to 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' But what is even more important, from our point of view, is that we can compute quantiles of the distribution of Pesp which will enable us to establish quality guarantees on the effective specificity of the longitudinal studies based on the design parameters n and m of the test-retest study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Expected value and bias According to (7), Pesp is given by Pesp = RC−1( ˆ RC(psp)) (8) = 1 − 2 · � 1 − Φ � Φ−1 � 1 − 1 − psp 2 � �wSD wSD �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (9) The function RC can be inverted as it is a continuous, monotonically increasing function on (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The expectation of this random quantity can be computed exactly using (3) or approximately using the central limit theorem (4), according to which the distribution of �wSD/wSD can be approximated with a normal distribution with expectation 1 and variance 1/(2n(m − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Hence, we get E[Pesp] =1 − 2 · � 1 − � ∞ 0 Φ � Φ−1 � 1 − 1 − psp 2 � w � fχ2 n(m−1)(n(m − 1)w2) 2wn(m − 1) dw � (10) ≈1 − 2 · � 1 − � ∞ −∞ Φ � Φ−1 � 1 − 1 − psp 2 � w � � n(m − 1) π exp � −n(m − 1)(w − 1)2� dw � (11) 4 A B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='950 20 40 60 n in test−retest study Expected value of effective specificity Sample size of test−retest study 10 30 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='0 Effective specificity Figure 1: A) Expected value of the effective specificity (E[Pesp]) as a function of n in the test- retest study for a target specificity psp of 95% and m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Already for a small case number of 10 (blue dot) the expected value is comparably high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' For a case number of 30 (green dot) the bias (E[Pesp] − psp) is below 1 percentage point and does not change substantially with an increase of the case number to 60 (red dot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' B) However, while E[Pesp] is already relatively high for a case number of 10, the tails of the corresponding PDF (blue area) are prominent, resulting in a high chance of obtaining a low Pesp in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The green and red area represent the PDF of the effective specificity for n = 30 and n = 60, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' where fχ2 n(m−1) denotes the probability density function (PDF) of a χ2-distributed random variable with n(m − 1) degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' By numerical evaluation of the terms in (10) and (11), the bias can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Quantiles of the distribution of Pesp We need to be aware that even if E[Pesp] is close to psp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' the bias is low, the probability for a substantial deviation of the actually realized specificity from the targeted specificity might be large (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Therefore, we want to know with which confidence pconf we can say that the effective specificity is larger than some lower bound pesp,lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This is expressed by the formula P [Pesp ≥ pesp,lb] = pconf ⇔P � ˆ RC(psp) ≥ RC(pesp,lb) � = pconf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (12) We want to introduce a new quality criterion based on this concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The quantity pconf is a function of pesp,lb and of course also depends on psp, n and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' For notational 5 convenience, however, we omit those arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' After some calculations, one obtains pconf =1 − Fχ2 n(m−1) � � � Φ−1 � 1 − 1−pesp,lb 2 �2 Φ−1 � 1 − 1−psp 2 �2 n(m − 1) � � � (13) ≈1 − Φ � � � � Φ−1 � 1 − 1−pesp,lb 2 � Φ−1 � 1 − 1−psp 2 � − 1 � � � 2n(m − 1) � � , (14) where Fχ2 n(m−1) denotes the cumulative distribution function of a χ2-distributed random variable with n(m − 1) degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This formulas can now be used in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In the above form, one can determine the confidence with which the effective specificity exceeds a fixed bound pesp,lb with given design parameters n and m of the test-retest study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Analogous considerations can be made for upper bounds by computing the probability of the complementary event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In the planning stage of the test-retest study it could be beneficial to choose the sample size n in such a way that a desired lower bound pesp,lb is achieved with a prespecified confidence pconf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' To this end, the asymptotic formula (14) can be solved explicitly for n: n ≥ 1 2(m − 1) � � Φ−1(1 − pconf)Φ−1 � 1 − 1−psp 2 � Φ−1 � 1 − 1−pesp,lb 2 � − Φ−1 � 1 − 1−psp 2 � � � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (15) The exact formula (13) cannot be explicitly solved for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' However one can numerically solve min � � � � � n ∈ N: 1 − Fχ2 n(m−1) � � � Φ−1 � 1 − 1−pesp,lb 2 �2 Φ−1 � 1 − 1−psp 2 �2 n(m − 1) � � � ≥ pconf � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (16) In our application example we will apply these formulas in the planning stage of a hypothetical test-retest study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' If one wants to identify the worst possible cases for given n and m, one could compute the lower bound of the effective specificity which is reached with confidence pconf: pesp,lb = 1 − 2 � � �1 − Φ � � � � � � �F −1 χ2 n(m−1)(1 − pconf) n(m − 1) Φ−1 � 1 − 1 − psp 2 � � � � � � � (17) ≈ 1 − 2 � �1 − Φ � � Φ−1(1 − pconf)Φ−1 � 1 − 1−psp 2 � � 2n(m − 1) + Φ−1 � 1 − 1 − psp 2 �� � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (18) Accordingly, in (1 − pconf) · 100% of all cases, the effective specificity will be even lower than the obtained pesp,lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' From our point of view, the probability of exceeding a lower bound pesp,lb is a valid criterion for evaluating the quality of assessment in a longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Different from the expected value of Pesp which has been previously proposed as a quality criterion [24], our criterion considers the tails of the distribution of Pesp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This allows to bound the probability of strongly deviating from the desired specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 4 Consideration of effective sensitivity Concerning the sensitivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' the ability to detect real change between two measurements of one patient in the longitudinal study, we can make similar considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Before coming back to 6 the problem of the uncertainty caused from the estimation of wSD, we first assume, that wSD and hence also RC(psp) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Of course, the sensitivity strongly depends on the difference between µpre and µpost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Also, such differences are more difficult to detect if wSD is large and a large target specificity is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' To be more precise, the sensitivity pse to detect a difference can be written as a function of µ∆ := µpost − µpre, wSD and the chosen specificity psp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It is given by pse(µ∆, wSD) :=P[Ypost − Ypre /∈ [−RC(psp), RC(psp)]] =1 − � Φ � Φ−1 � 1 − 1 − psp 2 � − µ∆ √ 2wSD � − Φ � Φ−1 �1 − psp 2 � − µ∆ √ 2wSD �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (19) In this form, the function can also be seen as a function of the effect size δ := µ∆/wSD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' pse(δ) :=1 − � Φ � Φ−1 � 1 − 1 − psp 2 � − δ √ 2 � − Φ � Φ−1 �1 − psp 2 � − δ √ 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (20) This dependence of the sensitivity from the effect size δ is visualized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='00 −6 −4 −2 0 2 4 6 δ Sensitivity Figure 2: Sensitivity as a function of effect size δ := µ∆/wSD for psp=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Note that we assume wSD to be known here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' As wSD is unknown and needs to be estimated by �wSD which will then be plugged in to compute ˆ RC(psp), the sensitivity computed in (20) will not be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Analogously to our considerations for the specificity, we introduce the effective sensitivity Pese which is the sensitivity which is actually achieved if a realisation of the estimate �wSD is plugged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Of course, it is also a random variable and does depend again on µ∆, wSD and psp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It can be defined by the equation Pese(µ∆, wSD) := P[Ypost − Ypre /∈ [− ˆ RC(psp), ˆ RC(psp)]| �wSD] = 1 − � Φ � Φ−1 � 1 − 1 − psp 2 � �wSD wSD − µ∆ √ 2wSD � −Φ � Φ−1 �1 − psp 2 � �wSD wSD − µ∆ √ 2wSD �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (21) With this expression and the exact distribution of �wSD given as in (3) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' the approximation of the distribution of � wSD wSD by a normal distribution from (4) we can now quantify the bias caused by the replacement of wSD by �wSD and compute quantiles of the distribution of Pese which will enable us to also give quality guarantees on the effective sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Unlike our considerations for the specificity, these values will also depend from the actual wSD and the difference µ∆ of the longitudinal study and hence will be regarded as functions of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 7 Bias To compute the bias in dependence from psp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' µ∆ and wSD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' we can take the expectation of the right hand side of (21) and use the exact distribution (3) and the central limit theorem (4) to obtain the result E[Pese(psp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' µ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' wSD)] =1 − � ∞ 0 Φ � Φ−1 � 1 − 1 − psp 2 � w − µ∆ √ 2wSD � fχ2 n(m−1)(n(m − 1)w2)2wn(m − 1) dw + � ∞ 0 Φ � Φ−1 �1 − psp 2 � w − µ∆ √ 2wSD � fχ2 n(m−1)(n(m − 1)w2)2wn(m − 1) dw ≈1 − � ∞ −∞ Φ � Φ−1 � 1 − 1 − psp 2 � w − µ∆ √ 2wSD � � n(m − 1) π exp � −n(m − 1)(w − 1)2� dw + � ∞ −∞ Φ � Φ−1 �1 − psp 2 � w − µ∆ √ 2wSD � � n(m − 1) π exp � −n(m − 1)(w − 1)2� dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (22) Please note that this can essentially be seen as a function of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Following (21) the bias of the effective sensitivity can be considered as a function of δ for any given psp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' E[Pese(psp, δ)] − pse(psp, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Quantiles of the distribution of Pese For the most accurate examination of the distribution of Pese we would need to consider both events Ypost − Ypre > ˆ RC(psp) and (23) Ypost − Ypre < − ˆ RC(psp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (24) However, this leads to expressions that are difficult to handle analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Actually, the two probabilities P[Ypost − Ypre > ˆ RC(psp)| �wSD] and (25) P[Ypost − Ypre < − ˆ RC(psp)| �wSD] (26) sum up to the effective sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' However, in the presence of an effect, one of them will be much larger than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In the case δ > 0, the probability in (25) is larger than that from (26) which is bounded from above by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='025 and quickly converges to 0 as δ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' To enable the derivation of analytical formulas, we will therefore restrict ourselves to the consideration of δ > 0 and the event (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It is nevertheless possible to circumvent this simplification by numerical inversion of the relationship given in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' But here, we will approximate Pese(psp, µ∆, wSD) ≈ P[Ypost − Ypre > ˆ RC(psp)| �wSD] (27) = 1 − Φ � Φ−1 � 1 − 1 − psp 2 � �wSD wSD − µ∆ √ 2wSD � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (28) In analogy to the previous section we can provide confidence levels pconf which indicate the prob- ability that the effective sensitivity for some effect δ exceeds the lower bound pese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='lb: pconf = P[Pese(psp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' δ) ≥ pese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='lb] ≈ Fχ2 n(m−1) � � � � �Φ−1(1 − pese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='lb) + δ/ √ 2 Φ−1 � 1 − 1−psp 2 � � � 2 n(m − 1) � � � ≈ Φ � � � �Φ−1(1 − pese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='lb) + δ/ √ 2 Φ−1 � 1 − 1−psp 2 � − 1 � � � 2n(m − 1) � � (29) 8 Of course,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' such considerations only make sense if pse > pese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='lb for the chosen effect size δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' As above, analogous considerations can be made for upper bounds by computing the probability of the complementary event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' While (29) allows to compute the confidence of reaching a certain lower bound of the sensitivity for an effect δ, this formula may also be transformed to be used in the planning stage of the test-retest study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' If one wants to achieve a fixed confidence with which the effective sensitivity for an effect size δ exceeds some lower bound, one can use the exact results from above or the approximations made thereafter to determine the sample size n of the test-retest study in which each patient is measured m times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It shall be chosen such that n = min � n ∈ N: P � 1 − � Φ � Φ−1 � 1 − 1 − psp 2 � �wSD wSD − µ∆ √ 2wSD � (30) −Φ � Φ−1 �1 − psp 2 � �wSD wSD − µ∆ √ 2wSD �� ≥ pese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='lb � ≥ pconf � (31) ≈ min � � � � � n ∈ N: Fχ2 n(m−1) � � � � �Φ−1(1 − pese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='lb) + δ/ √ 2 Φ−1 � 1 − 1−psp 2 � � � 2 n(m − 1) � � � ≥ pconf � � � � � (32) ≈ 1 2(m − 1) � � Φ−1(pconf)Φ−1 � 1 − 1−psp 2 � Φ−1 (1 − pese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='lb) + δ/ √ 2 − Φ−1 � 1 − 1−psp 2 � � � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (33) Analogous to the preceding section, we can use these results in the planning stage of a test-retest study, as we will demonstrate in the following application example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Even if a study is planned based on considerations of the specificity, the formulas allow to assess the distribution of the effective sensitivity for any given effect size of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Calculations for δ < 0 follow analogously to the considerations for δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 5 Application example To illustrate our considerations, we will discuss a hypothetical application for early treatment response assessment in recurrent or metastatic nasopharyngeal carcinoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' While some patients with recurrent nasopharyngeal carcinoma show response or stable disease to systemic treatment, many patients will have progressive disease, which is invariably lethal [13,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Nevertheless, futile treatments should be avoided due to associated toxicity [9, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' To suspend futile treatment as soon as possible an imaging biomarker is desirable which accurately classifies treatment response earlier than change in morphologic lesion size, the current standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' A promising biomarker in this context is diffusion weighted magnetic resonance imaging (DWI) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' DWI depends on the differences in the movement of water molecules based on Brownian motion, which can be quantified by the apparent diffusion coefficient (ADC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' An exemplary measurement of ADC is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Change in ADC has shown promise as an early treatment response marker in various tumors, including nasopharyngeal carcinoma [18,28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Prospective planning of test-retest studies As laid out above, before conducting a longitudinal study in which a biomarker is applied to assess treatment response, a test-retest study should be conducted to assess repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In our example, we will set psp to 95% and m = 2, as these are the usual values in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' We imagine the researcher would want to obtain a specificity of at least 90% (pesp,lb) with 95% certainty (pconf) in the longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' What sample size (n) is necessary in the test-retest study?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This question 9 Figure 3: Example case of a tumor in the right nose showing restricted diffusion (A) with a mean ADC of 610 · 10−6 mm2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' (B) Region of interest outlined in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' can be answered using the asymptotic formula (15): n ≥ 1 2(m − 1) � � Φ−1(1 − pconf)Φ−1 � 1 − 1−psp 2 � Φ−1 � 1 − 1−pesp,lb 2 � − Φ−1 � 1 − 1−psp 2 � � � 2 ⇔n ≥ 1 2(2 − 1) � Φ−1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='05)Φ−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='975) Φ−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='95) − Φ−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='975) �2 ⇔n ≥ 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='3 (34) This can also be concluded from Figure 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Numerical solution of the exact formula (16) yields a sample size of 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Resulting sample sizes for other values of pesp,lb can be taken from Figure 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The resulting scenario in terms of the distribution of the relative error in the estimation of �wSD and its effect on Pesp is displayed in Figure 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Analogous considerations can be made for the effective sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' We consider the sensitivity for an underlying true effect size of δ = 4 in a study with psp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='95 and m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' According to formula (20), a sensitivity of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='74% was achieved if wSD was a known quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' However, this will not be met in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' What is the minimum sample size (n) of the test-retest study such that we can be 95% (pconf) sure to achieve at least a sensitivity of 75% (pese,lb) for that effect size?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This question can be answered using the approximate formula (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' n ≈ 1 2(m − 1) � � Φ−1(pconf)Φ−1 � 1 − 1−psp 2 � Φ−1 (1 − pse,lb) + δ/ √ 2 − Φ−1 � 1 − 1−psp 2 � � � 2 (35) ⇒n ≈ � Φ−1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='95)Φ−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='975) Φ−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='25) + 4/ √ 2 − Φ−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='975) �2 (36) ⇔n ≈ 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='1 (37) Accordingly, a sample size of 139 patients in the test-retest study would be recommended to achieve the set targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Using the exact formula (17) or the asymptotic formula (18), an effective specificity of at least 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='25% resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='27% is reached with a certainty of 95%, in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This is also depicted by Figure 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 10 A BA B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9 0 10 20 30 40 50 60 70 n pesp,lb m 2 3 pesp,lb m n E[Pesp] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='7 2 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='8 2 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9 2 54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='925 2 164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='7 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='8 3 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9 3 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='925 3 82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9483 Figure 4: A) Lower bound of effective specificity reached with confidence of 95% as a function of sample size (n) and number of repeated measurements (m) of the test-retest study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' B) Sample size resulting from (16) for different values of the desired lower bound pesp,lb that shall be exceeded with a fixed confidence of 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Retrospective assessment of test-retest studies It is not always necessary to conduct a preceding test-retest study when planning a longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The RC used in the longitudinal study might be adopted from already published test- retest studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' If one intends to use the point estimate of the RC obtained in a previous study, it is advisable to retrospectively assess the resulting distribution of the effective specificity and sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This allows to evaluate the impact of the sample size of the used test-retest study on quality criteria of the longitudinal study, especially the probability of exceeding a given pesp,lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Common sample sizes in test-retest studies are around 10 and 20 [3,4,12,15,21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' If the point estimator of RC(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='95) resulting from a test-retest study with a sample size of 10 and two repeated measurements is used, the distribution of the effective specificity will have prominent tails as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' According to (17), the lower bound of the effective specificity obtained with 95% confidence is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='7814 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='8512 for a sample size of 10 and 20, respectively, which might be insufficient (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Note that for the recommendation by Obuchowski and Bullen [24] of a sample size of 35 for test-retest studies with m = 2 the probability of achieving an effective specificity below 94% is 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='74%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Such considerations are also possible for the effective sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 6 Discussion We have established a comprehensive framework for planning of test-retest studies concerning re- peatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It enables flexible calculation of sample size requirements and retrospective assessment of such studies with regard to different quality criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' To better discuss planning of test-retest studies we have introduced the notions of effective speci- ficity (Pesp) and effective sensitivity (Pese), allowing for clearer differentiation of the targeted specificity psp and sensitivity pse from the values actually achieved in the longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Both Pesp and Pese are random quantities and their actual values are unknown in practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 11 A B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='92 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='0 Figure 5: Visualization of the application example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The x-axis denotes the relative error between �wSD and wSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The solid black line represents the asymptotic PDF of the relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Note the near identity to the dashed red curve, representing the PDF of the exact χ2 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The violet line shows the effective specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Analogously, the blue line shows the effective sensitivity for an underlying effect size of δ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' A) For n = 53 and m = 2: The area shaded in light gray represents 95% of the area under the normal curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' there is a 95% chance of obtaining a �wSD from the test-retest study that will result in an effective specificity of greater than 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' B) For n = 139 and m = 2: The area shaded in light gray represents 95% of the area under the normal curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' there is a 95% chance of obtaining a �wSD from the test-retest study that will result in a specificity of greater than 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In this case, an effective specificity of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='27% will be reached with a certainty of 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' However, we can determine their distribution and thus can compute different characteristics which properly reflect the uncertainty caused by the estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Expanding on the work of Obuchowski and Bullen [24], we have introduced a new quality criterion for sample size calculation of test-retest studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In their work, Obuchowski and Bullen [24] de- mand that the mean effective specificity (E[Pesp]) deviates at most by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='01 from the fixed targeted specificity (psp) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' However, using the mean effective specificity as sole quality criterion has limitations, since the whole distribution of the effective specificity is not properly taken into ac- count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' As illustrated in Figure 1, there is a high probability that the actually achieved effective specificity deviates strongly from its target even if the mean effective specificity may be close to the targeted specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Therefore, we propose a quality criterion for sample size calculations based on the probability that the effective specificity exceeds a chosen lower bound, taking into account the tails of the distribution of Pesp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In contrast to previous works we expand our consideration also to issues of sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Here, of course, it must also be taken into account that the sensitivity depends on the underlying effect size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Nevertheless, we can determine the distribution of the effective sensitivity for any effect size and provide analogous sample size formulas as for the specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Finally, our study is the first to provide analytical rather than simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' This provides greater flexibility as the targeted specificity psp and number of repeated measurements m may be chosen freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Hence, it allows the readers to avoid conducting time-consuming simulation studies themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' While our formulas enable flexible calculations for all scenarios, for convenience of the reader we also provide a table with sample sizes for some exemplary scenarios in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 12 Sample sizes resulting from other choices of the parameters pconf, pesp,lb, psp and m can be found in Supplementary Tables S1-S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Our study has some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The field of application is restricted to test-retest studies in which true replicates of measurements are possible, for example in quantitative imaging markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Our considerations are not valid if the measurement process itself results in a change of the measurand (learning / practice effect) as has been described for some psychological assessments [14,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Our standard model (1) assumes independent and identically normally distributed errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It is therefore advisable to examine whether there is a relationship between the within-subject varia- tion and the level of the measured value before applying our approach [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' If the variability of the measurement error increases with the magnitude of the measured value, a log transformation might resolve the issue [1, 7, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Beyond that, non-normally distributed error terms are not cov- ered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' We also have not specifically considered the scenario of clustered data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' measuring multiple lesions per subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' However, if a hierarchical model structure with independent errors can be assumed, this does not pose a restriction to application of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' It should be noted that exact solutions based on the χ2 distribution for all our considerations are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In some cases, when an analytic solution is not possible, these exact solutions require the application of numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' In order to give completely analytic solutions, some of our formulas rely on asymptotic results and approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' The differences between exact and approximate results are most severe for small sample sizes and small effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Applying both exact and approximate formulas in our application example, it can be seen that these differences are negligible in practically relevant scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Implementations of exact and approximate solu- tions can be found in our supplementary R code [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' So far, our considerations are limited to repeatability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' assuming same measurement conditions for the repeated measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' However, for real world application of biomarkers, consideration of reproducibility is also important since longitudinal measurements are often performed under different measuring conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' varying readers or scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Therefore, our model should be perspectively enhanced to include aspects of reproducibility such as a fixed bias as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' in some models considered by Obuchowski and Bullen [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Nevertheless, since repeatability limits repro- ducibility, a good knowledge of the former is useful in order to interpret reproducibility studies properly [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Test-retest studies of repeatability should be well planned to guarantee for a sufficient quality of dependent longitudinal studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Our framework allows the derivation of analytical solutions for quality criteria that can be used to assess implications of the test-retest study design on subsequent longitudinal studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Acknowledgements B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' was funded as a clinician scientist by the Medical Faculty, University of M¨unster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' There was no dedicated funding for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Additional information Data availability No datasets were generated or analysed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Implementa- tions of exact and approximate formulas can be found in the Supplementary R Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Additionally, we provide sample sizes based on formula (16) in Supplementary Tables S1-S4 generated using our R code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Competing interests The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' References [1] Douglas G Altman and J Martin Bland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Measurement in medicine: the analysis of method comparison studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Stat.' 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Pace, Andrew N Priest, Rebecca A Quest, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Diffusion-weighted MRI in advanced epithelial ovarian cancer: apparent diffusion coef- ficient as a response marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Radiology, 293(2):374–383, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' 15 Supplementary Material In the following tables, we want to give sample sizes based on formula (16) in our main manuscript for different choices of parameters pconf, pesp,lb, psp and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' For further constellations that cannot be found in the following tables, we would like to refer to the sample size function provided in our Supplementary R Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content=' Supplementary Table S1: Sample sizes for different constellations of pconf, pesp,lb and psp for m = 2 m=2 pconf pesp,lb psp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='950 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='800 54 34 22 14 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='900 463 103 37 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='925 320 63 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='950 170 46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='975 159 16 Supplementary Table S2: Sample sizes for different constellations of pconf, pesp,lb and psp for m = 3 m=3 pconf pesp,lb psp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQf-C0r/content/2301.11690v1.pdf'} +page_content='800 0.' 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India +2Inter University Center for Astronomy and Astrophysics, Pune 411007, India +(Dated: February 1, 2023) +Gravitational-wave backgrounds are expected to arise from the superposition of gravitational +wave signals from a large number of unresolved sources and also from the stochastic processes that +occurred in the early universe. So far, we have not detected any gravitational wave background, but +with the improvements in the detectors’ sensitivities, such detection is expected in the near future. +The detection and inferences we draw from the detection of a gravitational-wave background will +depend on the source model, the type of search pipeline used, and the data generation process +in the gravitational-wave detectors. +In this work, we focus on the effect of the data generation +process, specifically the calibration of the detectors’ digital output into strain data used by the +search pipelines. Using the calibration model of the current LIGO detectors as an example, we +show that for power-law source models and for calibration uncertainties ≲ 10%, the detection of +gravitational wave background is not significantly affected. We also show that the source parameter +estimation and upper limits calculations get biased and must be corrected in the analyses. +I. +INTRODUCTION +Since the first detection in September 2015 [1], the +LIGO [2], and the Virgo [3] gravitational wave (GW) +detectors have detected nearly one-hundred compact bi- +nary merger signals [4–6]. They correspond to individual +merger signals with a high signal-to-noise ratio (SNR). In +addition to those high SNR signals, assuming the merger +events are outliers in a much larger population of com- +pact mergers, we also expect a large number of low SNR +signals that are hard to detect individually. The super- +position of such a large number of low SNR signals would +give rise to a gravitational-wave background (GWB) that +could be detected with the current or next generation of +GW detectors [7–10]. +Apart from the compact binary mergers signals, su- +perposition of other astrophysical GW signals such as +from core-collapse supernovae [11, 12], magnetars [13, 14] +could also give rise to GWB. In addition to these astro- +physical sources, various events that took place in the +early universe such as inflation and phase transitions +could also give rise to GWB [15]. The detection of GWB +from astrophysical sources can help us better understand +the population and the evolution of stars in the universe +[16–18] while the detection of GWB from cosmological +sources can provide information about the processes in +the very early universe which are otherwise difficult to +obtain [19]. +The LIGO-Virgo-KAGRA (LVK) collaboration, in +their recent analyses using data from the observing run +O3, did not find any evidence of GWBs and hence placed +upper limits on the amplitudes of possible isotropic [20] +and anisotropic GWBs [21]. With the proposed improve- +ments to the current GW detectors [22], it might be pos- +sible to detect the GWB from compact binary mergers +[10]. And, the proposed next-generation GW detectors +[23, 24] will certainly observe the GWB from compact bi- +nary mergers. The data generation and various aspects +of the search are expected to affect the GWB search re- +sults, and hence it is important to understand them. In +this paper, we focus on the effects of the data genera- +tion, specifically that of the calibration, on the analysis +results. Calibration is the process of converting the raw +digital outputs of the detectors into strain data that are +further used in the GW analyses. Any uncertainties in +that process could translate into biases in the final re- +sults, affecting our interpretations. +Typically cross-correlation-based searches, correlating +data from multiple detectors, are used to detect GWBs +[25]. In previous such searches using LIGO-Virgo data +[20, 26, 27], upper limits were calculated after marginal- +izing over calibration uncertainties as outlined in [28]. +However, that method does not capture any biases intro- +duced by the systematic errors in the calibration model. +In this work, we try to address that issue. In the past, +this has been studied mostly in the context of the search +for GW signals from individual compact binaries [29– +32]. Recently, such questions have also been addressed +for the detection, and parameter estimation of individual +compact binary merger signals [33–35]. We use a similar +simulation-based method [33, 34] to address the effects +of calibration uncertainties on the searches for GWB. In +addition, we also show that one could try to estimate the +GWB and calibration model parameters simultaneously +and get a reasonable signal recovery. +The remainder of this paper is organized as follows. +In Sec. II we briefly introduce the model and search for +GWB using data from GW detectors. In Sec. III, we dis- +cuss the calibration model used to convert the raw digital +output into strain data used in GW searches. In Sec. IV +we describe the method used to quantify the effects of +calibration uncertainties on GWB searches. In Sec. V we +show the results of our analyses, and in Sec. VI conclude +with the main results and future outlook. +arXiv:2301.13531v1 [gr-qc] 31 Jan 2023 + +2 +II. +MODELING AND SEARCH FOR +GRAVITATIONAL-WAVE BACKGROUND +A GWB is usually characterized in terms of fractional +energy density in gravitational waves Ωgw(f) [25], given +by, +Ωgw(f) = f +ρc +dρgw +df +, +(1) +where f is the frequency, dρgw is the energy in gravita- +tional waves in the frequency interval from f to f +df, ρc +is the critical energy density needed to close the universe. +The value of ρc is given by +ρc = 3c2H2 +0 +8πG +, +(2) +where c is the speed of light, G is the gravitational con- +stant and H is the Hubble constant. In this work, we +use the value of Hubble constant measured by the Plank +satellite, H0 = 67.9 km s−1 Mpc−1 [36]. However, the +conclusions drawn are independent of the actual value of +H0. +Typically Ωgw(f) is expressed in the form of a power +law, +Ωgw(f) = Ωα +� f +fref +�α +, +(3) +where fref is a reference frequency. For results reported +in this paper, we use a reference frequency of fref = 25 Hz +as used in the LVK analyses [20, 26, 27]. The value of the +power-law index α depends on the source of GWB we are +interested in. For cosmological GWB from inflationary +scenarios we typically expect α = 0 [15] while for astro- +physical GWB from the superposition of many compact +binary merger signals α = 2/3 [16]. +The optimal estimator of Ωα at a time t and at a fre- +quency bin f is given by [18, 37], +ˆΩα(t; f) = 2 +T +ℜ[d∗ +I(t; f)dJ(t; f)] +γIJ(f)Sα(f) +, +(4) +where d1(t; f) and d2(t; f) are short-time Fourier trans- +forms of the strain data from the two detectors (I, J) +evaluated at time t, T is the duration of the data seg- +ments used for Fourier transforms and γIJ(f) is the nor- +malized overlap reduction function for the given two de- +tectors (I, J). The function Sα(f) is proportional to the +assumed spectral shape α and is given by [18, 37], +Sα(f) = 3H2 +10π2 +1 +f 3 +� f +fref +�α +(5) +In the weak-signal limit, the variance of ˆΩα is given by +[18, 37], +σ2 +ˆΩα(t; f) = +1 +2T∆f +PI(f)PJ(f) +γ2 +IJ(f)S2α(f) +(6) +where PI(f), PJ(f) are the one-sided power spectral den- +sities of the strain data from the two detectors (I, J), and +∆f is the frequency resolution. For data spanning many +segments and a large frequency band, the final optimal +estimators are obtained by a weighted sum, +ˆΩα = +� +t,f σ−2 +ˆΩα(t; f)ˆΩα(t; f) +� +t,f σ−2 +ˆΩα(t; f) +, +σ−2 +ˆΩα = +� +t,f +σ−2 +ˆΩα(t; f), +(7) +where t runs over available time segments and f runs over +discrete frequency bins in the desired frequency band. +III. +CALIBRATION MODEL +The raw outputs of gravitational wave detectors are +digitized electrical signals from the photodetectors at +the output port. The process of converting these elec- +trical signals into strain data is called calibration. The +LIGO, Virgo and KAGRA detectors all have similar fun- +damentals in optical layout and control system topology +[2, 3, 38]. While their methods to describe and charac- +terize that system are different (sometimes only in sub- +tle ways that reflect their detailed differences), any of +those methods could be used to describe current GW de- +tectors. Thus, here, we follow and choose the methods +of the LIGO detectors [39, 40]. For details of different +calibration techniques used in the current generation of +gravitational wave detectors see [39, 41–43]. As shown in +[40], after detailed modeling of the detectors, a response +function R(f) is derived which is then used to convert +the digitized electrical output into strain h(f) using the +expression, +d(f) = 1 +Le(f)R(f) +(8) +where e(f) is the digitized signals from the output photo- +detectors, R(f) is the response function that converts e(f) +into the differential displacement of the two arms of the +detector and L is the average (macroscopic) length of the +two arms. +A typical response function in the frequency domain +can be written as [40, 44], +R(f) = 1 + A(f)D(f)C(f) +C(f) +(9) +where C(f) is the sensing function corresponding to the +response of the detector to differential changes in its two +arms without any feedback control, A(f) is the actuation +function used to control the positions of the mirrors and +D(f) is any digital filter(s) used in the control loop. + +3 +A. +Sensing function +The sensing function C(f) can be modeled in the fre- +quency domain as [40, 45], +C(f) = +� +κCHC +1 + iff −1 +cc +� � +f 2 +f 2 + f 2s − iffsQ−1 +� +× CR(f) +(10) +where optical gain HC +represents the overall gain, +coupled-cavity pole frequency fcc defines the detector +bandwidth, fs and Q correspond to optical anti-spring +pole frequency and its quality factor respectively. The +term CR represents the frequency dependencies not cap- +tured by the other terms (for example, the response of the +electronics chain used for the digitization, etc.), and κC +is a scale factor representing the changes in the sensing +function with respect to a reference time. An example +sensing function plot is shown in Fig. 1. +We use the +pyDARM package [46] to generate the calibration model +used in this work. For LIGO detectors, during the past +FIG. 1. Plot showing an example of sensing function C(f) +of LIGO Hanford detector during the observing run O3 [46]. +The unit of C(f) is the counts produced in the Analog-to- +Digital converter at the output port for a meter differential +length change in the two arms of the GW detector [40]. +observing runs and for frequencies ≳ 20 Hz, the optical +spring term (second term in Eq. 10) was usually close to +one (for example, see [47, 48]), +f 2 +f 2 + f 2s − iffsQ−1 ≈ 1. +Since in our work, we will be using 20 − 1726 Hz band +as done in LVK analyses [20, 26, 27], we will neglect the +optical spring term in Eq. 10 for the rest of the paper. +B. +Actuation function +The actuation function is modeled in the frequency +domain as [40, 45], +A(f) = κUAU(f) + κP AP (f) + κT AT (f) +(11) +where U, P, and T represent the lowest three stages of +suspensions (upper intermediate mass, penultimate, and +test mass stages) used to suspend the main optics [2, 40]. +Ai(f) (where i = U, P, T) are frequency-dependent ac- +tuation models of the three stages of the suspensions, +including digital filters in the control path and analog re- +sponses of the three stages of suspensions [40]. The scale +factors κi capture any changes in the reference actuation +model of each stage, and in general, they could be time- +and frequency-dependent [49]. +The plots of actuation +models for the three stages and the combined actuation +model are shown in Fig. 2. +FIG. 2. Plot showing an example of the actuation functions +of the bottom three stages (top, penultimate, and test mass +stages) as well as the combined actuation function of LIGO +Hanford’s main optic suspension during the observing run +O3 [46]. The unit of A(f) is the differential length change +produced in the two arms for a unit count in the Digital-to- +Analog converter that drives the actuators [40]. +C. +Total response function +Apart from the notch filters used to prevent the exci- +tation of resonances of the test mass suspensions, D(f) +is a smooth function of frequency that is decided by the +feedback control morphology used. The total response +function, as shown in 9, is a function of C(f), A(f), and +D(f). Fig. 3 shows an example response function of the +LIGO Hanford detector during the observing run O3. + +107 +Magnitude (ct/m) +106 +105 +102 +103 +Frequency (HzAu(f)(top) +Ap(f)(penultimate +10-16 +Magnitude (m/ct) +AT(f) (test mass +A(f) (total) +10-18 +10-20 +102 +103 +Frequency (Hz4 +FIG. 3. Plot showing an example of the response function +R(f) of the LIGO Hanford detector during the observing run +O3 [46]. +IV. +ANALYSIS METHOD +In this work, we look at the effects of calibration uncer- +tainties on the recovery of GWB and on the parameter +estimation of the recovered GWB. Specifically, we look +at GWBs described by power-law models with power-law +indices of α = 0, 2/3, 3 (see Sec. II). +If the response function used to calibrate the digitized +signal in Eq.8 is not the true response function, then we +get, +dtrue(f) = dcalc(f) × Rtrue(f) +Rcalc(f) +(12) += dcalc(f) × Λ(f) +(13) +where true and calc correspond to the true and calculated +quantities respectively. +In the above Eq. 12, we have +defined Λ(f) as, +Λ(f) = Rtrue(f) +Rcalc(f) +(14) +for convenience. The uncertainties in the calibration pro- +cess enter the GW analyses as Λ(f) shown above. We +note here that Rtrue(f), with measurement uncertainty, +can be calculated using a length (or frequency) reference +such as a photon calibrator [50], but due to difficulty in +the implementation Rcalc(f) is traditionally used in the +calibration process leading to the difference we see in the +Eq.12. The Rtrue(f) is usually in a non-parametric form +while Rcalc(f) is parameterized with a relatively small +number of parameters (Eq. 9). Hence from an implemen- +tation point of view, Rcalc(f) is more desirable. Because +of the simple parameterization, changes in Rcalc(f) can +also be easily tracked which is also important for the cal- +ibration. Moreover, the ratios Λ(f) are usually very close +to one and hence use of Rcalc(f) is well justified. +Due to the measurement uncertainties in Rtrue(f), the +estimation of the ratios Λ(f) has both systematic and +statistical uncertainties associated with it. Using Eq. 12 +in Eqs.4 and 6 we get, +ˆΩα(f) = 2 +T +ℜ +� +d∗ +I,calc(f)dJ,calc(f)Λ∗ +I(f)ΛJ(f) +� +γIJ(f)Sα(f) +(15) +and +σ2 +ˆΩα(f) = +1 +2T∆f +PI,calc(f)PJ,calc(f) +γ2 +IJ(f)S2α(f) +|ΛI|2|ΛJ|2. +(16) +The Eqs. 15 and 16 provide a way to estimate the effects +of calibration uncertainties on the signal estimate ˆΩα and +its variance σ2 +ˆΩα. If we further assume that the ratios +Λ(f) are real, i.e., the difference is only in the magnitude, +then we get, +ˆΩα(f) = ˆΩα,nocal(f)ΛI(f)ΛJ(f) , +(17) +σ2 +ˆΩα(f) = σ2 +ˆΩα,nocal(f)Λ2 +I(f)Λ2 +J(f), +(18) +where nocal subscript corresponds to the quantities cal- +culated in the absence of calibration uncertainties that +we want. With this assumption, the simulation becomes +a little bit easier. +We can start with ˆΩα,nocal(f) and +σ2 +ˆΩα,nocal(f) calculated from the simulated data and us- +ing Eqs. 17, 18 and 7 we can estimate the effects of cal- +ibration uncertainties on the calculation of ˆΩα(f) and +σ2 +ˆΩα(f). However, in Sec.V we also show the results with- +out using this assumption. Since the response functions, +RI,J themselves are functions of A (Eq. 11), C (Eq. 10) +and D the number of free parameters in the above equa- +tions becomes large. Due to the large number of param- +eters, it is difficult to calculate the effects analytically, +so we use numerical simulation to calculate the effects. +This method becomes more useful when one wants to +include a more complicated signal model and additional +calibration parameters. +For the results reported in this paper, we use one week +of simulated data for Hanford and Livingston detectors +using advanced LIGO design sensitivity [22]. Here, one +week of data is chosen to represent the traditional long- +duration analyses of GWB and to avoid complexities aris- +ing from large SNRs in individual segments [25]. We use +publicly available LVK codes to perform the searches [51]. +We also use standard search parameters of 192-sec seg- +ment duration and frequencies from 20 Hz to 1726 Hz +with a frequency resolution of 1/32 Hz [20, 26, 27]. We +use the same calibration model for both Hanford and +Livingston detectors. +We do the following to calculate the effects of cali- +bration uncertainties on the recovery of GWB signal. +As indicated in the Eqs. 17 and 18, we multiply the +ˆΩα,nocal(t; f) and σ2 +ˆΩα,nocal(t; f) estimators of each seg- +ment by distributions representing the ratios Λ(f). We +assume Gaussian distributions for Λ(f), centered at one +with standard deviations defined by the desired calibra- +tion uncertainty. We also truncate the Gaussian distri- +bution at 2-sigma points on both sides to avoid unreal- +istic values for Λ(f) (for example, values close to zero + +10-7 +102 +103 +Frequency (Hz)5 +or even negative). Then, using Eqs. 7, we combine the +segment-wise and frequency-dependent results of ˆΩα(t; f) +σˆΩα(t; f) to get the final estimate and its uncertainty. +Then we use SNR, defined in a frequentist approach [52], +given by, +SNR = +ˆΩα +σˆΩα +as the detection statistics for a GWB. We then compare +these results against the results obtained without any +calibration uncertainties. +Since the difference between +these results is just the application of calibration uncer- +tainties, the differences would typically show the effects +of calibration uncertainties on ˆΩα and σ2 +ˆΩα. +We further look at the effects of calibration uncertain- +ties on the parameter estimation, specifically on the Ωα +and α, by varying the values of various parameters in the +R(f) (see Eqs.9, 10. 11). +V. +RESULTS +In this section, we present the results of our studies. To +generate these results, we initially assume that the ratios +of response functions Λ(f) are real and hence use Eqs. +17 and 18. We note that this assumption is used in the +marginalization of calibration uncertainties in the LVK +GWB analyses [20, 26, 27]. However, for comparison, we +also produce results by additionally using 1-sigma phase +uncertainties of 5◦, the maximum of what was seen in +LIGO detectors during the observing run O3 [40]. This is +to show how much phase uncertainties, that are currently +not included in the GWB analyses, affect the final results. +At each frequency, we model the magnitude of Λ(f) by +a Gaussian distribution with a mean one and standard +deviation ϵ that is small compared to one and phase of +Λ(f) by a Gaussian distribution with a mean zero and +standard deviation of 5◦. As indicated earlier, we also +truncate the Gaussian distribution at 2-sigma values to +avoid unrealistic realizations of Λ(f). +A. +Effect of calibration uncertainties on the GWB +detection +The recovered values of the ˆΩα, σˆΩα and SNR at vari- +ous levels of calibration uncertainties for the three power +law models α = 0, 2/3, 3 are shown in Fig. 4. In these +plots, we increase the uncertainty from 0 % to 20 % in +steps of 2 %. We also repeat the analysis 20 times at each +uncertainty level to calculate the spread on the recovered +values. For comparison, we also show results assuming +1-sigma phase uncertainties of 5◦. +From the plots, we see that as we increase the values of +uncertainties, there are changes in the recovered values of +Ωα, σˆΩα, and SNR. However, the changes in the recovered +SNRs are small, almost negligible, below the calibration +uncertainties of ∼ 10%. Since SNR is generally used as +a detection statistic, this suggests that the detection of +a GWB is not significantly affected by the uncertainties +in the calibration. +The Ωα, σˆΩα change by ∼ 10% when we change the un- +certainty of response function by ∼ 20%. The reduction +in the estimated σˆΩα can be attributed to how we com- +bine different time segments and frequency bins. Since we +use weighted average method (see Eq. 7), any downward +fluctuations in individual σˆΩα(t; f) due to calibration un- +certainties will bring down the final σˆΩα. A similar effect +could be attributed to the reduction in the final Ωα. This +suggests that the recovered values of Ωα and σΩα are bi- +ased in the presence of calibration uncertainties. Since +the upper limits on Ωα, for example, 95 % upper limit in +the frequentist approach, can be written as +Ωα,95% ≈ ˆΩα + 2 σˆΩα, +calibration uncertainties are also expected to bias the +upper limit calculations. Such biases are not completely +taken into account when estimating Ωα or while calcu- +lating upper limits on Ωα in the analyses reported in the +literature [20, 26, 27] and need to be accounted for in +future analyses. +The plots also suggest that including +phase uncertainties does not change the results signifi- +cantly. +B. +Effects of the calibration uncertainties on the +recovery of parameters of GWB +The second part of the study is to see the effect of +calibration uncertainties on the estimation of parameters +of the GWB signals. Here we mainly focus on the es- +timation of Ωα and α (see Eq. 3). In Sec. V A, Fig. 4 +already shows the effect of the uncertainties of the re- +sponse function as a whole on the recovery of Ωα. In- +stead of the uncertainties of the total response function, +in this section, we look at the effects of individual cali- +bration parameters on the recoveries of Ωα and α. Since +we are using the parameters that make up the calibration +model, in the literature, this is considered as a physically +motivated approach to include calibration uncertainties +in the signal analyses [33, 34]. In this study we mainly +focus on the parameters κC, fcc (see Sec. III A), κU, κP +and κT (see Sec. III B). Other parameters in the response +function tend to be more or less constant during an ob- +serving run, and hence we do not include them here. We +also perform the analysis only for α = 2/3, whose detec- +tion is expected in the near future. +The maximum likelihood values of the recovered pa- +rameters Ωα and α as functions of errors on the various +calibration parameters are shown in Figs. 5 and 6. +We +use the maximum likelihood method described in [53] +and use dynesty [54] sampler in bilby [55] package for +sampling the likelihoods and estimating the parameters. +From the figure we see that κP , κT and κC have signifi- +cant effects on the recovery of Ωα and α while fcc and κU + +6 +α = 0 +α = 2/3 +α = 3 +FIG. 4. Plots showing the effect of calibration uncertainty on the recovery of Ωα, σˆΩα and SNR for injected GWB signals +described by α = 0, 2/3, 3. The calibration uncertainty is quantified in terms of the relative standard deviation of the varying +response function. The solid (blue) line corresponds to no phase uncertainty while dotted (red) line corresponds to 5◦ 1-sigma +phase uncertainty. +have very little effect. This is probably expected because +of the relative contributions of these terms to the total +response function. +Rewriting Eq. 9 into contributions +from different components, we get, +R(f) = 1/C(f) + κUD(f)AU(f) ++κP D(f)AP (f) + κT D(f)AT (f). +(19) +Fig. 7 shows the relative contribution of the different +terms in Eq. 19 to the response function and also 90 % +search sensitivity region for the α = 2/3 GWB signal. We +see that in the 90 % sensitivity region, penultimate and +test mass actuation and sensing functions have signifi- +cant contributions. In the sensing function (see Eq.10), +the dominant contribution comes from κC. +Since the +typical value of fcc of advanced LIGO detectors during +the O3 run was ∼ 400 Hz and the 90 % search sensi- +tivity region extents only up to ∼ 45 Hz, the effect of +fcc on the estimation of the parameters is minimal. Be- +cause of the non-trivial phase relationship between these +functions, we see that the relative contributions to the +response function from individual components can even +go above one. +We also try to simultaneously estimate the calibration +and GWB signal parameters to see how well we can do. +Here we use (simulated) uncalibrated raw digital signals +to extract all the parameters. Fig. 8 shows an example of +the simultaneous estimation of all the parameters. The +plot shows that, along with the GWB model parameters, +we can also infer the values κP , κT , and κC to some level, +but recoveries of fcc and κU are poor which are consistent +with the results in Figs. 5 and 6. For comparison, we also +show the recovery of GWB model parameters using cal- +ibrated data without any uncertainties. The plots also +have the Bayes factors, comparing the signal vs. noise +hypothesis for those two cases. We see that the Bayes +factors do not change significantly in the two cases (as +expected, it is lower when we estimate calibration param- +eters also). However, the posteriors of GWB parameters +are very broad and probably biased when we simultane- +ously estimate the GWB and calibration model param- + +11.50 +mag uncertaity +11.25 +mag + 5° phase uncertaity +11.00 +SNR2a +10.75 +10.50 +10.25 +10.00 +0.00 +0.05 +0.10 +0.15 +0.20 +Uncertainty of R(f)l (in %11.00 +mag uncertaity +10.75 +mag + 5° phase uncertaity +10.50 +10.25 +10.00 +9.75 +9.50 +0.00 +0.05 +0.10 +0.15 +0.20 +Uncertainty of R(f)l (in %10.50 +10.25 +10.00 +SNR2a +9.75 +9.50 +mag uncertaity +9.25 +mag + 5° phase uncertaity +9.00 +0.00 +0.05 +0.10 +0.15 +0.20 +Uncertainty of R(f)l (in %X10-8 +1.25 +1.20 +1.15 +1.10- +mag uncertaity +mag + 5° phase uncertainty +1.05 +0.00 +0.05 +0.10 +0.15 +0.20 +Uncertainty of IR(f)l (in %)X10-8 +1.00 - +°0.95 +0.90 +mag uncertaity +mag + 5° phase uncertaity +0.00 +0.05 +0.10 +0.15 +0.20 +Uncertainty of |R(f)l (in %)X10-9 +3.0 +2.9 +2.8 +2.7 +2.6 +mag uncertaity +mag + 5° phase uncertaity +2.5 +0.00 +0.05 +0.10 +0.15 +0.20 +Uncertainty of |R(f)l (in %)X10-9 +1.14 +1.12 +1.10 +1.06 +mag uncertaity +1.04 +mag + 5° phase uncertaity +0.00 +0.05 +0.10 +0.15 +0.20 +Uncertainty of |R(f)| (in %)X10-10 +9.8 +9.6 +9.4 +9.2 +mag uncertaity +mag + 5° phase uncertaity +9.0 +0.00 +0.05 +0.10 +0.15 +0.20 +Uncertainty of |R(f)l (in %)X10-10 +2.9 +2.8 +C +2.7- +mag uncertaity +mag + 5° phase uncertaity +0.00 +0.05 +0.10 +0.15 +0.20 +Uncertainty of |R(f)l (in %)7 +FIG. 5. Effect of the errors in various calibration model pa- +rameters on the recovery of the signal parameter Ωα. The +solid lines correspond to the maximum likelihood values, and +the shaded regions indicate 68 % confidence interval. +The +injected value of Ωα is 1.04 × 10−8 for α = 2/3. +FIG. 6. Effect of the errors in various calibration model pa- +rameters on the recovery of the signal parameter α. The solid +lines correspond to the maximum likelihood values, and the +shaded regions indicate 68 % confidence interval. The injected +value of α is 2/3. +eters. So it is important to have well-calibrated data to +get better posteriors on the signal parameters and also a +better Bayes factor. +VI. +CONCLUSIONS +In this work, we have studied the effect of calibration +uncertainties on the detection and parameter estimation +of GWB signals. +We focused on amplitude (Ωα) and +power law index (α) of power-law isotropic GWBs. We +find that, for the current generation of LIGO detectors, +FIG. 7. Contribution of various calibration parameters to the +total response function and 90 % search sensitivity region for +the α = 2/3 GWB search. +when the calibration uncertainties are less than ∼ 10%, +they do not significantly affect the detection of a GWB +signal. The calibration uncertainties of the LIGO detec- +tors reported during the last observing run O3 are well +within this ∼ 10% limit [40]. +We also find that the recovery of GWB model param- +eters could be significantly affected depending on which +calibration parameter is poorly constrained and its un- +certainty level. The recovered parameters are biased due +to errors in calibration model parameters. Even though +the current errors on the individual model parameters +of LIGO detectors are much smaller (≲ 1 %), the cu- +mulative effect of the different parameters could bias the +recovered GWB parameters. Currently, this bias is not +considered during the GWB parameter estimation or up- +per limit calculation. +For a calibration uncertainty of +∼ 5 % of the total response function (90 % maximum +reported for the LIGO detectors during O3), the bias in +the estimate of GWB amplitude or its upper limit could +be as large as ∼ 5 %. This might be significant when +we try to differentiate between different models of GWB. +We also tried to estimate the GWB and calibration model +parameters simultaneously and find that we could detect +the signal, albeit with some loss of Bayes factor (SNR). +However, the posteriors of the GWB signal parameters +become very broad and probably biased due to their cor- +relation with some of the calibration parameters. This +suggests the importance of well-calibrated data for de- +tecting and recovering GWB signals, which is expected +to be in the near future. +We also note that the analysis presented in this paper +highly depends on the GW detectors’ calibration model +(parameters). Hence, one might need to repeat this study +when the calibration model changes significantly, for ex- +ample, for future detectors. One could also extend this +study to estimate the effect of calibration uncertainties + +X10-8 +KC +2.0 +-100 +KU +Relative change in +KP +KT +1.5 +50 +1.0 ++0.0 ++10.0 ++20.0 +Error in the calibration parameter (in %)1.0 +0.5 +Recovered α +0.0 +-100 +KC +tttt +KU +-0.5 +KP +-200 +KT +-1.0 +CC ++0.0 ++10.0 ++20.0 +Error in the calibration parameter +(in %Relative contribution (abs) +Au(f) +100 +Ap(f) +10-1 +AT(f) +C(f) +10-2 +90 +102 +103 +Relative phase (deg) +100 +0 +100 +90 +102 +103 +Frequency (Hz8 +on the GWB with more complicated model parameters +or anisotropic GWB. +ACKNOWLEDGEMENTS +The authors thank Jeffrey S Kissel for providing use- +ful comments on the draft. +The authors acknowledge +the use of the IUCAA LDG cluster Sarathi for the +computational/numerical work. J. Yousuf also acknowl- +edges IUCAA for providing accommodation while car- +rying out this work. +J. Yousuf is thankful to the De- +partment of Science and Technology (DST), Government +of India, for providing financial assistance through IN- +SPIRE Fellowship. For this work, we used the software +packages pyDARM [46], bilby [55], stochastic [51] and +Matplotlib [56]. +[1] B. P. Abbott et al. (LIGO Scientific Collaboration and +Virgo Collaboration), Phys. Rev. Lett. 116, 061102 +(2016). +[2] J. Aasi et al. (LIGO Scientific Collaboration), Class. +Quant. Grav. 32, 074001 (2015). +[3] F. Acernese et al. (VIRGO), Class. Quant. Grav. 32, +024001 (2015), arXiv:1408.3978 [gr-qc]. +[4] B. P. Abbott et al. (LIGO Scientific Collaboration and +Virgo Collaboration), Phys. Rev. X 9, 031040 (2019). +[5] R. Abbott et al. (LIGO Scientific Collaboration and +Virgo Collaboration), Phys. Rev. X 11, 021053 (2021). +[6] R. Abbott et al. 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The Bayes factors comparing signal vs. noise hypothesis for the two cases have also +been shown. + +log Qα = -7.98±8:34 +Bayes factor ～ 47.7 +Bayes factor ～ 48.7 +KT = 1.07±8:233 +-12 +心 +600 +? +8.8 +log α +α +KC +fco +KT +Kp +Ku \ No newline at end of file diff --git a/ctFRT4oBgHgl3EQfTDc5/content/tmp_files/load_file.txt b/ctFRT4oBgHgl3EQfTDc5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f00f3c281f3b56c79c4e03989c603aae817ec622 --- /dev/null +++ b/ctFRT4oBgHgl3EQfTDc5/content/tmp_files/load_file.txt @@ -0,0 +1,859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf,len=858 +page_content='Effects of calibration uncertainties on the detection and parameter estimation of gravitational-wave backgrounds Junaid Yousuf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='1 Shivaraj Kandhasamy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='2 and Manzoor A Malik1 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' University of Kashmir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Srinagar 190006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' India 2Inter University Center for Astronomy and Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Pune 411007,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' India (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 2023) Gravitational-wave backgrounds are expected to arise from the superposition of gravitational wave signals from a large number of unresolved sources and also from the stochastic processes that occurred in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' So far, we have not detected any gravitational wave background, but with the improvements in the detectors’ sensitivities, such detection is expected in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The detection and inferences we draw from the detection of a gravitational-wave background will depend on the source model, the type of search pipeline used, and the data generation process in the gravitational-wave detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In this work, we focus on the effect of the data generation process, specifically the calibration of the detectors’ digital output into strain data used by the search pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Using the calibration model of the current LIGO detectors as an example, we show that for power-law source models and for calibration uncertainties ≲ 10%, the detection of gravitational wave background is not significantly affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We also show that the source parameter estimation and upper limits calculations get biased and must be corrected in the analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' INTRODUCTION Since the first detection in September 2015 [1], the LIGO [2], and the Virgo [3] gravitational wave (GW) detectors have detected nearly one-hundred compact bi- nary merger signals [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' They correspond to individual merger signals with a high signal-to-noise ratio (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In addition to those high SNR signals, assuming the merger events are outliers in a much larger population of com- pact mergers, we also expect a large number of low SNR signals that are hard to detect individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The super- position of such a large number of low SNR signals would give rise to a gravitational-wave background (GWB) that could be detected with the current or next generation of GW detectors [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Apart from the compact binary mergers signals, su- perposition of other astrophysical GW signals such as from core-collapse supernovae [11, 12], magnetars [13, 14] could also give rise to GWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In addition to these astro- physical sources, various events that took place in the early universe such as inflation and phase transitions could also give rise to GWB [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The detection of GWB from astrophysical sources can help us better understand the population and the evolution of stars in the universe [16–18] while the detection of GWB from cosmological sources can provide information about the processes in the very early universe which are otherwise difficult to obtain [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The LIGO-Virgo-KAGRA (LVK) collaboration, in their recent analyses using data from the observing run O3, did not find any evidence of GWBs and hence placed upper limits on the amplitudes of possible isotropic [20] and anisotropic GWBs [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' With the proposed improve- ments to the current GW detectors [22], it might be pos- sible to detect the GWB from compact binary mergers [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' And, the proposed next-generation GW detectors [23, 24] will certainly observe the GWB from compact bi- nary mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The data generation and various aspects of the search are expected to affect the GWB search re- sults, and hence it is important to understand them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In this paper, we focus on the effects of the data genera- tion, specifically that of the calibration, on the analysis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Calibration is the process of converting the raw digital outputs of the detectors into strain data that are further used in the GW analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Any uncertainties in that process could translate into biases in the final re- sults, affecting our interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Typically cross-correlation-based searches, correlating data from multiple detectors, are used to detect GWBs [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In previous such searches using LIGO-Virgo data [20, 26, 27], upper limits were calculated after marginal- izing over calibration uncertainties as outlined in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' However, that method does not capture any biases intro- duced by the systematic errors in the calibration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In this work, we try to address that issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In the past, this has been studied mostly in the context of the search for GW signals from individual compact binaries [29– 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Recently, such questions have also been addressed for the detection, and parameter estimation of individual compact binary merger signals [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We use a similar simulation-based method [33, 34] to address the effects of calibration uncertainties on the searches for GWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In addition, we also show that one could try to estimate the GWB and calibration model parameters simultaneously and get a reasonable signal recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' II we briefly introduce the model and search for GWB using data from GW detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' III, we dis- cuss the calibration model used to convert the raw digital output into strain data used in GW searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' IV we describe the method used to quantify the effects of calibration uncertainties on GWB searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' V we show the results of our analyses, and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' VI conclude with the main results and future outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='13531v1 [gr-qc] 31 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' MODELING AND SEARCH FOR GRAVITATIONAL-WAVE BACKGROUND A GWB is usually characterized in terms of fractional energy density in gravitational waves Ωgw(f) [25], given by, Ωgw(f) = f ρc dρgw df , (1) where f is the frequency, dρgw is the energy in gravita- tional waves in the frequency interval from f to f +df, ρc is the critical energy density needed to close the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The value of ρc is given by ρc = 3c2H2 0 8πG , (2) where c is the speed of light, G is the gravitational con- stant and H is the Hubble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In this work, we use the value of Hubble constant measured by the Plank satellite, H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='9 km s−1 Mpc−1 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' However, the conclusions drawn are independent of the actual value of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Typically Ωgw(f) is expressed in the form of a power law, Ωgw(f) = Ωα � f fref �α , (3) where fref is a reference frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For results reported in this paper, we use a reference frequency of fref = 25 Hz as used in the LVK analyses [20, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The value of the power-law index α depends on the source of GWB we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For cosmological GWB from inflationary scenarios we typically expect α = 0 [15] while for astro- physical GWB from the superposition of many compact binary merger signals α = 2/3 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The optimal estimator of Ωα at a time t and at a fre- quency bin f is given by [18, 37], ˆΩα(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) = 2 T ℜ[d∗ I(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f)dJ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f)] γIJ(f)Sα(f) , (4) where d1(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) and d2(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) are short-time Fourier trans- forms of the strain data from the two detectors (I, J) evaluated at time t, T is the duration of the data seg- ments used for Fourier transforms and γIJ(f) is the nor- malized overlap reduction function for the given two de- tectors (I, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The function Sα(f) is proportional to the assumed spectral shape α and is given by [18, 37], Sα(f) = 3H2 10π2 1 f 3 � f fref �α (5) In the weak-signal limit, the variance of ˆΩα is given by [18, 37], σ2 ˆΩα(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) = 1 2T∆f PI(f)PJ(f) γ2 IJ(f)S2α(f) (6) where PI(f), PJ(f) are the one-sided power spectral den- sities of the strain data from the two detectors (I, J), and ∆f is the frequency resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For data spanning many segments and a large frequency band, the final optimal estimators are obtained by a weighted sum, ˆΩα = � t,f σ−2 ˆΩα(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f)ˆΩα(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) � t,f σ−2 ˆΩα(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) , σ−2 ˆΩα = � t,f σ−2 ˆΩα(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f), (7) where t runs over available time segments and f runs over discrete frequency bins in the desired frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' CALIBRATION MODEL The raw outputs of gravitational wave detectors are digitized electrical signals from the photodetectors at the output port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The process of converting these elec- trical signals into strain data is called calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The LIGO, Virgo and KAGRA detectors all have similar fun- damentals in optical layout and control system topology [2, 3, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' While their methods to describe and charac- terize that system are different (sometimes only in sub- tle ways that reflect their detailed differences), any of those methods could be used to describe current GW de- tectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Thus, here, we follow and choose the methods of the LIGO detectors [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For details of different calibration techniques used in the current generation of gravitational wave detectors see [39, 41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' As shown in [40], after detailed modeling of the detectors, a response function R(f) is derived which is then used to convert the digitized electrical output into strain h(f) using the expression, d(f) = 1 Le(f)R(f) (8) where e(f) is the digitized signals from the output photo- detectors, R(f) is the response function that converts e(f) into the differential displacement of the two arms of the detector and L is the average (macroscopic) length of the two arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' A typical response function in the frequency domain can be written as [40, 44], R(f) = 1 + A(f)D(f)C(f) C(f) (9) where C(f) is the sensing function corresponding to the response of the detector to differential changes in its two arms without any feedback control, A(f) is the actuation function used to control the positions of the mirrors and D(f) is any digital filter(s) used in the control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Sensing function The sensing function C(f) can be modeled in the fre- quency domain as [40, 45], C(f) = � κCHC 1 + iff −1 cc � � f 2 f 2 + f 2s − iffsQ−1 � × CR(f) (10) where optical gain HC represents the overall gain, coupled-cavity pole frequency fcc defines the detector bandwidth, fs and Q correspond to optical anti-spring pole frequency and its quality factor respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The term CR represents the frequency dependencies not cap- tured by the other terms (for example, the response of the electronics chain used for the digitization, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' ), and κC is a scale factor representing the changes in the sensing function with respect to a reference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' An example sensing function plot is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We use the pyDARM package [46] to generate the calibration model used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For LIGO detectors, during the past FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Plot showing an example of sensing function C(f) of LIGO Hanford detector during the observing run O3 [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The unit of C(f) is the counts produced in the Analog-to- Digital converter at the output port for a meter differential length change in the two arms of the GW detector [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' observing runs and for frequencies ≳ 20 Hz, the optical spring term (second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 10) was usually close to one (for example, see [47, 48]), f 2 f 2 + f 2s − iffsQ−1 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Since in our work, we will be using 20 − 1726 Hz band as done in LVK analyses [20, 26, 27], we will neglect the optical spring term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 10 for the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Actuation function The actuation function is modeled in the frequency domain as [40, 45], A(f) = κUAU(f) + κP AP (f) + κT AT (f) (11) where U, P, and T represent the lowest three stages of suspensions (upper intermediate mass, penultimate, and test mass stages) used to suspend the main optics [2, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Ai(f) (where i = U, P, T) are frequency-dependent ac- tuation models of the three stages of the suspensions, including digital filters in the control path and analog re- sponses of the three stages of suspensions [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The scale factors κi capture any changes in the reference actuation model of each stage, and in general, they could be time- and frequency-dependent [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The plots of actuation models for the three stages and the combined actuation model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Plot showing an example of the actuation functions of the bottom three stages (top, penultimate, and test mass stages) as well as the combined actuation function of LIGO Hanford’s main optic suspension during the observing run O3 [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The unit of A(f) is the differential length change produced in the two arms for a unit count in the Digital-to- Analog converter that drives the actuators [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Total response function Apart from the notch filters used to prevent the exci- tation of resonances of the test mass suspensions, D(f) is a smooth function of frequency that is decided by the feedback control morphology used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The total response function, as shown in 9, is a function of C(f), A(f), and D(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 3 shows an example response function of the LIGO Hanford detector during the observing run O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 107 Magnitude (ct/m) 106 105 102 103 Frequency (HzAu(f)(top) Ap(f)(penultimate 10-16 Magnitude (m/ct) AT(f) (test mass A(f) (total) 10-18 10-20 102 103 Frequency (Hz4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Plot showing an example of the response function R(f) of the LIGO Hanford detector during the observing run O3 [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' ANALYSIS METHOD In this work, we look at the effects of calibration uncer- tainties on the recovery of GWB and on the parameter estimation of the recovered GWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Specifically, we look at GWBs described by power-law models with power-law indices of α = 0, 2/3, 3 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' If the response function used to calibrate the digitized signal in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='8 is not the true response function, then we get, dtrue(f) = dcalc(f) × Rtrue(f) Rcalc(f) (12) = dcalc(f) × Λ(f) (13) where true and calc correspond to the true and calculated quantities respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In the above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 12, we have defined Λ(f) as, Λ(f) = Rtrue(f) Rcalc(f) (14) for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The uncertainties in the calibration pro- cess enter the GW analyses as Λ(f) shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We note here that Rtrue(f), with measurement uncertainty, can be calculated using a length (or frequency) reference such as a photon calibrator [50], but due to difficulty in the implementation Rcalc(f) is traditionally used in the calibration process leading to the difference we see in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The Rtrue(f) is usually in a non-parametric form while Rcalc(f) is parameterized with a relatively small number of parameters (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Hence from an implemen- tation point of view, Rcalc(f) is more desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Because of the simple parameterization, changes in Rcalc(f) can also be easily tracked which is also important for the cal- ibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Moreover, the ratios Λ(f) are usually very close to one and hence use of Rcalc(f) is well justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Due to the measurement uncertainties in Rtrue(f), the estimation of the ratios Λ(f) has both systematic and statistical uncertainties associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 12 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='4 and 6 we get, ˆΩα(f) = 2 T ℜ � d∗ I,calc(f)dJ,calc(f)Λ∗ I(f)ΛJ(f) � γIJ(f)Sα(f) (15) and σ2 ˆΩα(f) = 1 2T∆f PI,calc(f)PJ,calc(f) γ2 IJ(f)S2α(f) |ΛI|2|ΛJ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' (16) The Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 15 and 16 provide a way to estimate the effects of calibration uncertainties on the signal estimate ˆΩα and its variance σ2 ˆΩα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' If we further assume that the ratios Λ(f) are real, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=', the difference is only in the magnitude, then we get, ˆΩα(f) = ˆΩα,nocal(f)ΛI(f)ΛJ(f) , (17) σ2 ˆΩα(f) = σ2 ˆΩα,nocal(f)Λ2 I(f)Λ2 J(f), (18) where nocal subscript corresponds to the quantities cal- culated in the absence of calibration uncertainties that we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' With this assumption, the simulation becomes a little bit easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We can start with ˆΩα,nocal(f) and σ2 ˆΩα,nocal(f) calculated from the simulated data and us- ing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 17, 18 and 7 we can estimate the effects of cal- ibration uncertainties on the calculation of ˆΩα(f) and σ2 ˆΩα(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' However, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='V we also show the results with- out using this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Since the response functions, RI,J themselves are functions of A (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 11), C (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 10) and D the number of free parameters in the above equa- tions becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Due to the large number of param- eters, it is difficult to calculate the effects analytically, so we use numerical simulation to calculate the effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' This method becomes more useful when one wants to include a more complicated signal model and additional calibration parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For the results reported in this paper, we use one week of simulated data for Hanford and Livingston detectors using advanced LIGO design sensitivity [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Here, one week of data is chosen to represent the traditional long- duration analyses of GWB and to avoid complexities aris- ing from large SNRs in individual segments [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We use publicly available LVK codes to perform the searches [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We also use standard search parameters of 192-sec seg- ment duration and frequencies from 20 Hz to 1726 Hz with a frequency resolution of 1/32 Hz [20, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We use the same calibration model for both Hanford and Livingston detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We do the following to calculate the effects of cali- bration uncertainties on the recovery of GWB signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' As indicated in the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 17 and 18, we multiply the ˆΩα,nocal(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) and σ2 ˆΩα,nocal(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) estimators of each seg- ment by distributions representing the ratios Λ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We assume Gaussian distributions for Λ(f), centered at one with standard deviations defined by the desired calibra- tion uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We also truncate the Gaussian distri- bution at 2-sigma points on both sides to avoid unreal- istic values for Λ(f) (for example, values close to zero 10-7 102 103 Frequency (Hz)5 or even negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Then, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 7, we combine the segment-wise and frequency-dependent results of ˆΩα(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) σˆΩα(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) to get the final estimate and its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Then we use SNR, defined in a frequentist approach [52], given by, SNR = ˆΩα σˆΩα as the detection statistics for a GWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We then compare these results against the results obtained without any calibration uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Since the difference between these results is just the application of calibration uncer- tainties, the differences would typically show the effects of calibration uncertainties on ˆΩα and σ2 ˆΩα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We further look at the effects of calibration uncertain- ties on the parameter estimation, specifically on the Ωα and α, by varying the values of various parameters in the R(f) (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='9, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' RESULTS In this section, we present the results of our studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' To generate these results, we initially assume that the ratios of response functions Λ(f) are real and hence use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 17 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We note that this assumption is used in the marginalization of calibration uncertainties in the LVK GWB analyses [20, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' However, for comparison, we also produce results by additionally using 1-sigma phase uncertainties of 5◦, the maximum of what was seen in LIGO detectors during the observing run O3 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' This is to show how much phase uncertainties, that are currently not included in the GWB analyses, affect the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' At each frequency, we model the magnitude of Λ(f) by a Gaussian distribution with a mean one and standard deviation ϵ that is small compared to one and phase of Λ(f) by a Gaussian distribution with a mean zero and standard deviation of 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' As indicated earlier, we also truncate the Gaussian distribution at 2-sigma values to avoid unrealistic realizations of Λ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Effect of calibration uncertainties on the GWB detection The recovered values of the ˆΩα, σˆΩα and SNR at vari- ous levels of calibration uncertainties for the three power law models α = 0, 2/3, 3 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In these plots, we increase the uncertainty from 0 % to 20 % in steps of 2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We also repeat the analysis 20 times at each uncertainty level to calculate the spread on the recovered values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For comparison, we also show results assuming 1-sigma phase uncertainties of 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' From the plots, we see that as we increase the values of uncertainties, there are changes in the recovered values of Ωα, σˆΩα, and SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' However, the changes in the recovered SNRs are small, almost negligible, below the calibration uncertainties of ∼ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Since SNR is generally used as a detection statistic, this suggests that the detection of a GWB is not significantly affected by the uncertainties in the calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The Ωα, σˆΩα change by ∼ 10% when we change the un- certainty of response function by ∼ 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The reduction in the estimated σˆΩα can be attributed to how we com- bine different time segments and frequency bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Since we use weighted average method (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 7), any downward fluctuations in individual σˆΩα(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' f) due to calibration un- certainties will bring down the final σˆΩα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' A similar effect could be attributed to the reduction in the final Ωα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' This suggests that the recovered values of Ωα and σΩα are bi- ased in the presence of calibration uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Since the upper limits on Ωα, for example, 95 % upper limit in the frequentist approach, can be written as Ωα,95% ≈ ˆΩα + 2 σˆΩα, calibration uncertainties are also expected to bias the upper limit calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Such biases are not completely taken into account when estimating Ωα or while calcu- lating upper limits on Ωα in the analyses reported in the literature [20, 26, 27] and need to be accounted for in future analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The plots also suggest that including phase uncertainties does not change the results signifi- cantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Effects of the calibration uncertainties on the recovery of parameters of GWB The second part of the study is to see the effect of calibration uncertainties on the estimation of parameters of the GWB signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Here we mainly focus on the es- timation of Ωα and α (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' V A, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 4 already shows the effect of the uncertainties of the re- sponse function as a whole on the recovery of Ωα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In- stead of the uncertainties of the total response function, in this section, we look at the effects of individual cali- bration parameters on the recoveries of Ωα and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Since we are using the parameters that make up the calibration model, in the literature, this is considered as a physically motivated approach to include calibration uncertainties in the signal analyses [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In this study we mainly focus on the parameters κC, fcc (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' III A), κU, κP and κT (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Other parameters in the response function tend to be more or less constant during an ob- serving run, and hence we do not include them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We also perform the analysis only for α = 2/3, whose detec- tion is expected in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The maximum likelihood values of the recovered pa- rameters Ωα and α as functions of errors on the various calibration parameters are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We use the maximum likelihood method described in [53] and use dynesty [54] sampler in bilby [55] package for sampling the likelihoods and estimating the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' From the figure we see that κP , κT and κC have signifi- cant effects on the recovery of Ωα and α while fcc and κU 6 α = 0 α = 2/3 α = 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Plots showing the effect of calibration uncertainty on the recovery of Ωα, σˆΩα and SNR for injected GWB signals described by α = 0, 2/3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The calibration uncertainty is quantified in terms of the relative standard deviation of the varying response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The solid (blue) line corresponds to no phase uncertainty while dotted (red) line corresponds to 5◦ 1-sigma phase uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' have very little effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' This is probably expected because of the relative contributions of these terms to the total response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Rewriting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 9 into contributions from different components, we get, R(f) = 1/C(f) + κUD(f)AU(f) +κP D(f)AP (f) + κT D(f)AT (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' (19) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 7 shows the relative contribution of the different terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 19 to the response function and also 90 % search sensitivity region for the α = 2/3 GWB signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We see that in the 90 % sensitivity region, penultimate and test mass actuation and sensing functions have signifi- cant contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' In the sensing function (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10), the dominant contribution comes from κC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Since the typical value of fcc of advanced LIGO detectors during the O3 run was ∼ 400 Hz and the 90 % search sensi- tivity region extents only up to ∼ 45 Hz, the effect of fcc on the estimation of the parameters is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Be- cause of the non-trivial phase relationship between these functions, we see that the relative contributions to the response function from individual components can even go above one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We also try to simultaneously estimate the calibration and GWB signal parameters to see how well we can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Here we use (simulated) uncalibrated raw digital signals to extract all the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 8 shows an example of the simultaneous estimation of all the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The plot shows that, along with the GWB model parameters, we can also infer the values κP , κT , and κC to some level, but recoveries of fcc and κU are poor which are consistent with the results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For comparison, we also show the recovery of GWB model parameters using cal- ibrated data without any uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The plots also have the Bayes factors, comparing the signal vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' noise hypothesis for those two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We see that the Bayes factors do not change significantly in the two cases (as expected, it is lower when we estimate calibration param- eters also).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' However, the posteriors of GWB parameters are very broad and probably biased when we simultane- ously estimate the GWB and calibration model param- 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='50 mag uncertaity 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='25 mag + 5° phase uncertaity 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 SNR2a 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='75 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='50 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='20 Uncertainty of R(f)l (in %11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 mag uncertaity 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='75 mag + 5° phase uncertaity 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='50 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='75 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='20 Uncertainty of R(f)l (in %10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='50 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 SNR2a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='75 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='50 mag uncertaity 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='25 mag + 5° phase uncertaity 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='20 Uncertainty of R(f)l (in %X10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10- mag uncertaity mag + 5° phase uncertainty 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='20 Uncertainty of IR(f)l (in %)X10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 - °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='90 mag uncertaity mag + 5° phase uncertaity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='6 mag uncertaity mag + 5° phase uncertaity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='20 Uncertainty of |R(f)l (in %)X10-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='06 mag uncertaity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='04 mag + 5° phase uncertaity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='20 Uncertainty of |R(f)| (in %)X10-10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='2 mag uncertaity mag + 5° phase uncertaity 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='20 Uncertainty of |R(f)l (in %)X10-10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='8 C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='7- mag uncertaity mag + 5° phase uncertaity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='20 Uncertainty of |R(f)l (in %)7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Effect of the errors in various calibration model pa- rameters on the recovery of the signal parameter Ωα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The solid lines correspond to the maximum likelihood values, and the shaded regions indicate 68 % confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The injected value of Ωα is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='04 × 10−8 for α = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Effect of the errors in various calibration model pa- rameters on the recovery of the signal parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The solid lines correspond to the maximum likelihood values, and the shaded regions indicate 68 % confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The injected value of α is 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' So it is important to have well-calibrated data to get better posteriors on the signal parameters and also a better Bayes factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' CONCLUSIONS In this work, we have studied the effect of calibration uncertainties on the detection and parameter estimation of GWB signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We focused on amplitude (Ωα) and power law index (α) of power-law isotropic GWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We find that, for the current generation of LIGO detectors, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Contribution of various calibration parameters to the total response function and 90 % search sensitivity region for the α = 2/3 GWB search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' when the calibration uncertainties are less than ∼ 10%, they do not significantly affect the detection of a GWB signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The calibration uncertainties of the LIGO detec- tors reported during the last observing run O3 are well within this ∼ 10% limit [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We also find that the recovery of GWB model param- eters could be significantly affected depending on which calibration parameter is poorly constrained and its un- certainty level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The recovered parameters are biased due to errors in calibration model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Even though the current errors on the individual model parameters of LIGO detectors are much smaller (≲ 1 %), the cu- mulative effect of the different parameters could bias the recovered GWB parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Currently, this bias is not considered during the GWB parameter estimation or up- per limit calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For a calibration uncertainty of ∼ 5 % of the total response function (90 % maximum reported for the LIGO detectors during O3), the bias in the estimate of GWB amplitude or its upper limit could be as large as ∼ 5 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' This might be significant when we try to differentiate between different models of GWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We also tried to estimate the GWB and calibration model parameters simultaneously and find that we could detect the signal, albeit with some loss of Bayes factor (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' However, the posteriors of the GWB signal parameters become very broad and probably biased due to their cor- relation with some of the calibration parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' This suggests the importance of well-calibrated data for de- tecting and recovering GWB signals, which is expected to be in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' We also note that the analysis presented in this paper highly depends on the GW detectors’ calibration model (parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Hence, one might need to repeat this study when the calibration model changes significantly, for ex- ample, for future detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' One could also extend this study to estimate the effect of calibration uncertainties X10-8 KC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 100 KU Relative change in KP KT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='5 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 +20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 Error in the calibration parameter (in %)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='5 Recovered α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 100 KC tttt KU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='5 KP 200 KT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 CC +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 +20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='0 Error in the calibration parameter (in %Relative contribution (abs) Au(f) 100 Ap(f) 10-1 AT(f) C(f) 10-2 90 102 103 Relative phase (deg) 100 0 100 90 102 103 Frequency (Hz8 on the GWB with more complicated model parameters or anisotropic GWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors thank Jeffrey S Kissel for providing use- ful comments on the draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The authors acknowledge the use of the IUCAA LDG cluster Sarathi for the computational/numerical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Yousuf also acknowl- edges IUCAA for providing accommodation while car- rying out this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Yousuf is thankful to the De- partment of Science and Technology (DST), Government of India, for providing financial assistance through IN- SPIRE Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For this work, we used the software packages pyDARM [46], bilby [55], stochastic [51] and Matplotlib [56].' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' (LIGO Scientific Collaboration and Virgo Collaboration and KAGRA Collaboration), (2021), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='03606, arXiv:2111.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Blair, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Gilmore, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 351, 1237 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Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 241, 27 (2019), arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='02042 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Hunter, Computing in Science & Engineering 9, 90 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' Corner plot (blue) showing the recovery of GWB and calibration model parameters using raw, uncalibrated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' For comparison, we also show the recovery of GWB parameters (green) using well-calibrated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The injected values are shown using the (red) cross and vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' The Bayes factors comparing signal vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' noise hypothesis for the two cases have also been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' log Qα = -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='98±8:34 Bayes factor ～ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='7 Bayes factor ～ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='7 KT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='07±8:233 12 心 600 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} +page_content='8 log α α KC fco KT Kp Ku' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFRT4oBgHgl3EQfTDc5/content/2301.13531v1.pdf'} diff --git a/d9AzT4oBgHgl3EQf3f6y/vector_store/index.faiss b/d9AzT4oBgHgl3EQf3f6y/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9f688aa1619e595450c3e432d5357f8376149427 --- /dev/null +++ b/d9AzT4oBgHgl3EQf3f6y/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e6b1109af2920e6c8d4073587dacb59cd90f886fb1c0cae5d008d6eaba421b98 +size 7405613 diff --git a/dNE0T4oBgHgl3EQfoAHN/content/tmp_files/2301.02521v1.pdf.txt b/dNE0T4oBgHgl3EQfoAHN/content/tmp_files/2301.02521v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ef75b1165eb3df4fc202d5ad0de6a61816169e44 --- /dev/null +++ b/dNE0T4oBgHgl3EQfoAHN/content/tmp_files/2301.02521v1.pdf.txt @@ -0,0 +1,1000 @@ +SAIDS: A Novel Approach for +Sentiment Analysis Informed of Dialect and Sarcasm +Abdelrahman Kaseb and Mona Farouk +Computer Engineering, Cairo University +Giza, Egypt +{abdelrahman.kaseb,mona_farouk}@eng.cu.edu.eg +Abstract +Sentiment analysis becomes an essential part +of every social network, as it enables decision- +makers to know more about users’ opinions +in almost all life aspects. +Despite its im- +portance, there are multiple issues it encoun- +ters like the sentiment of the sarcastic text +which is one of the main challenges of sen- +timent analysis. This paper tackles this chal- +lenge by introducing a novel system (SAIDS) +that predicts the sentiment, sarcasm and di- +alect of Arabic tweets. SAIDS uses its pre- +diction of sarcasm and dialect as known in- +formation to predict the sentiment. +It uses +MARBERT as a language model to generate +sentence embedding, then passes it to the sar- +casm and dialect models, and then the out- +puts of the three models are concatenated and +passed to the sentiment analysis model. Mul- +tiple system design setups were experimented +with and reported. SAIDS was applied to the +ArSarcasm-v2 dataset where it outperforms +the state-of-the-art model for the sentiment +analysis task. By training all tasks together, +SAIDS achieves results of 75.98 FPN, 59.09 +F1-score and 71.13 F1-score for sentiment +analysis, sarcasm detection, and dialect iden- +tification respectively. The system design can +be used to enhance the performance of any task +which is dependent on other tasks. +1 +Introduction +Sentiment analysis (SA) is one of the main tasks in +the natural language processing (NLP) field. It is +used for opinion mining which supports decision- +makers. Working on sentiment analysis starts rel- +atively early, for example, Pang et al. (2002) anal- +ysed the sentiment to positive and negative in movie +reviews. Following this paper, sentiment analysis +becomes one of the most important topics in NLP, +especially with the increasing number of reviews +on websites and social media platforms. Since then, +a lot of work has been done in English sentiment +analysis, while Arabic has relatively much less. +Since Abbasi et al. (2008) started their work on +Arabic SA, multiple researchers also began theirs. +Now there are well-known Arabic SA models like +(Alayba et al., 2018; Abdulla et al., 2013; Abu +Farha and Magdy, 2021; Elshakankery and Farouk, +2019). Of course, working with Arabic has many +challenges, one of the most challenging issues is +the complex morphology of the Arabic language +(Kaseb and Farouk, 2016; Abdul-Mageed, 2019). +Another challenge is the variety of Arabic dialects +(Abdul-Mageed, 2019). Moreover, one of the well- +known challenges in SA for all languages is sar- +casm, as the sarcastic person uses words and means +the opposite of it. For example, "I’d really truly +love going out in this weather!", does it reflect a +positive or negative sentiment? because of the sar- +casm, we cannot judge the sentiment correctly. +Several related works tackle English sarcasm de- +tection with sentiment analysis (Oprea and Magdy, +2020; Abercrombie and Hovy, 2016; Barbieri et al., +2014). On the other hand, there are only a few +works on both sentiment and sarcasm in Arabic. +There are two shared tasks on sarcasm detection +(Ghanem et al., 2019), but for both sarcasm and sen- +timent there was only one shared task Abu Farha +et al. (2021) but each sub-task is independent, +meaning that participating teams can submit a dif- +ferent model for each task. Some participants used +the same model for both sentiment and sarcasm +(El Mahdaouy et al., 2021). +Instead of training sentiment independently of +sarcasm, this work introduces a new model archi- +tecture that works with multi-task training which +trains both at the same time. There are other addi- +tions to the proposed architecture; firstly, it trains +with dialect also. Secondly, the sarcasm and di- +alect that are initially predicted are used in the +prediction of the sentiment. In other words, the +sentiment model is informed by the sarcasm and +dialect model output. The contributions offered by +this work are: +arXiv:2301.02521v1 [cs.CL] 6 Jan 2023 + +• Design a novel model architecture that can be +used for a complicated task that is dependent +on another task, e.g. sentiment analysis which +is dependent on sarcasm detection. +• Investigate the design setups for the new ar- +chitecture and find the best setup that could +be used. +• Train the model on ArSarcam-v2 dataset and +achieve the state-of-the-art results recorded as +75.98 FPN on sentiment analysis. +This paper is organized as follows Section 2 +shows the related work on sentiment analysis, sar- +casm detection, and dialect identification. Section +3 describes the dataset used in this work and shows +data statistics. Section 4 describes SAIDS model +and all the design setups. Section 5 shows the ex- +perimental results and finally section 6 concludes +the work. +2 +Related Work +SAIDS works on three tasks sentiment analysis, +sarcasm detection, and dialect identification. In +this section, the existing methods for each task are +discussed. +2.1 +Sentiment Analysis +Arabic sentiment analysis started with Abbasi et al. +(2008) work. Since then, it is developed by multiple +researchers. In the beginning, the main focus was +on modern standard Arabic (MSA), but over time +the researchers start to focus on dialectal Arabic +(Mourad and Darwish, 2013; Kaseb and Farouk, +2021). +Regarding the datasets, based on Alyafeai et al. +(2021), there are more than fifty datasets for senti- +ment analysis, including Elshakankery et al. (2021); +Kaseb and Farouk (2019); Kiritchenko et al. (2016); +Rosenthal et al. (2017); Elmadany et al. (2018) +datasets. +Because of the massive number of +datasets, there are a massive number of system +approaches for Arabic sentiments (Abu Farha and +Magdy, 2019; Alayba et al., 2018; El-Beltagy et al., +2017). Based on Abu Farha and Magdy (2021) +comparative study, using the word embedding with +deep learning models outperform, the classical ma- +chine learning models and the transformer-based +models outperform both of them. There is a reason- +able number of Arabic transformer-based models +like AraBERT (Antoun et al., 2020) and MAR- +BERT (Abdul-Mageed et al., 2021) which are used +by most Arabic sentiment analysis papers. +2.2 +Sarcasm Detection +Unlike Arabic sentiment analysis, Arabic sarcasm +detection has not gotten much attention yet. Only +a few research works tackle the problem and still +there is an obvious shortage of the Arabic sarcasm +datasets, like Karoui et al. (2017); Abu Farha et al. +(2022). Abbes et al. (2020) collected a dataset for +sarcastic tweets, they used hashtags to collect the +dataset for example #sarcasm. Then, they built +multiple classical machine learning models SVM, +Naive Bayes, and Logistic Regression, the best +F1-score was 0.73. +After that, Ghanem et al. (2019) organized a +shared task in a workshop on Arabic sarcasm detec- +tion. They built the dataset by collecting tweets on +different topics and using hashtags to set the class. +An additional step was added, by sampling some +of the datasets and manually annotating them. In +this shared task, eighteen teams were working on +sarcasm detection. Khalifa and Hussein (2019) was +the first team and achieved a 0.85 F1-score. +Then Abu Farha et al. (2021) made two tasks +based on the ArSarcasm-v2 dataset; sentiment anal- +ysis and sarcasm detection. They have 27 teams par- +ticipating in the workshop, the top teams achieved +62.25 F1-score and 74.80 FPN for sarcasm detec- +tion and sentiment analysis respectively. +2.3 +Dialect Identification +Arabic dialect identification is an NLP task to iden- +tify the dialect of a written text. It can be on three +levels, the first level is to identify MSA, classical +Arabic (CA), and dialectical Arabic (McWhorter, +2004). The second level is to identify the dialect +based on five main Arabic dialects EGY, LEV, +NOR, Gulf, and MSA (El-Haj, 2020; Khalifa et al., +2016; Sadat et al., 2014; Al-Sabbagh and Girju, +2012; Egan, 2010). The third level is to identify the +country-level dialect (Abdul-Mageed et al., 2020). +Regarding the datasets, there are datasets more +than twenty Arabic datasets labeled with dialect. +One of the most popular datasets is MADAR +(Bouamor et al., 2018) where the data is labeled +at the city-level for 25 Arab cities. Abdul-Mageed +et al. (2020) built a shared task to detect the dialect, +they published three different shared tasks. In the +2020 task, sixty teams participated, and the best + +results were 26.78 and 6.39 F1-score in the country- +level and the city-level dialects respectively. +3 +Dataset +ArSarcasm-v2 (Abu Farha et al., 2021) is the +main dataset used in this work, it was released +on WANLP 2021 shared task for two tasks sar- +casm and sentiment analysis. It has about 15k +tweets and is divided into 12k for training and +3k for testing, the same test set, as released on +WANLP 2021, was used. Each tweet was labelled +for the sentiment (positive (POS), neutral (NEU), +and negative (NEG)), sarcasm (true, and false), +and dialect (MSA, Egypt (EGY), Levantine (LEV), +Maghreb (NOR), and Gulf). The authors of the +dataset annotate it using a crowd-sourcing plat- +form. This dataset originally consisted of a combi- +nation of two datasets, the first one is ArSarcasm +(Abu Farha and Magdy, 2020) and the second one +is DAICT (Abbes et al., 2020), Abu Farha et al. +(2021) merged the two datasets. +3.1 +Dataset Statistics +In this subsection, we introduce some dataset statis- +tics that motivated us to work on SAIDS. The +ArSarcasm-v2 dataset has 15,548 tweets, 3000 +tweets are kept for testing and the rest of the tweets +for training. Table 1 shows the number of exam- +ples for all task labels on the training set, as we +can see, most of the data is labeled as MSA and +non-sarcastic in dialect and sarcasm respectively. +Task +Label +Count +Sentiment +Positive +2,180 +Neutral +5,747 +Negative +4,621 +Sarcasm +Sarcastic +2,168 +Non-sarcastic +10,380 +Dialect +MSA +8,562 +EGY +2,675 +Gulf +644 +LEV +624 +NOR +43 +Total +12,548 +Table 1: Number of labels of sentiment, sarcasm and +dialect on the training set +The relationship between sentiment labels and +both sarcasm and dialect independently can be +shown from Table 2. For the sentiment/sarcasm +part, we can see that about 90 percent of sarcastic +tweets are sentimentally labeled as negative, and +about 50 percent of non-sarcastic tweets are senti- +mentally labeled as neutral. On the other hand, for +the sentiment/dialect part, we can see that about 50 +percent of MSA tweets are sentimentally labeled +as neutral and about 50 percent of EGY tweets are +sentimentally labeled as negative. From this table, +we can conclude that the information we can get +on sarcasm and dialect will benefit the sentiment +analysis task. +POS +NEU +NEG +Non-sarcastic +2,122 +5,576 +2,682 +Sarcastic +58 +171 +1,939 +MSA +1,405 +4,486 +2,671 +EGY +506 +793 +1,376 +Gulf +121 +259 +264 +LEV +142 +197 +285 +NOR +6 +12 +25 +Table 2: Cross tabulation between sentiment labels and +both sarcasm and dialect labels on the training set +Table 3 shows the percentage of sarcastic tweets +on each dialect. As the number of NOR tweets is +limited, its percentage is not reliable, so we can +see that Egyptians’ tweets are the most sarcastic. +This supports the facts from table 2 that most EGY +tweets are negative and most of the sarcastic tweets +are negative tweets. +Dialect +Sarcasm percentage +MSA +10.83 % +EGY +34.77 % +Gulf +24.38 % +LEV +22.12 % +NOR +34.88 % +Table 3: Percentage of sarcastic tweets for each dialect +on the training set +4 +Proposed System +This section presents a detailed description of the +proposed system. SAIDS learns sentiment analy- +sis, sarcasm detection, and dialect identification at +the same time (multi-task training), in addition, it +uses the sarcasm detection and dialect outputs as +an additional input to the sentiment analysis model +which is called "informed decision". SAIDS de- +cides the sentiment class using the information of + +sarcasm and dialect class which are both outputs +itself. The main idea behind SAIDS is based on +analyzing the dataset statistics, as shown in section +3, which says that most sarcastic tweets are classi- +fied as negative tweets and most MSA tweets are +classified as neutral tweets. +4.1 +System Architecture +Figure 1 shows the SAIDS architecture. The ar- +chitecture consists of four main modules, the first +module is MARBERTv2 (Abdul-Mageed et al., +2021), it is a transformer-based model, its input +is the tweet, and its output is a sentence embedding +which is a vector of length 768. The second module +is the "Sarcasm Model", it is a binary classifier for +sarcasm, its input is the sentence embedding, and +its output is two values one for sarcastic tweets and +another for non-sarcastic tweets. The third module +is the "Dialect Model", which is identical to the +"Sarcasm Model" except that it outputs five classes +(EGY, LEV, NOR, Gulf, and MSA). The fourth +module is the "Sentiment Model", it is a classifier +for sentiment, its input is the concatenation of the +sentence embedding, sarcasm model outputs and +dialect model outputs. +Figure 1: SAIDS architecture +The loss function used is Cross-Entropy for sen- +timent and dialect. Of course, since sarcasm is +binary, we used binary Cross-Entropy for it. +4.2 +Training Setups +This subsection describes the multiple setups that +were used to arrive at the best model performance. +The experiments carried out utilized multiple se- +tups regarding the architecture and the training +strategies. +Modules Architecture Multiple architectures +were tested for the "Sentiment Model", "Sarcasm +Model" and "Dialect Model". As a proof of concept +for the idea, we first built a simple random forest +model in each task model (random forest version). +For the real scenario, we used multi-layer neural +network (MNN) models. The first and the simplest +is one output layer model and zero hidden layers. +The second is one or two hidden layers, then the +output layer. The third is one or two hidden layers +the output of the module is the output of the hidden +layer, which means that "Sentiment Model" inputs +is not the output layer of the "Sarcasm Model" but +the last hidden layer of it. The fourth setup is to +concatenate the last hidden layer with the output +layer and then pass it to "Sentiment Model". +What Should Be Informed The SAIDS archi- +tecture Figure 1 shows that the "Sentiment Model" +inputs are "Sarcasm Model" and "Dialect Model" +outputs but we experimented with multiple settings +in this part; sentiment analysis informed of sarcasm +only, dialect only, and both sarcasm and dialect. +Limited Backpropagation We limited the back- +propagation over the dotted lines in Figure 1. It is +used to ensure that the "Sarcasm Model" and the +"Dialect Model" learn their main target correctly. +When the model predicts sentiment incorrectly, its +loss propagates directly to the MARBERTv2 model +via the solid line and does not propagate via the +dotted lines. Also, we evaluate SAIDS without lim- +iting backpropagation which means the loss prop- +agates everywhere, and with partial limiting. The +partial limiting can be only set when the "Sarcasm +Model" has hidden layers. We then limit the back- +propagation through the sarcasm model’s output +layer but propagate it through the hidden layers. +Activation Function The experiments were car- +ried out with Softmax as the activation function for +the output of all modules. However, for the sake +of comparison, we run the training without Soft- +max for the modules outputs, which means that the +values are not from one to zero. +Task By Task Training As we train all the three +tasks together with the same model, we experi- +mented to train the first layer models, "Sarcasm +Model" and "Dialect Model", for some epochs +first, then train the full system together for mul- +tiple epochs. The motivation behind this idea is +that as long as the first layer models work correctly, +the sentiment analysis will correspondingly work +correctly. We train in multiple orders like alternat- +ing between first layer models and full system and +so on. +Other Training Parameters In our experi- + +Sentiment +Model +Sarcasm +Dialect +Model +Model +MARBERT +Tweetments, we built SAIDS and used the MARBERTv2 +model provided by HuggingFace’s transformers li- +brary (Wolf et al., 2020). Most of the experiments +trained for five epochs except for a low learning +rate where it was twenty epochs. For the learning +rate, we used a range from 1e−4 to 1e−6. The se- +quence was truncated to a maximum length of 128 +tokens. Adam (Kingma and Ba, 2015) was used as +an optimizer for all models. +5 +Results +In this section, the results achieved with SAIDS +are discussed. For the sake of comparison, base- +lines were built for the system. To initially evaluate +the idea itself, a random forest model baseline was +built and compared with the random forest version +of SAIDS. Baselines for real scenario are baseline +one (B1) which is identical to BERTModelForSe- +quenceClassification class in HuggingFace’s (Wolf +et al., 2020), which takes the MARBERTv2 sen- +tence embedding and passes it to the output layer +for classification, and baseline two (B2) which uses +two hidden layers before the classification layer, the +hidden layer size is equal to the "Sentiment Model" +hidden layer size, and baseline three (B3) which +uses a larger hidden layer size to match the total +number of trained parameters of SAIDS model. +For evaluation, we used the original metrics de- +scribed for the dataset (Abu Farha et al., 2021). +For sentiment analysis, the metric is the average of +the F1-score for the negative and positive classes +(FPN). For sarcasm detection, the metric is F1- +score for the sarcastic class only (FSar). For dialect +identification, we used the weighted average of the +F1-score for all dialects (WFS). +5.1 +Results of Different Training Setups +This subsection presents the results of the training +setups and describes the best setup that was chosen +for the proposed model. For each part of this sub- +section, every other setup was not changed to make +the comparison fair. +Modules Architecture As a proof of concept +for our system, the random forest (RF) model base- +line was compared with the informed random for- +est (IRF) which is the random forest version of +SAIDS. Table 4 shows that IRF outperforms RF +where the FPN is improved by 3 percent which is +due to the proposed architecture. The information +gained from the new inputs, "outputs of sarcasm +model" and "outputs of dialect model", was 5 and +4 percent respectively. This means that about 10 +percent of the sentiment analysis decision came +from the newly added information. +Model +FPN +Random Forest +59.36 +Informed Random Forest +62.34 +Table 4: Performance comparison for the proof of con- +cept on the validation set +For the MNN architecture of the modules, multi- +ple numbers of hidden layers were trained. At each +experiment, all the modules have the same number +of hidden layers. Table 5 shows that using zero +hidden layers gives the best results. So no hidden +layer setup was used in SAIDS. +Model +FPN +0 Hidden Layer +75.23 +1 Hidden Layer +74.90 +2 Hidden Layer +74.89 +Table 5: Performance comparison for the number of +hidden layers in modules on the validation set +What Should Be Informed Experiments were +also done to find the best features to use while +analysing sentiment. Table 6 shows that using both +dialect and sarcasm is better than using only one +of them and of course better than not using any of +them which is the baseline. With a quick obser- +vation, it was found out that the dialect benefits +the sentiment more than the sarcasm, this can be +obvious when speaking about MSA tweets because +most of them are labeled as neutral on sentiment. +Accordingly, sarcasm and dialect information was +used in SAIDS. +Model +FPN +Not Informed (B1) +72.40 +Informed of sarcasm +73.67 +Informed of dialect +74.41 +Informed of sarcasm and dialect +75.23 +Table 6: Performance comparison for what should be +informed on the validation set +Limited Backpropagation Experiments were +also done to find the best path for backpropagation +to work with. "Full limit" is when the loss does not +propagate through the "Sarcasm model" and "Di- +alect Model", "Partial limit" is when it propagates + +through some layers, and "Unlimited" is when it +propagates through all layers. The model was com- +posed of two hidden layers while running these +experiments. Table 7 shows that "Partial limit" gets +better results than the others, but on SAIDS we did +not use it as we used a no hidden layer setup, so we +used the "Full limit" backpropagation. +Model +FPN +Full limit +74.23 +Partial limit +74.89 +Unlimited +72.31 +Table 7: Performance comparison for limiting back- +propagation on the validation set +Activation Function For the sake of compari- +son, the Softmax layer was removed from the out- +put layer of the model in the experiments. Table 8 +compares both setups, it shows that, as expected, +using Softmax is better than not using it, as it quan- +tify the probability of being sarcasm or being a +certain dialect. So in SAIDS, Softmax was used on +each module. +Model +FPN +With Softmax +75.23 +Without Softmax +72.15 +Table 8: Performance comparison for the activation +function setting on the validation set +Task By Task Training Experiments were also +done with training the three tasks together at the +same time (All tasks), and multiple sets of the train- +ing sequence. The first is one epoch of training for +sarcasm and dialect, and the rest for the full system +(Seq 1). The second is odd epochs for sarcasm and +dialect and even epochs for the full system (Seq +2). The third is two epochs of training for sarcasm +and dialect and the rest for sentiment only (Seq 3). +Table 9 shows that Seq 1 performs better than the +other sequences, so we used it for the final model +training. +Model +FPN +All tasks +74.35 +Seq 1 +75.23 +Seq 2 +73.49 +Seq 3 +73.01 +Table 9: Performance comparison for different model +training sequences on the validation set +Summary of Used Setups SAIDS used infor- +mation from sarcasm and dialect models, which +are both one classification layer with no hidden lay- +ers, the sentiment loss does not propagate through +sarcasm and dialect models, and the Softmax ac- +tivation function was used on each model output. +The used training sequence was one epoch of train- +ing for sarcasm and dialect, and the rest epochs for +the full system. +5.2 +Results comparison with literature +SAIDS was trained and compared to the baselines +we built and also the state-of-the-art models. Ta- +ble 10 shows that SAIDS outperforms the existing +state-of-the-art models on the sentiment analysis +task. SAIDS’s main task is sentiment analysis, the +sarcasm detection and dialect identification are con- +sidered secondary outputs. Although the FSar score +for SAIDS is considerably high, it is ranked third +in the state-of-the-art models. On the other hand, +most works that achieve state-of-the-art results are +using different models for each task but in the pro- +posed architecture, one model is used for both. The +model also outputs the dialect, it achieves 71.13 +percent on the weighted F1-score metric, but the +literature has not reported the dialect performance +so it is not included in the table. +Model +FPN +FSar +Baseline 1 +71.60 +58.41 +Baseline 2 +72.53 +58.61 +Baseline 3 +73.11 +58.62 +El Mahdaouy et al. (2021) +74.80 +60.00 +Song et al. (2021) +73.92 +61.27 +Abdel-Salam (2021) +73.21 +56.62 +Wadhawan (2021) +72.55 +58.72 +SAIDS +75.98 +59.09 +Table 10: Performance comparison for the state-of-the- +art models and SAIDS on the test set +6 +Conclusion +Sentiment analysis is an important system that is be- +ing used extensively in decision-making, though it +has different drawbacks like dealing with sarcastic +sentences. In this work, we propose SAIDS which +is a novel model architecture to tackle this prob- +lem. SAIDS essentially improves the sentiment +analysis results while being informed of sarcasm +and dialect of the sentence. This was achieved by +training on the ArSarcasm-v2 dataset which is la- + +beled for sentiment, sarcasm, and dialect. SAIDS’s +main target is to predict the sentiment of a tweet. It +is trained to predict dialect and sarcasm, and then +make use of them to predict the sentiment of the +tweets. This means that while the model is pre- +dicting the sentiment, it is informed of its sarcasm +and dialect prediction. SAIDS achieved state-of- +the-art performance on the ArSarcasm-v2 dataset +for predicting the sentiment; 75.98 percent average +F1-score for negative and positive sentiment. For +sarcasm detection, SAIDS achieved a 59.09 percent +F1-score for the sarcastic class, whereas for dialect +identification it achieved a 71.13 percent weighted +F1-score for all the dialects. We believe that this +model architecture could be used as a starting point +to tackle every challenge in sentiment analysis. Not +only sentiment analysis but also this is a general +architecture that can be used in any context where +the prediction of a task depends on other tasks. 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As- +sociation for Computational Linguistics. +Bo Pang, Lillian Lee, and Shivakumar Vaithyanathan. +2002. Thumbs up? sentiment classification using +machine learning techniques. In Proceedings of the +2002 Conference on Empirical Methods in Natural +Language Processing (EMNLP 2002), pages 79–86. +Association for Computational Linguistics. +Sara Rosenthal, Noura Farra, and Preslav Nakov. 2017. +SemEval-2017 task 4: Sentiment analysis in Twit- +ter. +In Proceedings of the 11th International +Workshop on Semantic Evaluation (SemEval-2017), +pages 502–518, Vancouver, Canada. Association for +Computational Linguistics. +Fatiha Sadat, Farnazeh Kazemi, and Atefeh Farzindar. +2014. +Automatic identification of arabic dialects +in social media. In Proceedings of the First Inter- +national Workshop on Social Media Retrieval and +Analysis, SoMeRA ’14, page 35–40, New York, NY, +USA. Association for Computing Machinery. +Bingyan Song, Chunguang Pan, Shengguang Wang, +and Zhipeng Luo. 2021. DeepBlueAI at WANLP- +EACL2021 task 2: A deep ensemble-based method +for sarcasm and sentiment detection in Arabic. +In Proceedings of the Sixth Arabic Natural Lan- +guage Processing Workshop, pages 390–394, Kyiv, +Ukraine (Virtual). Association for Computational +Linguistics. +Anshul Wadhawan. 2021. +AraBERT and farasa seg- +mentation based approach for sarcasm and sentiment +detection in Arabic tweets. In Proceedings of the +Sixth Arabic Natural Language Processing Work- +shop, pages 395–400, Kyiv, Ukraine (Virtual). As- +sociation for Computational Linguistics. +Thomas Wolf, Lysandre Debut, Victor Sanh, Julien +Chaumond, Clement Delangue, Anthony Moi, Pier- +ric Cistac, Tim Rault, Remi Louf, Morgan Funtow- +icz, Joe Davison, Sam Shleifer, Patrick von Platen, +Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, +Teven Le Scao, Sylvain Gugger, Mariama Drame, +Quentin Lhoest, and Alexander Rush. 2020. Trans- +formers: State-of-the-art natural language process- +ing. In Proceedings of the 2020 Conference on Em- +pirical Methods in Natural Language Processing: +System Demonstrations, pages 38–45, Online. Asso- +ciation for Computational Linguistics. + diff --git a/dNE0T4oBgHgl3EQfoAHN/content/tmp_files/load_file.txt b/dNE0T4oBgHgl3EQfoAHN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..90935d1dc0aa5f0af05a7f4979714b560f5218d6 --- /dev/null +++ b/dNE0T4oBgHgl3EQfoAHN/content/tmp_files/load_file.txt @@ -0,0 +1,530 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf,len=529 +page_content='SAIDS: A Novel Approach for Sentiment Analysis Informed of Dialect and Sarcasm Abdelrahman Kaseb and Mona Farouk Computer Engineering, Cairo University Giza, Egypt {abdelrahman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='kaseb,mona_farouk}@eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='eg Abstract Sentiment analysis becomes an essential part of every social network, as it enables decision- makers to know more about users’ opinions in almost all life aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Despite its im- portance, there are multiple issues it encoun- ters like the sentiment of the sarcastic text which is one of the main challenges of sen- timent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' This paper tackles this chal- lenge by introducing a novel system (SAIDS) that predicts the sentiment, sarcasm and di- alect of Arabic tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' SAIDS uses its pre- diction of sarcasm and dialect as known in- formation to predict the sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' It uses MARBERT as a language model to generate sentence embedding, then passes it to the sar- casm and dialect models, and then the out- puts of the three models are concatenated and passed to the sentiment analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Mul- tiple system design setups were experimented with and reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' SAIDS was applied to the ArSarcasm-v2 dataset where it outperforms the state-of-the-art model for the sentiment analysis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' By training all tasks together, SAIDS achieves results of 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='98 FPN, 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='09 F1-score and 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='13 F1-score for sentiment analysis, sarcasm detection, and dialect iden- tification respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The system design can be used to enhance the performance of any task which is dependent on other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 1 Introduction Sentiment analysis (SA) is one of the main tasks in the natural language processing (NLP) field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' It is used for opinion mining which supports decision- makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Working on sentiment analysis starts rel- atively early, for example, Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2002) anal- ysed the sentiment to positive and negative in movie reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Following this paper, sentiment analysis becomes one of the most important topics in NLP, especially with the increasing number of reviews on websites and social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Since then, a lot of work has been done in English sentiment analysis, while Arabic has relatively much less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Since Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2008) started their work on Arabic SA, multiple researchers also began theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Now there are well-known Arabic SA models like (Alayba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Abdulla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Abu Farha and Magdy, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Elshakankery and Farouk, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Of course, working with Arabic has many challenges, one of the most challenging issues is the complex morphology of the Arabic language (Kaseb and Farouk, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Abdul-Mageed, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Another challenge is the variety of Arabic dialects (Abdul-Mageed, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Moreover, one of the well- known challenges in SA for all languages is sar- casm, as the sarcastic person uses words and means the opposite of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For example, "I’d really truly love going out in this weather!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' ", does it reflect a positive or negative sentiment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' because of the sar- casm, we cannot judge the sentiment correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Several related works tackle English sarcasm de- tection with sentiment analysis (Oprea and Magdy, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Abercrombie and Hovy, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Barbieri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' On the other hand, there are only a few works on both sentiment and sarcasm in Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' There are two shared tasks on sarcasm detection (Ghanem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2019), but for both sarcasm and sen- timent there was only one shared task Abu Farha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2021) but each sub-task is independent, meaning that participating teams can submit a dif- ferent model for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Some participants used the same model for both sentiment and sarcasm (El Mahdaouy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Instead of training sentiment independently of sarcasm, this work introduces a new model archi- tecture that works with multi-task training which trains both at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' There are other addi- tions to the proposed architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' firstly, it trains with dialect also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Secondly, the sarcasm and di- alect that are initially predicted are used in the prediction of the sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' In other words, the sentiment model is informed by the sarcasm and dialect model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The contributions offered by this work are: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='02521v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='CL] 6 Jan 2023 Design a novel model architecture that can be used for a complicated task that is dependent on another task, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' sentiment analysis which is dependent on sarcasm detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Investigate the design setups for the new ar- chitecture and find the best setup that could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Train the model on ArSarcam-v2 dataset and achieve the state-of-the-art results recorded as 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='98 FPN on sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' This paper is organized as follows Section 2 shows the related work on sentiment analysis, sar- casm detection, and dialect identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Section 3 describes the dataset used in this work and shows data statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Section 4 describes SAIDS model and all the design setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Section 5 shows the ex- perimental results and finally section 6 concludes the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 2 Related Work SAIDS works on three tasks sentiment analysis, sarcasm detection, and dialect identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' In this section, the existing methods for each task are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='1 Sentiment Analysis Arabic sentiment analysis started with Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2008) work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Since then, it is developed by multiple researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' In the beginning, the main focus was on modern standard Arabic (MSA), but over time the researchers start to focus on dialectal Arabic (Mourad and Darwish, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Kaseb and Farouk, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Regarding the datasets, based on Alyafeai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2021), there are more than fifty datasets for senti- ment analysis, including Elshakankery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Kaseb and Farouk (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Kiritchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Elmadany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2018) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Because of the massive number of datasets, there are a massive number of system approaches for Arabic sentiments (Abu Farha and Magdy, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Alayba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' El-Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Based on Abu Farha and Magdy (2021) comparative study, using the word embedding with deep learning models outperform, the classical ma- chine learning models and the transformer-based models outperform both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' There is a reason- able number of Arabic transformer-based models like AraBERT (Antoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2020) and MAR- BERT (Abdul-Mageed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2021) which are used by most Arabic sentiment analysis papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='2 Sarcasm Detection Unlike Arabic sentiment analysis, Arabic sarcasm detection has not gotten much attention yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Only a few research works tackle the problem and still there is an obvious shortage of the Arabic sarcasm datasets, like Karoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Abu Farha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Abbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2020) collected a dataset for sarcastic tweets, they used hashtags to collect the dataset for example #sarcasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Then, they built multiple classical machine learning models SVM, Naive Bayes, and Logistic Regression, the best F1-score was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' After that, Ghanem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2019) organized a shared task in a workshop on Arabic sarcasm detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' They built the dataset by collecting tweets on different topics and using hashtags to set the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' An additional step was added, by sampling some of the datasets and manually annotating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' In this shared task, eighteen teams were working on sarcasm detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Khalifa and Hussein (2019) was the first team and achieved a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='85 F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Then Abu Farha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2021) made two tasks based on the ArSarcasm-v2 dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' sentiment anal- ysis and sarcasm detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' They have 27 teams par- ticipating in the workshop, the top teams achieved 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='25 F1-score and 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='80 FPN for sarcasm detec- tion and sentiment analysis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='3 Dialect Identification Arabic dialect identification is an NLP task to iden- tify the dialect of a written text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' It can be on three levels, the first level is to identify MSA, classical Arabic (CA), and dialectical Arabic (McWhorter, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The second level is to identify the dialect based on five main Arabic dialects EGY, LEV, NOR, Gulf, and MSA (El-Haj, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Khalifa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Sadat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Al-Sabbagh and Girju, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Egan, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The third level is to identify the country-level dialect (Abdul-Mageed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Regarding the datasets, there are datasets more than twenty Arabic datasets labeled with dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' One of the most popular datasets is MADAR (Bouamor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2018) where the data is labeled at the city-level for 25 Arab cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Abdul-Mageed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2020) built a shared task to detect the dialect, they published three different shared tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' In the 2020 task, sixty teams participated, and the best results were 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='78 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='39 F1-score in the country- level and the city-level dialects respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 3 Dataset ArSarcasm-v2 (Abu Farha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2021) is the main dataset used in this work, it was released on WANLP 2021 shared task for two tasks sar- casm and sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' It has about 15k tweets and is divided into 12k for training and 3k for testing, the same test set, as released on WANLP 2021, was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Each tweet was labelled for the sentiment (positive (POS), neutral (NEU), and negative (NEG)), sarcasm (true, and false), and dialect (MSA, Egypt (EGY), Levantine (LEV), Maghreb (NOR), and Gulf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The authors of the dataset annotate it using a crowd-sourcing plat- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' This dataset originally consisted of a combi- nation of two datasets, the first one is ArSarcasm (Abu Farha and Magdy, 2020) and the second one is DAICT (Abbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2020), Abu Farha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2021) merged the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='1 Dataset Statistics In this subsection, we introduce some dataset statis- tics that motivated us to work on SAIDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The ArSarcasm-v2 dataset has 15,548 tweets, 3000 tweets are kept for testing and the rest of the tweets for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Table 1 shows the number of exam- ples for all task labels on the training set, as we can see, most of the data is labeled as MSA and non-sarcastic in dialect and sarcasm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Task Label Count Sentiment Positive 2,180 Neutral 5,747 Negative 4,621 Sarcasm Sarcastic 2,168 Non-sarcastic 10,380 Dialect MSA 8,562 EGY 2,675 Gulf 644 LEV 624 NOR 43 Total 12,548 Table 1: Number of labels of sentiment, sarcasm and dialect on the training set The relationship between sentiment labels and both sarcasm and dialect independently can be shown from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For the sentiment/sarcasm part, we can see that about 90 percent of sarcastic tweets are sentimentally labeled as negative, and about 50 percent of non-sarcastic tweets are senti- mentally labeled as neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' On the other hand, for the sentiment/dialect part, we can see that about 50 percent of MSA tweets are sentimentally labeled as neutral and about 50 percent of EGY tweets are sentimentally labeled as negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' From this table, we can conclude that the information we can get on sarcasm and dialect will benefit the sentiment analysis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' POS NEU NEG Non-sarcastic 2,122 5,576 2,682 Sarcastic 58 171 1,939 MSA 1,405 4,486 2,671 EGY 506 793 1,376 Gulf 121 259 264 LEV 142 197 285 NOR 6 12 25 Table 2: Cross tabulation between sentiment labels and both sarcasm and dialect labels on the training set Table 3 shows the percentage of sarcastic tweets on each dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' As the number of NOR tweets is limited, its percentage is not reliable, so we can see that Egyptians’ tweets are the most sarcastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' This supports the facts from table 2 that most EGY tweets are negative and most of the sarcastic tweets are negative tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Dialect Sarcasm percentage MSA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='83 % EGY 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='77 % Gulf 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='38 % LEV 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='12 % NOR 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='88 % Table 3: Percentage of sarcastic tweets for each dialect on the training set 4 Proposed System This section presents a detailed description of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' SAIDS learns sentiment analy- sis, sarcasm detection, and dialect identification at the same time (multi-task training), in addition, it uses the sarcasm detection and dialect outputs as an additional input to the sentiment analysis model which is called "informed decision".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' SAIDS de- cides the sentiment class using the information of sarcasm and dialect class which are both outputs itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The main idea behind SAIDS is based on analyzing the dataset statistics, as shown in section 3, which says that most sarcastic tweets are classi- fied as negative tweets and most MSA tweets are classified as neutral tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='1 System Architecture Figure 1 shows the SAIDS architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The ar- chitecture consists of four main modules, the first module is MARBERTv2 (Abdul-Mageed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2021), it is a transformer-based model, its input is the tweet, and its output is a sentence embedding which is a vector of length 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The second module is the "Sarcasm Model", it is a binary classifier for sarcasm, its input is the sentence embedding, and its output is two values one for sarcastic tweets and another for non-sarcastic tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The third module is the "Dialect Model", which is identical to the "Sarcasm Model" except that it outputs five classes (EGY, LEV, NOR, Gulf, and MSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The fourth module is the "Sentiment Model", it is a classifier for sentiment, its input is the concatenation of the sentence embedding, sarcasm model outputs and dialect model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Figure 1: SAIDS architecture The loss function used is Cross-Entropy for sen- timent and dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Of course, since sarcasm is binary, we used binary Cross-Entropy for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='2 Training Setups This subsection describes the multiple setups that were used to arrive at the best model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The experiments carried out utilized multiple se- tups regarding the architecture and the training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Modules Architecture Multiple architectures were tested for the "Sentiment Model", "Sarcasm Model" and "Dialect Model".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' As a proof of concept for the idea, we first built a simple random forest model in each task model (random forest version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For the real scenario, we used multi-layer neural network (MNN) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The first and the simplest is one output layer model and zero hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The second is one or two hidden layers, then the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The third is one or two hidden layers the output of the module is the output of the hidden layer, which means that "Sentiment Model" inputs is not the output layer of the "Sarcasm Model" but the last hidden layer of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The fourth setup is to concatenate the last hidden layer with the output layer and then pass it to "Sentiment Model".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' What Should Be Informed The SAIDS archi- tecture Figure 1 shows that the "Sentiment Model" inputs are "Sarcasm Model" and "Dialect Model" outputs but we experimented with multiple settings in this part;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' sentiment analysis informed of sarcasm only, dialect only, and both sarcasm and dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Limited Backpropagation We limited the back- propagation over the dotted lines in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' It is used to ensure that the "Sarcasm Model" and the "Dialect Model" learn their main target correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' When the model predicts sentiment incorrectly, its loss propagates directly to the MARBERTv2 model via the solid line and does not propagate via the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Also, we evaluate SAIDS without lim- iting backpropagation which means the loss prop- agates everywhere, and with partial limiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The partial limiting can be only set when the "Sarcasm Model" has hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' We then limit the back- propagation through the sarcasm model’s output layer but propagate it through the hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Activation Function The experiments were car- ried out with Softmax as the activation function for the output of all modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' However, for the sake of comparison, we run the training without Soft- max for the modules outputs, which means that the values are not from one to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Task By Task Training As we train all the three tasks together with the same model, we experi- mented to train the first layer models, "Sarcasm Model" and "Dialect Model", for some epochs first, then train the full system together for mul- tiple epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The motivation behind this idea is that as long as the first layer models work correctly, the sentiment analysis will correspondingly work correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' We train in multiple orders like alternat- ing between first layer models and full system and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Other Training Parameters In our experi- Sentiment Model Sarcasm Dialect Model Model MARBERT Tweetments, we built SAIDS and used the MARBERTv2 model provided by HuggingFace’s transformers li- brary (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Most of the experiments trained for five epochs except for a low learning rate where it was twenty epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For the learning rate, we used a range from 1e−4 to 1e−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The se- quence was truncated to a maximum length of 128 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Adam (Kingma and Ba, 2015) was used as an optimizer for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 5 Results In this section, the results achieved with SAIDS are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For the sake of comparison, base- lines were built for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' To initially evaluate the idea itself, a random forest model baseline was built and compared with the random forest version of SAIDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Baselines for real scenario are baseline one (B1) which is identical to BERTModelForSe- quenceClassification class in HuggingFace’s (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2020), which takes the MARBERTv2 sen- tence embedding and passes it to the output layer for classification, and baseline two (B2) which uses two hidden layers before the classification layer, the hidden layer size is equal to the "Sentiment Model" hidden layer size, and baseline three (B3) which uses a larger hidden layer size to match the total number of trained parameters of SAIDS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For evaluation, we used the original metrics de- scribed for the dataset (Abu Farha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For sentiment analysis, the metric is the average of the F1-score for the negative and positive classes (FPN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For sarcasm detection, the metric is F1- score for the sarcastic class only (FSar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For dialect identification, we used the weighted average of the F1-score for all dialects (WFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='1 Results of Different Training Setups This subsection presents the results of the training setups and describes the best setup that was chosen for the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For each part of this sub- section, every other setup was not changed to make the comparison fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Modules Architecture As a proof of concept for our system, the random forest (RF) model base- line was compared with the informed random for- est (IRF) which is the random forest version of SAIDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Table 4 shows that IRF outperforms RF where the FPN is improved by 3 percent which is due to the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The information gained from the new inputs, "outputs of sarcasm model" and "outputs of dialect model", was 5 and 4 percent respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' This means that about 10 percent of the sentiment analysis decision came from the newly added information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Model FPN Random Forest 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='36 Informed Random Forest 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='34 Table 4: Performance comparison for the proof of con- cept on the validation set For the MNN architecture of the modules, multi- ple numbers of hidden layers were trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' At each experiment, all the modules have the same number of hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Table 5 shows that using zero hidden layers gives the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' So no hidden layer setup was used in SAIDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Model FPN 0 Hidden Layer 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='23 1 Hidden Layer 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='90 2 Hidden Layer 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='89 Table 5: Performance comparison for the number of hidden layers in modules on the validation set What Should Be Informed Experiments were also done to find the best features to use while analysing sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Table 6 shows that using both dialect and sarcasm is better than using only one of them and of course better than not using any of them which is the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' With a quick obser- vation, it was found out that the dialect benefits the sentiment more than the sarcasm, this can be obvious when speaking about MSA tweets because most of them are labeled as neutral on sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Accordingly, sarcasm and dialect information was used in SAIDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Model FPN Not Informed (B1) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='40 Informed of sarcasm 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='67 Informed of dialect 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='41 Informed of sarcasm and dialect 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='23 Table 6: Performance comparison for what should be informed on the validation set Limited Backpropagation Experiments were also done to find the best path for backpropagation to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' "Full limit" is when the loss does not propagate through the "Sarcasm model" and "Di- alect Model", "Partial limit" is when it propagates through some layers, and "Unlimited" is when it propagates through all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The model was com- posed of two hidden layers while running these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Table 7 shows that "Partial limit" gets better results than the others, but on SAIDS we did not use it as we used a no hidden layer setup, so we used the "Full limit" backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Model FPN Full limit 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='23 Partial limit 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='89 Unlimited 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='31 Table 7: Performance comparison for limiting back- propagation on the validation set Activation Function For the sake of compari- son, the Softmax layer was removed from the out- put layer of the model in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Table 8 compares both setups, it shows that, as expected, using Softmax is better than not using it, as it quan- tify the probability of being sarcasm or being a certain dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' So in SAIDS, Softmax was used on each module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Model FPN With Softmax 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='23 Without Softmax 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='15 Table 8: Performance comparison for the activation function setting on the validation set Task By Task Training Experiments were also done with training the three tasks together at the same time (All tasks), and multiple sets of the train- ing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The first is one epoch of training for sarcasm and dialect, and the rest for the full system (Seq 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The second is odd epochs for sarcasm and dialect and even epochs for the full system (Seq 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The third is two epochs of training for sarcasm and dialect and the rest for sentiment only (Seq 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Table 9 shows that Seq 1 performs better than the other sequences, so we used it for the final model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Model FPN All tasks 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='35 Seq 1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='23 Seq 2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='49 Seq 3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='01 Table 9: Performance comparison for different model training sequences on the validation set Summary of Used Setups SAIDS used infor- mation from sarcasm and dialect models, which are both one classification layer with no hidden lay- ers, the sentiment loss does not propagate through sarcasm and dialect models, and the Softmax ac- tivation function was used on each model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The used training sequence was one epoch of train- ing for sarcasm and dialect, and the rest epochs for the full system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='2 Results comparison with literature SAIDS was trained and compared to the baselines we built and also the state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Ta- ble 10 shows that SAIDS outperforms the existing state-of-the-art models on the sentiment analysis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' SAIDS’s main task is sentiment analysis, the sarcasm detection and dialect identification are con- sidered secondary outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Although the FSar score for SAIDS is considerably high, it is ranked third in the state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' On the other hand, most works that achieve state-of-the-art results are using different models for each task but in the pro- posed architecture, one model is used for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The model also outputs the dialect, it achieves 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='13 percent on the weighted F1-score metric, but the literature has not reported the dialect performance so it is not included in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Model FPN FSar Baseline 1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='60 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='41 Baseline 2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='53 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='61 Baseline 3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='11 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='62 El Mahdaouy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2021) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='80 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='00 Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' (2021) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='92 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='27 Abdel-Salam (2021) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='21 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='62 Wadhawan (2021) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='55 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='72 SAIDS 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='98 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='09 Table 10: Performance comparison for the state-of-the- art models and SAIDS on the test set 6 Conclusion Sentiment analysis is an important system that is be- ing used extensively in decision-making, though it has different drawbacks like dealing with sarcastic sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' In this work, we propose SAIDS which is a novel model architecture to tackle this prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' SAIDS essentially improves the sentiment analysis results while being informed of sarcasm and dialect of the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' This was achieved by training on the ArSarcasm-v2 dataset which is la- beled for sentiment, sarcasm, and dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' SAIDS’s main target is to predict the sentiment of a tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' It is trained to predict dialect and sarcasm, and then make use of them to predict the sentiment of the tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' This means that while the model is pre- dicting the sentiment, it is informed of its sarcasm and dialect prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' SAIDS achieved state-of- the-art performance on the ArSarcasm-v2 dataset for predicting the sentiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='98 percent average F1-score for negative and positive sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' For sarcasm detection, SAIDS achieved a 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='09 percent F1-score for the sarcastic class, whereas for dialect identification it achieved a 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content='13 percent weighted F1-score for all the dialects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' We believe that this model architecture could be used as a starting point to tackle every challenge in sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' Not only sentiment analysis but also this is a general architecture that can be used in any context where the prediction of a task depends on other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' The idea behind the architecture is intuitive, train for both tasks and inform the model of the dependent task with the output of the independent task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE0T4oBgHgl3EQfoAHN/content/2301.02521v1.pdf'} +page_content=' References Ahmed Abbasi, Hsinchun Chen, and Arab 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Zhang, and Lingling Zhang +Abstract—Logical reasoning task involves diverse types of complex reasoning over text, based on the form of multiple-choice question +answering. Given the context, question and a set of options as the input, previous methods achieve superior performances on the +full-data setting. However, the current benchmark dataset has the ideal assumption that the reasoning type distribution on the train split is +close to the test split, which is inconsistent with many real application scenarios. To address it, there remain two problems to be studied: +(1) How is the zero-shot capability of the models (train on seen types and test on unseen types)? (2) How to enhance the perception of +reasoning types for the models? For problem 1, we propose a new benchmark for generalized zero-shot logical reasoning, named ZsLR. +It includes six splits based on the three type sampling strategies. For problem 2, a type-aware model TaCo is proposed. It utilizes both the +heuristic input reconstruction and the contrastive learning to improve the type perception in the global representation. Extensive +experiments on both the zero-shot and full-data settings prove the superiority of TaCo over the state-of-the-art methods. Also, we +experiment and verify the generalization capability of TaCo on other logical reasoning dataset. +Index Terms—Natural Language Processing, Logical Reasoning, Question Answering, Generalized Zero-shot. +! +1 +INTRODUCTION +Logical reasoning over text has aroused wide interest in +the area of Machine Reading Comprehension (MRC) [1] and +Natural Language Processing (NLP) [2] [3] recently. In the +form of the traditional multiple-choice question answering +(MCQA) [4] [5] [6], the task of logical reasoning requires +the model to perform complex reasoning and generalization. +One of the main difficulties of the task lies in addressing +diverse reasoning types. Fig. 1 shows some examples of +reasoning types in the logical reasoning task. Given questions +with different reasoning types, humans tend to focus on the +respective aspects of interactions between the context and +the option. For instance, for the type Identify the flaw (a), the +option is strongly related to the detailed logic flaws within +the global idea. While for the type Necessary assumption (b), +the focus may be switched to the premise of the arguments +and detect the missing assumption reflected by the option. +Also, for the reasoning type of Parallel reasoning (c), it is +required to consider the corresponding logical structure of +the context and the option, rather than the specific entities +or events. Therefore, the modeling of the specific reasoning +type is intuitive and necessary to the logical reasoning task. +Recent works have witnessed improvements in the logical +reasoning tasks. Generally, they can be categorized into +two folds: graph-based and data-based. In the graph-based +family, DAGN [7], FocalReasoner [8] and Logiformer [9] +attempt to construct the context graphs from different levels, +• +Fangzhi Xu, Qika Lin, Tianzhe Zhao, Jian Zhang are with School of +Computer Science and Technology, Xi’an Jiaotong University, Xi’an, China. +• +Jun Liu and Lingling Zhang are with Shaanxi Province Key Laboratory of +Satellite and Terrestrial Network Tech. R&D, National Engineering lab for +Big Data Analytics, Xi’an, China. +• +The corresponding author is Jun Liu. +Fig. 1: Examples of different reasoning types. +such as causal and co-occurrence. And AdaLoGN [10] +proposes a neural-symbolic system in an adaptive manner. +In the data-based family, previous works explore various +data augmentation strategies. For example, LReasoner [11] +extracts the symbols from the text and extends them with +logical rules. MERIt [12] designs several data generation +arXiv:2301.02983v1 [cs.AI] 8 Jan 2023 + +Context +logic flam +Option +If +and hence +logic flaw.-. A. The argument fails +, though +However, +Thus, if +B... +C... +D... +logic flaw +Question +Reasoning Type +Which one of the following most accurately +Identify the flam +describes a flaw in the reasoning of the argument ? +(a) An example of Identify the flam +Option +Context +misssing assumption +If +then +A. If +Only if +Therefore, +In all, +B... +C... +D... +Reasoning Type +Question +Which one of the following is an assumption +Necessary assumption +(b) An example of Necessary assumption +Context +Option +macth logics, ignore facts +, only(if +Al (f) +because +will hot) +However +(unless +f +.(lf) +And(if) +Similar +.(If) +then +Therefore +Logic +5 +Thus +Question +Reasoning Type +The pattern of reasoning displayed above most +Parallel reasoning +closely parallels which of the following ? +(c) An example of Parallel reasoningJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +methods to facilitate the training process. However, all of +the methods in the two families lack the modeling of the +reasoning type features. Although the above mentioned +methods achieve superior results over strong baselines, the +current researches are based on the ideal setting that train +and test splits share the similar distribution of reasoning +types. Nevertheless, in the real application, we are mainly +exposed to the common types of questions while unfamiliar +with the reasoning on the novel and uncommon types. In +another word, there is an obvious gap between the ideal +setting and the real scenarios. To address this issue, there +exist two problems to be studied: 1) What about the test +performance on the unseen types during training and how is +the zero-shot capability of the models? 2) How to enhance +the perception of reasoning types for the models? +As for Problem 1, we propose a new benchmark for +the generalized Zero-shot Logical Reasoning, named ZsLR. +Based on the ReClor dataset [13] with 17 reasoning types, +we form 6 zero-shot data splits according to 3 strategies (i.e., +amount, randomness, and difficulty). To make comprehen- +sive assessments, we introduce the generalized zero-shot +setting [14], which test on both seen types and unseen types +with two defined metrics. The necessity and meaning of +ZsLR is verified by the pilot experiments and it encourages +more future works to mind reasoning manners. +As for Problem 2, we propose a Type-aware reasoning +network based on Contrastive learning, named TaCo. First, +we design a keyword-based extractor to output the reasoning +type. Then, through the two designed heuristic strategies, +we merge each question and the option into a unified Q-A +pair. Based on the co-occurrence node extraction algorithm +proposed in Logiformer [9], we form the topology within +context part and Q-A pair part respectively. To model the +interaction of the context and Q-A pair for different reasoning +types, we add a global node to connect all the nodes of +the two parts. Through the self-attention aggregation, the +final representation of the global node is obtained and +utilized to predict the answer. Meanwhile, we employ the +sentence-BERT [15] to obtain type embeddings based on +their descriptions. The margin loss is applied to model +the reasoning type, where the global node serves as the +anchor, ground truth type as positive example and other +types as negative examples. In this way, the global node +representation contains both the context and the reasoning +type semantics. Therefore, the perception of the reasoning +types can facilitate the zero-shot capability. +In all, the main contributions of this paper are summa- +rized as follows: +(1) We are the first to focus on the issue of type-oriented +reasoning manners. To address the issue, we propose the first +benchmark: generalized ZsLR.1 It can well reflect the real +scenarios and test the zero-shot capabilities of the models. +(2) We propose to tackle the zero-shot task through the +heuristic input reconstruction and type-aware contrastive +learning. The proposed model TaCo can function as a strong +baseline for the future works on the task of ZsLR. +(3) Extensive experiments on both zero-shot and full-data +settings prove the huge potential of the proposed issue, as +1The ZsLR dataset and implementation of TaCo are public at github. +com/xufangzhi/TaCo +well as the superiority of TaCo. Also, we conduct additional +experiments to further verify the generalization capability of +TaCo on other setting and dataset. +2 +RELATED WORK +2.1 +Logical Reasoning +The logical reasoning task, which aims at testing the rea- +soning capability of models, has aroused wide interest. +Several representative datasets have been proposed, such as +ReClor [13] and LogiQA [16]. Current methods on the logical +reasoning task can be divided into graph-based methods and +data-based methods. The former focuses on the construction +of the text graphs and leverage the node connection to +model the logical relations. Among this category, DAGN +[7] is the first work to split the text into EDUs and perform +the reasoning with graph neural networks [17] [18], but it +only builds a chain-type graph and ignores the long distance +interaction between nodes. FocalReasoner [8] proposes to +attend to the fact triplets [19] within the text and constructs +a supergraph based on the extracted triplets. However, it +lacks the modeling of the logical information in the text. +To fully explore the logic, AdaLoGN [10] is proposed to +construct the adaptive neural-symbolic system to improve +the performances. However, its reasoning process is complex +and costly. Considering the above drawbacks, Logiformer [9] +stresses much importance on both the causal relations and the +co-occurrence relations in a two-branch graph transformer +network. Yet it still lacks the perception of different question +types, which limits the zero-shot reasoning capability of +the model. The latter one is the data-based methods. These +works aim to improve the performance through the data aug- +mentation strategy. One of the representatives is LReasoner +[11], which extends the symbolic expressions by logical rules +and templates. In addition, MERIt [12] designs a meta-path- +guided contrastive learning method to facilitate the training +process, by utilizing the extra data. However, both of them +fail to model the reasoning type of each question and lack +the application value in zero-shot setting. +2.2 +Current Machine Reading Comprehension Settings +Current machine reading comprehension (MRC) tasks [20] +can be categorized into two settings: full-data setting and +low-resource (i.e., few-shot, zero-shot) setting. +Full-data setting has aroused wide concerns in the area +of MRC in recent years. Popular benchmark SQuAD [21] +[22], which is sourced from from Wikipedia articles, contains +over 100,000 questions. It forms an abundant database for +the model training. Similarly, HotpotQA [23] includes 113K +questions with a variety of reasoning strategies. It stresses +more on the multi-hop reasoning abilities of the model. RACE +dataset [24] is specially designed to improve the reading +skills for both middle school and high school students, which +contains about 9.7K examples. Also, some domain-specific +datasets like NewsQA [25], TextbookQA [26], PubMedQA +[27], aim at the area of journalism, education and biology +respectively. All of them have rich training examples to +improve the model capability. However, such full-data setting +encourages the models to depend on the ideal scenes with +abundant training data, which may not fit in some real cases. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +Based on such concern, some previous works focus on +the low-resource setting MRC, including few-shot or zero- +shot setting. For example, to evaluate the robustness of the +model, [28] reconstructs the data from knowledge sources +for zero-shot commonsense QA. Also, [29] proposes a neuro- +symbolic approach to boost the performance of zero-shot +commonsense QA. Both of the methods are not exposed +to the training data, forming a zero-shot setting. However, +they rely on the extra knowledge bases, which limits their +applications. There are also some works related to the few- +shot QA. In [30], a new pretraining strategy is proposed to +explore the realistic few-shot setting. Also, [31] tackles the +few-shot challenge on the task of Visual Question Answering. +However, all of the above methods simply obtain the few- +shot splits according to the amount, which treat all the +samples equally. +To sum up, current MRC benchmarks mainly focus on +the common types of questions. However, in reality, it is +more challenging when faced with some uncommon types of +questions. To make up for these drawbacks, we attend to the +reasoning types in the logical reasoning tasks and propose +the first benchmark for zero-shot logical reasoning based on +type attributes. +3 +THE BENCHMARK OF GENERALIZED ZSLR +In this section, we introduce the generalized ZsLR bench- +mark. At the very beginning, we obtain the statistical +distribution of the number of reasoning types on the train, +development and test splits, shown in Fig. 2a. We arrange +them in descending order of the number. The distributions +are in the similar form. It demonstrates that the full-data +setting is based on such an ideal assumption, which is +insufficient to verify the zero-shot generalization capability +of the models. Therefore, we propose the benchmark of +generalized ZsLR and conduct the pilot experiments to verify +the necessity. +3.1 +Zero-shot Data Construction +The zero-shot logical reasoning datasets are split based on +ReClor [13] without shuffling. Considering the situation in +reality, it is easier to learn to reason on common types of +samples while struggle with the rare ones. To this end, we +design three sampling strategies, namely amount, randomness +and difficulty. +For amount, we select top-k reasoning types as the seen +types, merely by the amount. It can be seen as a simple +implementation to filter the uncommon types of samples. +For randomness, we arrange the reasoning types in de- +scending order of amount and select the seen ones based on +the geometric distribution. The discrete form of the geometric +distribution is, +P (X = k) = (1 − p)k−1p, +(1) +where k is the sorted index of the reasoning type, p is the +hyper-parameter set to 0.1 in our implementation. In this +random setting, the type with more training samples has a +higher probability to be selected, which is in parallel with +the real situation. +�� � +� +� +� +� +� �� � �� � �� �� � �� �� � +���������� +� +� +� +� +� +�� +�� +��������� +����� +��� +���� +(a) The distribution of the occurrence frequency of different +reasoning types in the ReClor dataset. +(b) Pilot comparisons between full and zero-shot settings. +Fig. 2: The visualization of the pilot experiments. +For difficulty, we first rank the difficulty of the reasoning +types, based on their performance with RoBERTa-Large +single model. On one split, we select some of the most +difficult reasoning types as the seen ones. On the other split, +we select a part of the easiest types as the seen ones. +In total, 6 zero-shot splits are obtained (2 for each strategy) +for ReClor. The details of the splits are presented in Table 1. +3.2 +Generalized Zero-shot Setting +We propose the zero-shot setting for the logical reasoning +task. Given the context Ci, question Qi (type(Qi) ∈ T ) and +option set Ai of the ith question, the model is required to +predict the correct answer a ∈ Ai. In the definition, type(Qi) +denotes the reasoning type of the question Qi and T is the +type set of the zero-shot splits. For each zero-shot split, a +part of the types are sampled as the seen types, while others +are viewed as the unseen ones. Only the questions of seen +types exist during training and they consist of the training +examples. The type set seen during the training stage and +the set for the test stage are T train and T test respectively, +thus they satisfy T train ∪ T test = T . +To be closer to the real scenes, we consider the generalized +zero-shot setting. That is to say, the test scope is not limited +to the unseen types. To this end, we employ two metrics for +the generalized zero-shot setting, Test-All and Test-Unseen. +The former one denotes the exact match results on the full- +data test split, which contains both seen and unseen types, +i.e., T train ⊆ T test. The latter one is the exact match results +only on the unseen types of the test split, which ignores the +performance on the seen types, i.e., T train ∩ T test = ∅. In +this way, we expect to achieve comprehensive assessments +of the model capabilities. + +60 +Test-All (Full) +Test-Unseen (Full) +Test-All (Zs) +Test-Unseen (Zs) +55 +Performance +45 +40 +35 +30 +v3 +V +V4 +Zero-shot splitsJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +TABLE 1: Details of the split datasets for the zero-shot setting. The third column presents the index of the seen types and +the fourth column shows the number of the seen and unseen types. The last two columns present the number of training +samples and test samples respectively. Especially, test samples contain both seen and unseen types for generalized zero-shot +setting. +Strategy +Split +Seen Type +# Type +Seen (Unseen) +# Train +Seen +# Test +Seen (Unseen) +Amount +v1 +{0,3,4,8,13} +5 (12) +2,190 +475 (525) +v2 +{0,1,2,3,8,9,14,16} +8 (9) +2,700 +595 (405) +Randomness +v3 +{0,2,3,13} +4 (13) +1,928 +435 (565) +v4 +{0,2,3,5,7,8,13} +7 (10) +2,896 +645 (355) +Difficulty +v5 +{0,2,4,6,8,13,15} +7 (10) +2,175 +473 (527) +v6 +{1,3,5,7,9,10,11,12,14,16} +10 (7) +2,463 +527 (473) +3.3 +Pilot Experiments +Also, we are going to verify the necessity of the zero-shot +setting from the pilot experiments. For comparison with each +split, we randomly sample the same amount of training +examples on both the seen and unseen types, forming +the comparison group. Take the zero-shot split v1 as an +example, it has 2, 190 seen samples for training. Thus, its +comparison training group also includes 2, 190 examples, +but are distributed over all types. The purpose of the pilot +experiments is to observe the performance changes of the +two metrics on the test split. +In the implementation, we utilize the RoBERTa single +model [32] to conduct the reasoning and maintain the same +hyper-parameters of the zero-shot splits with comparison +groups. Especially, to avoid the noise brought by the random +sampling, we conduct the comparison experiments five times +with different random seeds. Final results are the average +values of the five experiments. +We select the results on four of the split versions for +illustration (i.e., v1-v4), shown in Fig. 2b. The first and the +third column in blue are the performance of the comparison +group, on the full-data test and the unseen types of test +respectively. The second and the fourth column represent the +performance on the zero-shot setting. With the same number +of training examples, training only on seen types (zero- +shot setting) witnesses obvious drops on the performance, +especially on the unseen types of the test split. +Such observations are consistent among all the zero-shot +splits. Zero-shot pilot experiments based on reasoning types +uncover the obvious drawback of the current full-data setting. +In another word, it verifies the necessity to propose a new +benchmark for the generalized ZsLR. +4 +METHODS +In this section, we introduce the proposed methods. To tackle +the zero-shot challenges, we propose a model named TaCo, +which focuses on the reasoning type perception in the logical +reasoning task. The architecture of TaCo is shown in Fig. 3. +It mainly consists of three parts: (a) heuristic reconstruction +to acquire the type-aware input sequences; (b) text graph +construction and reasoning for the QA problems; (c) type- +aware contrastive learning. +4.1 +Heuristic Input Reconstruction +One of the common practices for MCQA problems is to +concatenate three sequences as inputs: context, question and +option. But it is insufficient for the zero-shot setting. There +exist two main drawbacks: 1) the modeling of the reasoning +type is implicit in the sequence; 2) it is difficult to bridge +the interaction between context and options via the question +sequence in the middle since it is not natural for the language +model (LM). To this end, this paper introduces the heuristic +reconstruction to the inputs based on the type extraction. +To address the first drawback, we are required to label +the reasoning type of each question based on the limited +inputs. Inspired by LReasoner [11], we propose a simple but +effective type extractor through keywords. The procedure of +the heuristic type extractor is presented in the pseudo-code +of Algorithm 1. +Since the ReClor dataset only makes the reasoning types +on the test split public, we are required to collect the +classification method from it. Before the extraction, we +conclude the keywords and phrases for each reasoning +type, forming the keywords base B. Also, we include the +maximum window size W and the question sentence as the +inputs. In Line 1, we first split the question into words and +form the word sequence S. Then we start the iteration in the +descending order of the window size. We slide the window +over the question sequence to obtain the sub-sequence (Line +5). Then we match the sub-sequence C with keywords base +Bk for each reasoning type k to derive the number of exact +matches (Line 7). After each window size iteration, we judge +the exit condition (Line 11). If there exists a unique type with +the maximum number of matches, we label the type as the +ground truth. Meanwhile, if we can not extract the type at +the last round of iteration, we label the instance as Others +type (Line 16, 17). +Then, we convert the type index of each question into the +natural language (e.g., Implication, Conclusion). Meanwhile, +to equip the LM with the type information, we add a type- +related prefix at the beginning of the sequence: +prefix = This is the task of [R-Type] +(2) +where [R-Type] denotes the natural language label of the +specific type. Thus, the type semantic is merged into the +inputs in an explicit manner. +For the second drawback, one intuitive idea is to re- +construct the inputs of question and option sequences, +transforming them into a single sequence in a declarative +form, defined as Q-A pair. In another word, we are required +to fill the option into the proper position in the question. +According to our observation, the question sentence is led by +a trigger span. For example, in the question Which one of the + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +5 +Fig. 3: The architecture of TaCo. Taken the three basic parts (i.e., context, question and option) as the inputs, we reconstruct +them with the heuristic strategy (a). The text graph is built in module (b1) and the reasoning based on the graph transformer +is conducted in module (b2). The updated representation of the global node is employed to perform the final inference. +Meanwhile in module (c), the contrastive learning is introduced to improve the awareness of the reasoning type in the global +node. +Algorithm 1: Reasoning Type Extraction +Input: keywords base B, window size W, question +sentence q +Output: reasoning type T +1 S ← SplitByWords (q) +2 for i = W, ..., 2, 1 do +3 +Count ← {} +4 +for j = 0, ..., length(S)− W do +5 +C ← S[j : W] +6 +for k = 0, 1, ..., 16 do +7 +num ← NumOfMatch(C, Bk) +8 +Count[k] += num +9 +end +10 +end +11 +if max(Count) > 0 and is unique then +12 +T ← type of the max match count +13 +return reasoning type T +14 +end +15 +if i == 1 then +/* classify to ‘Others’ +*/ +16 +T ← ‘Others′ +17 +return reasoning type T +18 +end +19 end +following most weakens the arguments above ?, the span Which +one of the following serves as the trigger. Then, we can replace +the trigger span with the option sequence to obtain the Q-A +pair, formulated as [Option] most weakens the arguments above. +Therefore, the core of the heuristic strategy is to pinpoint +the precise position of the triggers. Although the trigger span +is similar to the basic form of Which of the following, it is not +always the same. To further improve the diversity of the +trigger base, we propose two heuristic strategies: +• +Combination: We predefine a set of basic words (i.e., +{which, one, of, the, following}), and then randomly +combine all or parts of them to form the new triggers. +For instances, of which the following and which of +following are two newly constructed triggers. +• +Tolerance: More trigger spans are not limited to the +combination of the predefined words. To this end, +we propose a tolerance strategy to contain the extra +words within the trigger span. For example, in the +trigger of the following claims, which, we include the +extra word claims to the span. +In this way, the trigger base can be expanded to adapt to +the complex situations in reality. So far, the declarative Q-A +pair is obtained. +Concatenating the context sequence Ci and Q-A pair +sequence Pi of the ith question, we send it into the pre- +trained LM. In this paper, we employ the RoBERTa-Large +model [32] to obtain the token-level representations as +follows, +�h; hc0; . . . ; h; hp0; . . . ; h +� += RoBERTa (< s > c0 · · · p0 · · · < /s >) , +(3) +where the tokens {c0, c1, . . . , c|Ci|} make up Ci, and +{p0, p1, . . . , p|Pi|} make up Pi. The representation of Ci, Pi +and the whole sequence Si (concatenation of prefix, Ci and +Pi) can be obtained through the mean pooling strategy, +hCi = +1 +|Ci| +|Ci| +� +k=1 +hck, hPi = +1 +|Pi| +|Pi| +� +k=1 +hpk, +hSi = +1 +|Si| +|Si| +� +k=1 +hsk. +(4) + +(a) Heuristic Input +(b1) Graph Construction +Reconstruction +(b2) Graph Reasoning +Extract +Context +Context Ques.Opt. +Node +initialize +Concat +Heuristic Extractor +h +P +Node( +Classify +R-Type +PrefixP +Pos (oD C1 +h +Adjacent +Global +S + Q-A Pair +PContext +Relations +Node +Transformer +Extract +d人 +M +Layers +Q-A Pair +h +MO +Node +g +Text Encoder +Updated +Global Node + h +(c) Type-aware Contrastive Learning +Anchor +R-Type +Positive +push +pull +farther +(Ground-truth) +Type +Text +Description +Encoder +con +* one sentence +for each type +Positive +Negative +Negative +(Other Types)JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +6 +Fig. 4: An example of the construction of context subgraph. +The input context can be split into 6 units (i.e., U1-U6). Based +on the order relation and overlap relation, the topology of +the subgraph can be obtained. +4.2 +Text Graph Construction and Reasoning +According to the previous analysis, the context and Q-A +pair may interact differently under different reasoning types. +Therefore, we consider to build the topology structure of +the two parts respectively and finally learn their interactive +semantic. For clearer illustration, we take the context part as +an example and present the graph construction process in +Fig.4. +Firstly, we split the text sequence into units (e.g., U1- +U6 in Fig. 4) based on the predefined explicit connectives +or punctuation (marked in blue in Fig. 4). These text units +function as the nodes during the graph reasoning process. +Therefore, two node sets for the context and Q-A pair parts +are obtained respectively. +To form the dual subgraphs of the input sequence, we are +also required to define the edge relations, including a) order +relation and b) overlap relation. The former one models the +original text order, where nodes are connected in sequence. +It forms a chain-type structure which maintains the position +perception. The latter one models the similarity between +nodes, where nodes with more vocabulary overlaps or similar +semantics will be connected. In the implementation, we will +transform the text units into word sets, where the stopwords +are excluded. Through the calculation of the overlap ratio +between sets, we connect the node pairs that exceed the +threshold. +In this way, the independent topology of the context part +and the Q-A pair part are formed. To make the interaction +of the two parts learnable, we add a global node to link all +the extracted nodes. The global node is initialized with hSi +and aggregates the two-way information flow. Therefore, this +node representation is expected to contain the type semantic +to improve the zero-shot performance. +Further, to overcome over-smoothing in graph neural +networks [33] and fully conduct node interactions, we +employ the Graph Transformer Network [9], [34]. Let N +be the node set in one graph, with the number of |N|. We +feed the node sequence into the Transformer architecture +[35]. For each layer l, the updated information ej of the node +Nj is formulated as follows: +e(l) +j += σ(W(l−1) +self e(l−1) +j ++ +� +Nk∈N \{Nj} +αj,ke(l−1) +k +). +(5) +where N\{Nj} denotes the set without jth node. +The core of the equation is the computation for the +weighted attention αj,k. It is obtained by the two matrices +query Q and key K: +A = softmax(QKT +√ +d ++ M), +(6) +where αj,k is one of the elements in the matrix A. d is +the dimension of the hidden states. M ∈ R|N |×|N | is the +attention bias matrix to encode the structural semantic of the +whole graph, which is the adjacent matrix of the graph. +Thus, the final feature of the global node hg is updated +through the graph reasoning network, containing the global +semantic of the text sequence. Before predicting the correct +answer, we concatenate h(j) +g , h(j) +Ci , h(j) +Pi and h(j) +Si to obtain the +score of the jth option followed by softmax: +z(j) +i += softmax +� +W · +� +h(j) +g ; h(j) +Ci ; h(j) +Pi ; h(j) +Si +�� +, +(7) +where W ∈ R4d×1 denotes the linear projection. Same with +most of the MCQA methods, we adopt the cross-entropy loss +function for the optimization: +LMC = CrossEntropy (zi, yi) , +(8) +where zi denotes the stack of each option score and yi is the +ground-truth label of the ith example. +4.3 +Type-aware Contrastive Learning +In the upper part of the architecture, the interactive semantic +of the context and Q-A pair is learned through one global +node. Besides the implicit awareness of the type, we also +expect the model to distinguish the ground-truth type from +the negative ones for zero-shot logical reasoning. +Based on the heuristic extractor mentioned above, the +reasoning type of each example can be derived. For each +reasoning type, the detailed description in the form of natural +language is utilized (listed in Table 2). +Feeding all 17 reasoning type descriptions into the +LM (i.e., Sentence-BERT [15]), we obtain the sentence-level +embeddings, which are fixed during training. +For each example, we propose to attend to the type infor- +mation through contrastive learning. The final representation +of the global node hg serves as the anchor. The ground- +truth type functions as the positive sample, while others are +negative samples, with the embedding of hgt and hn (n is +the index of the negative samples) respectively. Our purpose +is to close the distance between the global node hg and the +positive sample hgt, while distancing the negative samples. + +Context +[U1 Paula will visit the dentist tomorrow morning only if [U2] +Bill goes golfing in the morning. [U3i Bill will not go golfing +unless [U4l Damien agrees to go golfing too. However, [U5] +Damien has decided not to go golfing. Therefore, IU6l Paula +will not be visiting the dentist tomorrow morning. +Subgraph Construction +a) order relation +b) overlap relation +U4 +U4 +U2 +U2 +J5 +U5 +U3 +U3 +U6 +U6 +U4 +U2 +05 +U3 +U6 +Context SubgraphJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +7 +TABLE 2: The defined index and descriptions of 17 logical reasoning types in the dataset. +Id +Type +Descriptions +0 +Necessary Assumptions +identify the claim that must be true or is required in order for +the argument to work. +1 +Sufficient Assumptions +identify a sufficient assumption, that is, an assumption that, if +added to the argument, would make it logically valid. +2 +Strengthen +identify information that would strengthen an argument. +3 +Weaken +identify information that would weaken an argument. +4 +Evaluation +identify information that would be useful to know to evaluate +an argument. +5 +Implication +identify something that follows logically from a set of premises. +6 +Conclusion and Main Point +identify the conclusion/main point of a line of reasoning. +7 +Most Strongly Supported +find the choice that is most strongly supported by a stimulus. +8 +Explain or Resolve +identify information that would explain or resolve a situation. +9 +Principle +identify the principle, or find a situation that conforms to a +principle, or match the principles. +10 +Dispute +identify or infer an issue in dispute. +11 +Technique +identify the technique used in the reasoning of an argument. +12 +Role +describe the individual role that a statement is playing in a +larger argument. +13 +Identify a Flaw +identify a flaw in an arguments reasoning. +14 +Match Flaws +find a choice containing an argument that exhibits the same +flaws as the passages argument. +15 +Match the Structure +match the structure of an argument in a choice to the structure +of the argument in the passage. +16 +Others +other types of questions which are not included by the above. +For simplicity, we model the score of two vectors using +Hadamard product. That is: +score+ = hg ⊙ hgt, +score− +n = hg ⊙ hn, +(9) +where score+ and score− +n represent the score of the positive +sample and the scores of the negative ones respectively. We +employ the margin loss as the auxiliary optimization function +for each example: +Lcon = max{0, γ − score+ + max +n (score− +n )}, +(10) +where γ > 0 controls the difference between positive and +negative scores. We select the maximum score of the negative +one for the loss computation. +To maximize the joint optimization performance of the +two loss functions, we set a trade-off coefficient α and +transform the final loss function into: +L = LMC + αLcon. +(11) +In this way, the global semantic for reasoning reinforces +mutually with the type-aware semantic, which benefits +the zero-shot performance. In addition, it provides the +interpretability for the test process, which will distinguish +the correct reasoning type from others. +5 +MAIN EXPERIMENTS +In this section, we will introduce the details of the experiment +setup. Also, extensive experiments on both the zero-shot and +full-data setting will be analyzed. +5.1 +Current Full-data Benchmark +Currently, there exist two datasets in the logical reasoning +task, ReClor [13] and LogiQA [16]. Since the LogiQA datasets +contain only 5 reasoning types, we consider the ReClor +dataset in the zero-shot experiments. And we take both +of them into account for the full-data setting to prove the +generalization capability of TaCo. The detailed information +of the two datasets is presented below. +ReClor is sourced from some standardized graduate ad- +mission examinations. It consists of 6,138 examples in total, +including 4,638 training samples, 500 validation samples, and +1,000 test samples. As is listed in Table 2, ReClor contains 17 +reasoning types. Especially, its test split is divided into two +parts, which are Test-E and Test-H. The former one is the easy +splits, which can be addressed without context. The latter +one is the hard splits. +LogiQA is collected from National Civil Servants Exami- +nations of China, including 8,678 samples in total. Among +them, 7,376 are training samples while validation and test +splits both contain 651 samples. Different from the diverse +reasoning types in ReClor, LogiQA contains only 5 reasoning +types. +5.2 +Baselines +We include all the previous SOTA baselines in the main paper +for comparison, as well as the results from some classical +language models. +Random The results in this setting are based on the random +predictions. +RoBERTa-Large [32] The results are obtained simply utiliz- +ing the RoBERTa language model for the predictions. It is +similar to the baseline of BERT-Large [36] and XLNet-Large +[37]. +Human Performance [13] In ReClor, the average score of +different graduate students on the test split is utilized as the +human performance. +DAGN [7] It is the first work to propose the construction of +the text graph based on the extracted EDUs. It mainly relies +on the RoBERTa-Large [32] to encode the tokens and graph +neural network [17] to update the features. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +8 +TABLE 3: The tuned hyper-parameters with search scopes. +‘GT’ denotes the graph transformer network. +Name of Param. +Search Scope +Best +# epoch +{10,15,20,25,30} +30 +# head in GT +{3,4,5,6} +6 +# layer in GT +{3,4,5,6} +4 +max sequence len +{128,256} +256 +learning rate +{4e-6. 5e-6, 6e-6} +5e-6 +margin γ +{8,10,12,14} +12 +trade-off α +{0.1,0.2,0.5,1} +0.2 +FocalReasoner [8] It constructs a supergraph for reasoning, +which consists of the fact units extracted from the text. +LReasoner [11] It explores the context extension based on +the defined logical rules (e.g., De Morgan’s laws). Also, +it employs the data augmentation method to improve the +performance. Since constructing more training data is proved +to be of great help to the zero-shot performance of most of the +current models, we do not consider the data augmentation +strategy when reproducing. +MERIt [12] It proposes a meta-path-guided contrastive +learning method to reason over the text. The supervised +pretraining is performed on the abundant unlabeled text +data. Due to the employment of the external data, we do not +consider this method in the zero-shot baselines for the fair +comparison. +AdaLoGN [10] It proposes a neuro-symbolic system to adap- +tively update the text graph. Additionally, a novel subgraph- +to-node mechanism is utilized aggregate the information. +Logiformer [9] It introduces two different strategies to +construct the logical graph and the syntax graph respectively. +Through the two-branch graph transformer network, the text +features are updated to conduct the reasoning. +5.3 +Implementation Details +In this paper, all experiments run on a single GPU of Tesla +A100. We employ RoBERTa-large model [32] as the text +encoder for all previous methods for fair comparison. For +each split of the zero-shot setting, we train on the seen +types and select the best epoch on the seen types of the +development set for the test. We rerun other baselines on +the zero-shot setting with the original configuration and +maintain their reported results on the full-data setting. As for +our proposed model, the hyper-parameters keep the same for +both settings. The training epoch is set to 30 and the batchsize +is fixed to 1. For optimization, we take Adam [38], with the +peak learning rate as 5e-6. The number of heads and layers +in the graph transformer is set to 6 and 4 respectively. γ in +the margin loss is tuned as 12. The loss trade-off α is 0.2. The +selected five random seeds for the comparison groups are +{42, 12, 23, 234, 1234}. Additionally, some important hyper- +parameters are tuned for the best within a search scope. The +details are included in the Table 3. +5.4 +Main Results +The model TaCo is designed to bridge the gaps of the zero- +shot logical reasoning setting, thus we are going to evaluate +its performance. We present the results of 6 zero-shot splits +in Table 4. We rerun five strong baselines on the zero-shot +setting, including (1) BERT-Large, (2) RoBERTa-Large, (3) +graph-based representative method DAGN [7], (4) data- +based representative model LReasoner [11], (5) SOTA model +Logiformer [9]. +It shows that TaCo outperforms these strong baselines +with large margins on all the zero-shot splits. Compared +with the previous SOTA model Logiformer, the averaged +improvements among all the splits are 3.80% and 4.54% +respectively for the two metrics Test-All and Test-Unseen. +And compared with all suboptimal results on the two +metrics, TaCo still has great superiority of 2.50% and 3.65% +respectively. To explore the respective roles of the two metrics, +Test-Unseen witnesses the greater improvements among all +the splits, especially 6.14% and 5.04% over the sub-optimal +methods in split v1 and v4 respectively. It illustrates that +TaCo performs better on tackling the unseen types of samples +than the current methods, while maintaining the superior +performances on the seen ones. In addition, among all the +zero-shot splits, the performances within each method vary +greatly. For example, the previous SOTA Logiformer shows +good performances in v2 but struggles a lot in v1. It proves +that the benchmark provides diverse distributions of the +data splits and encourages the consistent performances of +the model. And considering the experiment results, TaCo can +function as a strong baseline for the future work on the ZsLR +benchmark. +5.5 +Ablation Studies +To illustrate the effectiveness of each module of TaCo in the +zero-shot setting, we conduct the following ablation studies. +The results are listed in Fig. 5 +Detailedly, in part a), w/o type prefix and w/o reconstruction +are related to the heuristic input reconstruction, where the +former one ablates the type-oriented prefix, and the latter +one maintains the simple concatenation of the question +and option. Generally, the heuristic type prefix and input +reconstruction bring more improvements on the metric of +Test-Unseen with an average of 3.95% and 3.26% respectively +respectively on all six splits. It shows that type-oriented +prefix is more effective than input reconstruction in most +cases (i.e., v2-v5). +In part b), w/o graph reason ablates the whole module of +graph construction and reasoning, and replaces the global node +feature with the pooled output of the whole sequence. Since +the reasoning process is the core of this task, it contributes +most to the performance on both the metrics, with the +average gains of 2.58% and 4.23%. While w/o global node +simply replaces the global node feature with the pooled +representation of all the other nodes. It can be regarded as a +small part of graph construction and reasoning. From the results, +it works on the splits v2, v4 and v6. As the training set of +these three splits are obviously larger, it can be concluded +that global node is especially effective with more training +samples. +In part c), we ablate the type-aware contrastive learning +module. From the experiments, the modeling of reasoning +types does great help to the unseen types of samples. In +the unseen types of v1, v2 and v4, the margin loss brings +significant improvements, with gains of 4.81%, 4.17% and +3.70% respectively. +Overall, the three key modules (i.e., type prefix, input +reconstruction, type contrast) proposed to enhance the type + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +9 +TABLE 4: Experimental results on 6 zero-shot splits of ReClor dataset. The percentage signs (%) of accuracy values are +omitted. The optimal and sub-optimal results are marked in bold and underline respectively (same for the following tables). +Test-A and Test-U denote the abbreviations of the metrics Test-All and Test-Unseen respectively. +Model +v1 +v2 +v3 +v4 +v5 +v6 +Test-A +Test-U +Test-A +Test-U +Test-A +Test-U +Test-A +Test-U +Test-A +Test-U +Test-A +Test-U +BERT-Large +38.00 +34.36 +42.00 +33.39 +37.50 +31.61 +38.00 +33.26 +29.60 +28.02 +28.80 +32.24 +RoBERTa-Large +47.70 +39.47 +50.60 +39.90 +46.10 +40.58 +50.40 +42.45 +53.00 +43.66 +49.90 +50.92 +DAGN +49.20 +41.37 +52.70 +43.56 +49.60 +39.73 +52.50 +44.51 +52.40 +42.63 +48.50 +49.15 +LReasoner +46.90 +40.60 +50.20 +43.49 +48.40 +42.76 +49.20 +44.12 +51.90 +42.02 +46.30 +44.93 +Logiformer +43.50 +39.31 +54.80 +46.30 +48.80 +42.24 +52.10 +44.85 +52.10 +40.88 +51.50 +51.44 +TaCo +52.20 +47.51 +55.80 +48.79 +52.20 +44.26 +54.70 +49.89 +56.00 +46.67 +54.70 +55.17 +TABLE 5: Ablation Studies on zero-shot splits. The drops on the performances are marked in red. The darker ones represent +the greater decline in values. +Model +v1 +v2 +v3 +v4 +v5 +v6 +Test-A +Test-U +Test-A +Test-U +Test-A +Test-U +Test-A +Test-U +Test-A +Test-U +Test-A +Test-U +TaCo +52.20 +47.51 +55.80 +48.79 +52.20 +44.26 +54.70 +49.89 +56.00 +46.67 +54.70 +55.17 +a) Input Reconstruction +w/o type prefix +51.10 +42.30 +55.30 +44.01 +49.90 +41.08 +53.80 +45.60 +53.60 +42.04 +53.50 +53.67 +∆ +-1.10 +-5.21 +-0.50 +-4.78 +-2.30 +-3.18 +-0.90 +-4.39 +-2.40 +-4.63 +-1.20 +-1.50 +w/o input reconstruction +50.20 +42.17 +55.30 +45.39 +51.20 +42.70 +53.60 +46.62 +54.40 +43.74 +52.30 +52.14 +∆ +-2.00 +-5.34 +-0.30 +-3.40 +-1.00 +-1.56 +-1.10 +-3.27 +-1.60 +-2.93 +-2.40 +-3.03 +b) Graph Construction & Reasoning +w/o graph reasoning +48.70 +42.28 +54.10 +42.19 +50.60 +41.99 +52.70 +45.55 +53.10 +44.25 +50.90 +50.63 +∆ +-3.50 +-5.23 +-1.70 +-6.60 +-1.60 +-2.27 +-2.00 +-4.34 +-2.90 +-2.42 +-3.80 +-4.54 +w/o global node +51.50 +46.96 +53.70 +44.78 +51.40 +43.51 +53.50 +46.16 +55.10 +46.57 +53.50 +52.53 +∆ +-0.70 +-0.55 +-2.10 +-4.01 +-0.80 +-0.75 +-1.20 +-3.73 +-0.90 +-0.10 +-1.20 +-2.64 +c) Type-aware Contrastive Learning +w/o type contrast +51.20 +42.70 +55.00 +44.62 +50.90 +43.96 +54.40 +46.19 +55.70 +45.89 +53.80 +53.94 +∆ +-1.00 +-4.81 +-0.80 +-4.17 +-1.30 +-0.30 +-0.30 +-3.70 +-0.30 +-0.78 +-0.90 +-1.23 +perception all have positive effects on the model perfor- +mances, especially on the unseen parts of the test split. +5.6 +Parameter Analysis +Further, we present the visualization of different selections +for hyper-parameters. All the hyper-parameters are selected +in the full-data setting, since the zero-shot splits are diverse +and hard to unify. The first one is the number of layers and +the number of heads in the graph transformer network, +shown in Fig. 5. To make a comprehensive comparison, +we search both the head number and the layer number +within {3,4,5,6}. As can be seen from the heatmap, the darker +color represents the higher performance. TaCo reaches the +optimal performance when the head number is 6 and the +layer number is 4. +In addition, we present the model performances un- +der different margins γ. The search scope of γ is within +{8,10,12,14,16}. And we select the optimal performance on the +test split. The optimal value of γ is 12. +6 +ANALYSIS OF MODEL GENERALIZATION +In this section, we will discuss the generalization capability +of the proposed model. Since TaCo has achieved superior +performances on the zero-shot setting, we further extend it +to the full-data setting. Experiments are conducted on two +mainstream logical reasoning dataset, ReClor and LogiQA. +In Fig. 6, we present the comparison results. +6.1 +Generalization on Full-data Setting of ReClor +Compared with all the current methods, TaCo achieves +competitive results. In comparison with the SOTA model +Logiformer, TaCo is better on the metric of Dev and Test-E +with 0.60% and 2.27% gains. And it is only slightly lower +TABLE 6: Experimental results on ReClor with the full-data +setting. Test-E and Test-H denote the easy part and the hard +part of the ReClor test split respectively. +Model +ReClor +Dev +Test +Test-E +Test-H +Random +25.00 +25.00 +25.00 +25.00 +Human Performance [13] +- +63.00 +57.10 +67.20 +BERT-Large [13] +53.80 +49.80 +72.00 +32.30 +XLNet-Large [13] +62.00 +56.00 +75.70 +40.50 +RoBERTa-Large [13] +62.60 +55.60 +75.50 +40.00 +DAGN [7] +65.80 +58.30 +75.91 +44.46 +FocalReasoner [8] +66.80 +58.90 +77.05 +44.64 +LReasoner [11] +66.20 +62.40 +81.40 +47.50 +MERIt [12] +66.80 +59.60 +78.10 +45.20 +AdaLoGN [10] +65.20 +60.20 +79.32 +45.18 +Logiformer [9] +68.40 +63.50 +79.09 +51.25 +TaCo (Ours) +69.00 +63.30 +81.36 +49.11 +than SOTA with 0.20% in the test split. We argue that in the +full-data setting, the ideal distribution of types leads to the +weakening of the type modeling. Combined with previous +experiments on the zero-shot setting, though Logiformer +does good at over-fitting the full-data setting, it loses +some generalization capability. From this perspective, TaCo +also shows great superiority on the model generalization, +performing consistently for both settings. +Further we conduct the ablation studies to analyze the +effectiveness of the proposed modules in the full-data setting. +From the results, the graph reasoning contributes most to the +performance on the test split. Also, we only witness 0.40% +improvements for the consideration of type-aware margin +loss. Since in the full-data setting, the train and test splits +share the same distributions, the effects of the reasoning +types are reduced. Therefore, the slight gain of the contrastive +learning is reasonable. In all, the generalization capability of + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +10 +TABLE 7: Ablation Studies on the full-data setting. The drops on the performances are marked in red. The darker ones +represent the greater decline in values. +Model +ReClor +Valid +∆ +Test +∆ +Test-E +Test-H +TaCo (Ours) +69.00 +- +63.30 +- +81.36 +49.11 +a) Heuristic Input Reconstruction +w/o type prefix +67.60 +-1.40 +61.30 +-2.00 +80.45 +46.25 +w/o input reconstruction +67.80 +-1.20 +61.10 +-2.20 +79.09 +51.25 +b) Graph Construction & Reasoning +w/o graph reasoning +64.40 +-4.60 +59.30 +-4.00 +79.55 +43.39 +w/o global node +65.60 +-3.40 +61.90 +-1.40 +81.82 +46.25 +c) Type-aware Contrastive Learning +w/o type contrast +67.80 +-1.20 +62.90 +-0.40 +79.55 +49.82 +� +� +� +� +������� +� +� +� +� +������ +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +�� +�� +�� +�� +(a) The performance on the validation split. +� +� +� +� +������� +� +� +� +� +������ +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +����� +�� +�� +�� +�� +(b) The performance on the test split. +Fig. 5: The performances on the different margin γ. +TaCo is well verified. +6.2 +Generalization on Other Dataset +To discuss the generalization capability of TaCo, we further +conduct some experiments on another logical reasoning +dataset LogiQA [16]. Following the method utilized for +ReClor, we also label each example of LogiQA with one +of the 17 reasoning types, and consider the zero-shot setting. +We apply the above mentioned strategies to obtain three +zero-shot splits (i.e., z1, z2 and z3) for LogiQA. +In comparison, we take some previous logical reasoning +models into account, including two typical language model +� +�� +�� +�� +�� +γ +�� +�� +�� +�� +�� +��������������������� +(a) The performance on the validation split. +� +�� +�� +�� +�� +γ +�� +�� +�� +�� +�� +�� +�������������������� +(b) The performance on the test split. +Fig. 6: The performances on the different margin γ. +TABLE 8: Experiments results on the zero-shot splits of the +LogiQA dataset. +Model +z1 +z2 +z3 +Test-A +Test-U +Test-A +Test-U +Test-A +Test-U +BERT-L +26.26 +28.64 +27.80 +27.06 +28.26 +27.14 +RoBERTa-L +27.65 +33.98 +29.49 +31.76 +28.73 +29.43 +DAGN +34.41 +40.78 +35.48 +34.12 +33.79 +32.00 +Logiformer +35.33 +36.41 +33.18 +35.29 +34.56 +33.43 +TaCo (Ours) +35.58 +37.38 +36.76 +35.29 +36.41 +35.14 +BERT-Large and RoBERTa-Large, the graph-based method +DAGN and previous SOTA model Logiformer. We rerun the +zero-shot setting for each baseline. +From the results shown in TABLE 8, TaCo shows its supe- +riority in most cases, only except the Test-Unseen metric of z1 +split. Specifically, compared with previous SOTA Logiformer, +TaCo achieves an average gain of 1.89% and 0.89% on the Test- +All and Test-Unseen respectively. In general, experiments on +the LogiQA dataset prove the good generalization capability +of TaCo, which satisfies our expectations. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +11 +(a) A successful case. +(b) A failure case. +Fig. 7: Case Study. +7 +CASE STUDY +In this section, we discuss the interpretability of TaCo for +the perception of reasoning types. Fig. 7 shows a successful +case and a failure case. For each case study, we present the +visualization of the type perception via dimension reduction +and make the comparison with the SOTA model Logiformer. +For the successful case, TaCo can well distance the ground +truth type Implication with others and thus make the correct +prediction. In this case, Logiformer which lacks the type +perception fails. For the failure one, TaCo obviously struggles +at distinguishing Necessary assumption from other reasoning +types. It leads to the wrong prediction, which is in same with +Logiformer. It encourages us to explore the deeper modeling +of types from the graph construction. In all, the type-aware +contrastive learning is helpful to the answer prediction and +provides the interpretability of the model. +8 +CONCLUSION AND FUTURE WORK +To study the zero-shot capability of the logical reasoning +models, we propose the first benchmark for the generalized +zero-shot logical reasoning, named ZsLR. 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Ba, “Adam: A method for stochastic optimiza- +tion,” arXiv preprint arXiv:1412.6980, 2014. + diff --git a/fdE1T4oBgHgl3EQfMAND/content/tmp_files/load_file.txt b/fdE1T4oBgHgl3EQfMAND/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6ebcb2c6e8c4e2ab7cae7460db63170216da3ca --- /dev/null +++ b/fdE1T4oBgHgl3EQfMAND/content/tmp_files/load_file.txt @@ -0,0 +1,1240 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf,len=1239 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 1 Mind Reasoning Manners: Enhancing Type Perception for Generalized Zero-shot Logical Reasoning over Text Fangzhi Xu, Jun Liu, Senior Member, IEEE, Qika Lin, Tianzhe Zhao, Jian Zhang, and Lingling Zhang Abstract—Logical reasoning task involves diverse types of complex reasoning over text, based on the form of multiple-choice question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Given the context, question and a set of options as the input, previous methods achieve superior performances on the full-data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, the current benchmark dataset has the ideal assumption that the reasoning type distribution on the train split is close to the test split, which is inconsistent with many real application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To address it, there remain two problems to be studied: (1) How is the zero-shot capability of the models (train on seen types and test on unseen types)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (2) How to enhance the perception of reasoning types for the models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For problem 1, we propose a new benchmark for generalized zero-shot logical reasoning, named ZsLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It includes six splits based on the three type sampling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For problem 2, a type-aware model TaCo is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It utilizes both the heuristic input reconstruction and the contrastive learning to improve the type perception in the global representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Extensive experiments on both the zero-shot and full-data settings prove the superiority of TaCo over the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, we experiment and verify the generalization capability of TaCo on other logical reasoning dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Index Terms—Natural Language Processing, Logical Reasoning, Question Answering, Generalized Zero-shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 1 INTRODUCTION Logical reasoning over text has aroused wide interest in the area of Machine Reading Comprehension (MRC) [1] and Natural Language Processing (NLP) [2] [3] recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In the form of the traditional multiple-choice question answering (MCQA) [4] [5] [6], the task of logical reasoning requires the model to perform complex reasoning and generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' One of the main difficulties of the task lies in addressing diverse reasoning types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 1 shows some examples of reasoning types in the logical reasoning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Given questions with different reasoning types, humans tend to focus on the respective aspects of interactions between the context and the option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For instance, for the type Identify the flaw (a), the option is strongly related to the detailed logic flaws within the global idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' While for the type Necessary assumption (b), the focus may be switched to the premise of the arguments and detect the missing assumption reflected by the option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, for the reasoning type of Parallel reasoning (c), it is required to consider the corresponding logical structure of the context and the option, rather than the specific entities or events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Therefore, the modeling of the specific reasoning type is intuitive and necessary to the logical reasoning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Recent works have witnessed improvements in the logical reasoning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Generally, they can be categorized into two folds: graph-based and data-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In the graph-based family, DAGN [7], FocalReasoner [8] and Logiformer [9] attempt to construct the context graphs from different levels, Fangzhi Xu, Qika Lin, Tianzhe Zhao, Jian Zhang are with School of Computer Science and Technology, Xi’an Jiaotong University, Xi’an, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Jun Liu and Lingling Zhang are with Shaanxi Province Key Laboratory of Satellite and Terrestrial Network Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' R&D, National Engineering lab for Big Data Analytics, Xi’an, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The corresponding author is Jun Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 1: Examples of different reasoning types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' such as causal and co-occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' And AdaLoGN [10] proposes a neural-symbolic system in an adaptive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In the data-based family, previous works explore various data augmentation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For example, LReasoner [11] extracts the symbols from the text and extends them with logical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' MERIt [12] designs several data generation arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='02983v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='AI] 8 Jan 2023 Context logic flam Option If and hence logic flaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The argument fails , though However, Thus, if B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' logic flaw Question Reasoning Type Which one of the following most accurately Identify the flam describes a flaw in the reasoning of the argument ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (a) An example of Identify the flam Option Context misssing assumption If then A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' If Only if Therefore, In all, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Reasoning Type Question Which one of the following is an assumption Necessary assumption (b) An example of Necessary assumption Context Option macth logics, ignore facts , only(if Al (f) because will hot) However (unless f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (lf) And(if) Similar .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (If) then Therefore Logic 5 Thus Question Reasoning Type The pattern of reasoning displayed above most Parallel reasoning closely parallels which of the following ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (c) An example of Parallel reasoningJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 2 methods to facilitate the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, all of the methods in the two families lack the modeling of the reasoning type features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Although the above mentioned methods achieve superior results over strong baselines, the current researches are based on the ideal setting that train and test splits share the similar distribution of reasoning types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Nevertheless, in the real application, we are mainly exposed to the common types of questions while unfamiliar with the reasoning on the novel and uncommon types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In another word, there is an obvious gap between the ideal setting and the real scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To address this issue, there exist two problems to be studied: 1) What about the test performance on the unseen types during training and how is the zero-shot capability of the models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 2) How to enhance the perception of reasoning types for the models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' As for Problem 1, we propose a new benchmark for the generalized Zero-shot Logical Reasoning, named ZsLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Based on the ReClor dataset [13] with 17 reasoning types, we form 6 zero-shot data splits according to 3 strategies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', amount, randomness, and difficulty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To make comprehen- sive assessments, we introduce the generalized zero-shot setting [14], which test on both seen types and unseen types with two defined metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The necessity and meaning of ZsLR is verified by the pilot experiments and it encourages more future works to mind reasoning manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' As for Problem 2, we propose a Type-aware reasoning network based on Contrastive learning, named TaCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' First, we design a keyword-based extractor to output the reasoning type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Then, through the two designed heuristic strategies, we merge each question and the option into a unified Q-A pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Based on the co-occurrence node extraction algorithm proposed in Logiformer [9], we form the topology within context part and Q-A pair part respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To model the interaction of the context and Q-A pair for different reasoning types, we add a global node to connect all the nodes of the two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Through the self-attention aggregation, the final representation of the global node is obtained and utilized to predict the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Meanwhile, we employ the sentence-BERT [15] to obtain type embeddings based on their descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The margin loss is applied to model the reasoning type, where the global node serves as the anchor, ground truth type as positive example and other types as negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In this way, the global node representation contains both the context and the reasoning type semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Therefore, the perception of the reasoning types can facilitate the zero-shot capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In all, the main contributions of this paper are summa- rized as follows: (1) We are the first to focus on the issue of type-oriented reasoning manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To address the issue, we propose the first benchmark: generalized ZsLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1 It can well reflect the real scenarios and test the zero-shot capabilities of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (2) We propose to tackle the zero-shot task through the heuristic input reconstruction and type-aware contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The proposed model TaCo can function as a strong baseline for the future works on the task of ZsLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (3) Extensive experiments on both zero-shot and full-data settings prove the huge potential of the proposed issue, as 1The ZsLR dataset and implementation of TaCo are public at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' com/xufangzhi/TaCo well as the superiority of TaCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, we conduct additional experiments to further verify the generalization capability of TaCo on other setting and dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1 Logical Reasoning The logical reasoning task, which aims at testing the rea- soning capability of models, has aroused wide interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Several representative datasets have been proposed, such as ReClor [13] and LogiQA [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Current methods on the logical reasoning task can be divided into graph-based methods and data-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The former focuses on the construction of the text graphs and leverage the node connection to model the logical relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Among this category, DAGN [7] is the first work to split the text into EDUs and perform the reasoning with graph neural networks [17] [18], but it only builds a chain-type graph and ignores the long distance interaction between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' FocalReasoner [8] proposes to attend to the fact triplets [19] within the text and constructs a supergraph based on the extracted triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, it lacks the modeling of the logical information in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To fully explore the logic, AdaLoGN [10] is proposed to construct the adaptive neural-symbolic system to improve the performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, its reasoning process is complex and costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Considering the above drawbacks, Logiformer [9] stresses much importance on both the causal relations and the co-occurrence relations in a two-branch graph transformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Yet it still lacks the perception of different question types, which limits the zero-shot reasoning capability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The latter one is the data-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' These works aim to improve the performance through the data aug- mentation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' One of the representatives is LReasoner [11], which extends the symbolic expressions by logical rules and templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In addition, MERIt [12] designs a meta-path- guided contrastive learning method to facilitate the training process, by utilizing the extra data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, both of them fail to model the reasoning type of each question and lack the application value in zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='2 Current Machine Reading Comprehension Settings Current machine reading comprehension (MRC) tasks [20] can be categorized into two settings: full-data setting and low-resource (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', few-shot, zero-shot) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Full-data setting has aroused wide concerns in the area of MRC in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Popular benchmark SQuAD [21] [22], which is sourced from from Wikipedia articles, contains over 100,000 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It forms an abundant database for the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Similarly, HotpotQA [23] includes 113K questions with a variety of reasoning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It stresses more on the multi-hop reasoning abilities of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' RACE dataset [24] is specially designed to improve the reading skills for both middle school and high school students, which contains about 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='7K examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, some domain-specific datasets like NewsQA [25], TextbookQA [26], PubMedQA [27], aim at the area of journalism, education and biology respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' All of them have rich training examples to improve the model capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, such full-data setting encourages the models to depend on the ideal scenes with abundant training data, which may not fit in some real cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 3 Based on such concern, some previous works focus on the low-resource setting MRC, including few-shot or zero- shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For example, to evaluate the robustness of the model, [28] reconstructs the data from knowledge sources for zero-shot commonsense QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, [29] proposes a neuro- symbolic approach to boost the performance of zero-shot commonsense QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Both of the methods are not exposed to the training data, forming a zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, they rely on the extra knowledge bases, which limits their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' There are also some works related to the few- shot QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In [30], a new pretraining strategy is proposed to explore the realistic few-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, [31] tackles the few-shot challenge on the task of Visual Question Answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, all of the above methods simply obtain the few- shot splits according to the amount, which treat all the samples equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To sum up, current MRC benchmarks mainly focus on the common types of questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, in reality, it is more challenging when faced with some uncommon types of questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To make up for these drawbacks, we attend to the reasoning types in the logical reasoning tasks and propose the first benchmark for zero-shot logical reasoning based on type attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 3 THE BENCHMARK OF GENERALIZED ZSLR In this section, we introduce the generalized ZsLR bench- mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' At the very beginning, we obtain the statistical distribution of the number of reasoning types on the train, development and test splits, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We arrange them in descending order of the number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The distributions are in the similar form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It demonstrates that the full-data setting is based on such an ideal assumption, which is insufficient to verify the zero-shot generalization capability of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Therefore, we propose the benchmark of generalized ZsLR and conduct the pilot experiments to verify the necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1 Zero-shot Data Construction The zero-shot logical reasoning datasets are split based on ReClor [13] without shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Considering the situation in reality, it is easier to learn to reason on common types of samples while struggle with the rare ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To this end, we design three sampling strategies, namely amount, randomness and difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For amount, we select top-k reasoning types as the seen types, merely by the amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It can be seen as a simple implementation to filter the uncommon types of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For randomness, we arrange the reasoning types in de- scending order of amount and select the seen ones based on the geometric distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The discrete form of the geometric distribution is, P (X = k) = (1 − p)k−1p, (1) where k is the sorted index of the reasoning type, p is the hyper-parameter set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1 in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In this random setting, the type with more training samples has a higher probability to be selected, which is in parallel with the real situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' �� � � � � � � �� � �� � �� �� � �� �� � ���������� � � � � � �� �� ��������� ����� ��� ���� (a) The distribution of the occurrence frequency of different reasoning types in the ReClor dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (b) Pilot comparisons between full and zero-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 2: The visualization of the pilot experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For difficulty, we first rank the difficulty of the reasoning types, based on their performance with RoBERTa-Large single model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' On one split, we select some of the most difficult reasoning types as the seen ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' On the other split, we select a part of the easiest types as the seen ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In total, 6 zero-shot splits are obtained (2 for each strategy) for ReClor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The details of the splits are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='2 Generalized Zero-shot Setting We propose the zero-shot setting for the logical reasoning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Given the context Ci, question Qi (type(Qi) ∈ T ) and option set Ai of the ith question, the model is required to predict the correct answer a ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In the definition, type(Qi) denotes the reasoning type of the question Qi and T is the type set of the zero-shot splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For each zero-shot split, a part of the types are sampled as the seen types, while others are viewed as the unseen ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Only the questions of seen types exist during training and they consist of the training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The type set seen during the training stage and the set for the test stage are T train and T test respectively, thus they satisfy T train ∪ T test = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To be closer to the real scenes, we consider the generalized zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' That is to say, the test scope is not limited to the unseen types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To this end, we employ two metrics for the generalized zero-shot setting, Test-All and Test-Unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The former one denotes the exact match results on the full- data test split, which contains both seen and unseen types, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', T train ⊆ T test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The latter one is the exact match results only on the unseen types of the test split, which ignores the performance on the seen types, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', T train ∩ T test = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In this way, we expect to achieve comprehensive assessments of the model capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 60 Test-All (Full) Test-Unseen (Full) Test-All (Zs) Test-Unseen (Zs) 55 Performance 45 40 35 30 v3 V V4 Zero-shot splitsJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 4 TABLE 1: Details of the split datasets for the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The third column presents the index of the seen types and the fourth column shows the number of the seen and unseen types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The last two columns present the number of training samples and test samples respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Especially, test samples contain both seen and unseen types for generalized zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Strategy Split Seen Type # Type Seen (Unseen) # Train Seen # Test Seen (Unseen) Amount v1 {0,3,4,8,13} 5 (12) 2,190 475 (525) v2 {0,1,2,3,8,9,14,16} 8 (9) 2,700 595 (405) Randomness v3 {0,2,3,13} 4 (13) 1,928 435 (565) v4 {0,2,3,5,7,8,13} 7 (10) 2,896 645 (355) Difficulty v5 {0,2,4,6,8,13,15} 7 (10) 2,175 473 (527) v6 {1,3,5,7,9,10,11,12,14,16} 10 (7) 2,463 527 (473) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='3 Pilot Experiments Also, we are going to verify the necessity of the zero-shot setting from the pilot experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For comparison with each split, we randomly sample the same amount of training examples on both the seen and unseen types, forming the comparison group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Take the zero-shot split v1 as an example, it has 2, 190 seen samples for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Thus, its comparison training group also includes 2, 190 examples, but are distributed over all types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The purpose of the pilot experiments is to observe the performance changes of the two metrics on the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In the implementation, we utilize the RoBERTa single model [32] to conduct the reasoning and maintain the same hyper-parameters of the zero-shot splits with comparison groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Especially, to avoid the noise brought by the random sampling, we conduct the comparison experiments five times with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Final results are the average values of the five experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We select the results on four of the split versions for illustration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', v1-v4), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The first and the third column in blue are the performance of the comparison group, on the full-data test and the unseen types of test respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The second and the fourth column represent the performance on the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' With the same number of training examples, training only on seen types (zero- shot setting) witnesses obvious drops on the performance, especially on the unseen types of the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Such observations are consistent among all the zero-shot splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Zero-shot pilot experiments based on reasoning types uncover the obvious drawback of the current full-data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In another word, it verifies the necessity to propose a new benchmark for the generalized ZsLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 4 METHODS In this section, we introduce the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To tackle the zero-shot challenges, we propose a model named TaCo, which focuses on the reasoning type perception in the logical reasoning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The architecture of TaCo is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It mainly consists of three parts: (a) heuristic reconstruction to acquire the type-aware input sequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (b) text graph construction and reasoning for the QA problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (c) type- aware contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1 Heuristic Input Reconstruction One of the common practices for MCQA problems is to concatenate three sequences as inputs: context, question and option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' But it is insufficient for the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' There exist two main drawbacks: 1) the modeling of the reasoning type is implicit in the sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 2) it is difficult to bridge the interaction between context and options via the question sequence in the middle since it is not natural for the language model (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To this end, this paper introduces the heuristic reconstruction to the inputs based on the type extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To address the first drawback, we are required to label the reasoning type of each question based on the limited inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Inspired by LReasoner [11], we propose a simple but effective type extractor through keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The procedure of the heuristic type extractor is presented in the pseudo-code of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Since the ReClor dataset only makes the reasoning types on the test split public, we are required to collect the classification method from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Before the extraction, we conclude the keywords and phrases for each reasoning type, forming the keywords base B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, we include the maximum window size W and the question sentence as the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In Line 1, we first split the question into words and form the word sequence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Then we start the iteration in the descending order of the window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We slide the window over the question sequence to obtain the sub-sequence (Line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Then we match the sub-sequence C with keywords base Bk for each reasoning type k to derive the number of exact matches (Line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' After each window size iteration, we judge the exit condition (Line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' If there exists a unique type with the maximum number of matches, we label the type as the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Meanwhile, if we can not extract the type at the last round of iteration, we label the instance as Others type (Line 16, 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Then, we convert the type index of each question into the natural language (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', Implication, Conclusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Meanwhile, to equip the LM with the type information, we add a type- related prefix at the beginning of the sequence: prefix = This is the task of [R-Type] (2) where [R-Type] denotes the natural language label of the specific type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Thus, the type semantic is merged into the inputs in an explicit manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For the second drawback, one intuitive idea is to re- construct the inputs of question and option sequences, transforming them into a single sequence in a declarative form, defined as Q-A pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In another word, we are required to fill the option into the proper position in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' According to our observation, the question sentence is led by a trigger span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For example, in the question Which one of the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 3: The architecture of TaCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Taken the three basic parts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', context, question and option) as the inputs, we reconstruct them with the heuristic strategy (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The text graph is built in module (b1) and the reasoning based on the graph transformer is conducted in module (b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The updated representation of the global node is employed to perform the final inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Meanwhile in module (c), the contrastive learning is introduced to improve the awareness of the reasoning type in the global node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Algorithm 1: Reasoning Type Extraction Input: keywords base B, window size W, question sentence q Output: reasoning type T 1 S ← SplitByWords (q) 2 for i = W, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', 2, 1 do 3 Count ← {} 4 for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', length(S)− W do 5 C ← S[j : W] 6 for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', 16 do 7 num ← NumOfMatch(C, Bk) 8 Count[k] += num 9 end 10 end 11 if max(Count) > 0 and is unique then 12 T ← type of the max match count 13 return reasoning type T 14 end 15 if i == 1 then /* classify to ‘Others’ / 16 T ← ‘Others′ 17 return reasoning type T 18 end 19 end following most weakens the arguments above ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', the span Which one of the following serves as the trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Then, we can replace the trigger span with the option sequence to obtain the Q-A pair, formulated as [Option] most weakens the arguments above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Therefore, the core of the heuristic strategy is to pinpoint the precise position of the triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Although the trigger span is similar to the basic form of Which of the following, it is not always the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To further improve the diversity of the trigger base, we propose two heuristic strategies: Combination: We predefine a set of basic words (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', {which, one, of, the, following}), and then randomly combine all or parts of them to form the new triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For instances, of which the following and which of following are two newly constructed triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Tolerance: More trigger spans are not limited to the combination of the predefined words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To this end, we propose a tolerance strategy to contain the extra words within the trigger span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For example, in the trigger of the following claims, which, we include the extra word claims to the span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In this way, the trigger base can be expanded to adapt to the complex situations in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' So far, the declarative Q-A pair is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Concatenating the context sequence Ci and Q-A pair sequence Pi of the ith question, we send it into the pre- trained LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In this paper, we employ the RoBERTa-Large model [32] to obtain the token-level representations as follows, �h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' hc0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' hp0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' h � = RoBERTa (< s > c0 · · · p0 · · · < /s >) , (3) where the tokens {c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' , c|Ci|} make up Ci, and {p0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' , p|Pi|} make up Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The representation of Ci, Pi and the whole sequence Si (concatenation of prefix, Ci and Pi) can be obtained through the mean pooling strategy, hCi = 1 |Ci| |Ci| � k=1 hck, hPi = 1 |Pi| |Pi| � k=1 hpk, hSi = 1 |Si| |Si| � k=1 hsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (4) (a) Heuristic Input (b1) Graph Construction Reconstruction (b2) Graph Reasoning Extract Context Context Ques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='initialize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Concat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Heuristic Extractor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Node( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Classify ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='R-Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='PrefixP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Pos (oD C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Adjacent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Global ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Q-A Pair ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='PContext ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Relations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Transformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Extract ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='d人 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Q-A Pair ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='MO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Text Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Updated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Global Node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='(c) Type-aware Contrastive Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Anchor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='R-Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Positive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='push ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='pull ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='farther ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='(Ground-truth) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='con ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='one sentence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='for each type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Positive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Negative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='Negative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='(Other Types)JOURNAL OF LATEX CLASS FILES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 4: An example of the construction of context subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The input context can be split into 6 units (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', U1-U6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Based on the order relation and overlap relation, the topology of the subgraph can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='2 Text Graph Construction and Reasoning According to the previous analysis, the context and Q-A pair may interact differently under different reasoning types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Therefore, we consider to build the topology structure of the two parts respectively and finally learn their interactive semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For clearer illustration, we take the context part as an example and present the graph construction process in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Firstly, we split the text sequence into units (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', U1- U6 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 4) based on the predefined explicit connectives or punctuation (marked in blue in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' These text units function as the nodes during the graph reasoning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Therefore, two node sets for the context and Q-A pair parts are obtained respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To form the dual subgraphs of the input sequence, we are also required to define the edge relations, including a) order relation and b) overlap relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The former one models the original text order, where nodes are connected in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It forms a chain-type structure which maintains the position perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The latter one models the similarity between nodes, where nodes with more vocabulary overlaps or similar semantics will be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In the implementation, we will transform the text units into word sets, where the stopwords are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Through the calculation of the overlap ratio between sets, we connect the node pairs that exceed the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In this way, the independent topology of the context part and the Q-A pair part are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To make the interaction of the two parts learnable, we add a global node to link all the extracted nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The global node is initialized with hSi and aggregates the two-way information flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Therefore, this node representation is expected to contain the type semantic to improve the zero-shot performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Further, to overcome over-smoothing in graph neural networks [33] and fully conduct node interactions, we employ the Graph Transformer Network [9], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Let N be the node set in one graph, with the number of |N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We feed the node sequence into the Transformer architecture [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For each layer l, the updated information ej of the node Nj is formulated as follows: e(l) j = σ(W(l−1) self e(l−1) j + � Nk∈N \\{Nj} αj,ke(l−1) k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (5) where N\\{Nj} denotes the set without jth node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The core of the equation is the computation for the weighted attention αj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It is obtained by the two matrices query Q and key K: A = softmax(QKT √ d + M), (6) where αj,k is one of the elements in the matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' d is the dimension of the hidden states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' M ∈ R|N |×|N | is the attention bias matrix to encode the structural semantic of the whole graph, which is the adjacent matrix of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Thus, the final feature of the global node hg is updated through the graph reasoning network, containing the global semantic of the text sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Before predicting the correct answer, we concatenate h(j) g , h(j) Ci , h(j) Pi and h(j) Si to obtain the score of the jth option followed by softmax: z(j) i = softmax � W · � h(j) g ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' h(j) Ci ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' h(j) Pi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' h(j) Si �� , (7) where W ∈ R4d×1 denotes the linear projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Same with most of the MCQA methods, we adopt the cross-entropy loss function for the optimization: LMC = CrossEntropy (zi, yi) , (8) where zi denotes the stack of each option score and yi is the ground-truth label of the ith example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='3 Type-aware Contrastive Learning In the upper part of the architecture, the interactive semantic of the context and Q-A pair is learned through one global node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Besides the implicit awareness of the type, we also expect the model to distinguish the ground-truth type from the negative ones for zero-shot logical reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Based on the heuristic extractor mentioned above, the reasoning type of each example can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For each reasoning type, the detailed description in the form of natural language is utilized (listed in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Feeding all 17 reasoning type descriptions into the LM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', Sentence-BERT [15]), we obtain the sentence-level embeddings, which are fixed during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For each example, we propose to attend to the type infor- mation through contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The final representation of the global node hg serves as the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The ground- truth type functions as the positive sample, while others are negative samples, with the embedding of hgt and hn (n is the index of the negative samples) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Our purpose is to close the distance between the global node hg and the positive sample hgt, while distancing the negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Context [U1 Paula will visit the dentist tomorrow morning only if [U2] Bill goes golfing in the morning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' [U3i Bill will not go golfing unless [U4l Damien agrees to go golfing too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' However, [U5] Damien has decided not to go golfing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Therefore, IU6l Paula will not be visiting the dentist tomorrow morning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Subgraph Construction a) order relation b) overlap relation U4 U4 U2 U2 J5 U5 U3 U3 U6 U6 U4 U2 05 U3 U6 Context SubgraphJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 7 TABLE 2: The defined index and descriptions of 17 logical reasoning types in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Id Type Descriptions 0 Necessary Assumptions identify the claim that must be true or is required in order for the argument to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 1 Sufficient Assumptions identify a sufficient assumption, that is, an assumption that, if added to the argument, would make it logically valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 2 Strengthen identify information that would strengthen an argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 3 Weaken identify information that would weaken an argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 4 Evaluation identify information that would be useful to know to evaluate an argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5 Implication identify something that follows logically from a set of premises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 6 Conclusion and Main Point identify the conclusion/main point of a line of reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 7 Most Strongly Supported find the choice that is most strongly supported by a stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8 Explain or Resolve identify information that would explain or resolve a situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 9 Principle identify the principle, or find a situation that conforms to a principle, or match the principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 10 Dispute identify or infer an issue in dispute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 11 Technique identify the technique used in the reasoning of an argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 12 Role describe the individual role that a statement is playing in a larger argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 13 Identify a Flaw identify a flaw in an arguments reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14 Match Flaws find a choice containing an argument that exhibits the same flaws as the passages argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 15 Match the Structure match the structure of an argument in a choice to the structure of the argument in the passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 16 Others other types of questions which are not included by the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For simplicity, we model the score of two vectors using Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' That is: score+ = hg ⊙ hgt, score− n = hg ⊙ hn, (9) where score+ and score− n represent the score of the positive sample and the scores of the negative ones respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We employ the margin loss as the auxiliary optimization function for each example: Lcon = max{0, γ − score+ + max n (score− n )}, (10) where γ > 0 controls the difference between positive and negative scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We select the maximum score of the negative one for the loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To maximize the joint optimization performance of the two loss functions, we set a trade-off coefficient α and transform the final loss function into: L = LMC + αLcon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (11) In this way, the global semantic for reasoning reinforces mutually with the type-aware semantic, which benefits the zero-shot performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In addition, it provides the interpretability for the test process, which will distinguish the correct reasoning type from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5 MAIN EXPERIMENTS In this section, we will introduce the details of the experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, extensive experiments on both the zero-shot and full-data setting will be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1 Current Full-data Benchmark Currently, there exist two datasets in the logical reasoning task, ReClor [13] and LogiQA [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Since the LogiQA datasets contain only 5 reasoning types, we consider the ReClor dataset in the zero-shot experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' And we take both of them into account for the full-data setting to prove the generalization capability of TaCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The detailed information of the two datasets is presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' ReClor is sourced from some standardized graduate ad- mission examinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It consists of 6,138 examples in total, including 4,638 training samples, 500 validation samples, and 1,000 test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' As is listed in Table 2, ReClor contains 17 reasoning types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Especially, its test split is divided into two parts, which are Test-E and Test-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The former one is the easy splits, which can be addressed without context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The latter one is the hard splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' LogiQA is collected from National Civil Servants Exami- nations of China, including 8,678 samples in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Among them, 7,376 are training samples while validation and test splits both contain 651 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Different from the diverse reasoning types in ReClor, LogiQA contains only 5 reasoning types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='2 Baselines We include all the previous SOTA baselines in the main paper for comparison, as well as the results from some classical language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Random The results in this setting are based on the random predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' RoBERTa-Large [32] The results are obtained simply utiliz- ing the RoBERTa language model for the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It is similar to the baseline of BERT-Large [36] and XLNet-Large [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Human Performance [13] In ReClor, the average score of different graduate students on the test split is utilized as the human performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' DAGN [7] It is the first work to propose the construction of the text graph based on the extracted EDUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It mainly relies on the RoBERTa-Large [32] to encode the tokens and graph neural network [17] to update the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 8 TABLE 3: The tuned hyper-parameters with search scopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' ‘GT’ denotes the graph transformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Name of Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Search Scope Best # epoch {10,15,20,25,30} 30 # head in GT {3,4,5,6} 6 # layer in GT {3,4,5,6} 4 max sequence len {128,256} 256 learning rate {4e-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5e-6, 6e-6} 5e-6 margin γ {8,10,12,14} 12 trade-off α {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='5,1} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='2 FocalReasoner [8] It constructs a supergraph for reasoning, which consists of the fact units extracted from the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' LReasoner [11] It explores the context extension based on the defined logical rules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', De Morgan’s laws).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, it employs the data augmentation method to improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Since constructing more training data is proved to be of great help to the zero-shot performance of most of the current models, we do not consider the data augmentation strategy when reproducing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' MERIt [12] It proposes a meta-path-guided contrastive learning method to reason over the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The supervised pretraining is performed on the abundant unlabeled text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Due to the employment of the external data, we do not consider this method in the zero-shot baselines for the fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' AdaLoGN [10] It proposes a neuro-symbolic system to adap- tively update the text graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Additionally, a novel subgraph- to-node mechanism is utilized aggregate the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Logiformer [9] It introduces two different strategies to construct the logical graph and the syntax graph respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Through the two-branch graph transformer network, the text features are updated to conduct the reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='3 Implementation Details In this paper, all experiments run on a single GPU of Tesla A100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We employ RoBERTa-large model [32] as the text encoder for all previous methods for fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For each split of the zero-shot setting, we train on the seen types and select the best epoch on the seen types of the development set for the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We rerun other baselines on the zero-shot setting with the original configuration and maintain their reported results on the full-data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' As for our proposed model, the hyper-parameters keep the same for both settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The training epoch is set to 30 and the batchsize is fixed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For optimization, we take Adam [38], with the peak learning rate as 5e-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The number of heads and layers in the graph transformer is set to 6 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' γ in the margin loss is tuned as 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The loss trade-off α is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The selected five random seeds for the comparison groups are {42, 12, 23, 234, 1234}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Additionally, some important hyper- parameters are tuned for the best within a search scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The details are included in the Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='4 Main Results The model TaCo is designed to bridge the gaps of the zero- shot logical reasoning setting, thus we are going to evaluate its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We present the results of 6 zero-shot splits in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We rerun five strong baselines on the zero-shot setting, including (1) BERT-Large, (2) RoBERTa-Large, (3) graph-based representative method DAGN [7], (4) data- based representative model LReasoner [11], (5) SOTA model Logiformer [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It shows that TaCo outperforms these strong baselines with large margins on all the zero-shot splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Compared with the previous SOTA model Logiformer, the averaged improvements among all the splits are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='54% respectively for the two metrics Test-All and Test-Unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' And compared with all suboptimal results on the two metrics, TaCo still has great superiority of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='65% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To explore the respective roles of the two metrics, Test-Unseen witnesses the greater improvements among all the splits, especially 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='14% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='04% over the sub-optimal methods in split v1 and v4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It illustrates that TaCo performs better on tackling the unseen types of samples than the current methods, while maintaining the superior performances on the seen ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In addition, among all the zero-shot splits, the performances within each method vary greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For example, the previous SOTA Logiformer shows good performances in v2 but struggles a lot in v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It proves that the benchmark provides diverse distributions of the data splits and encourages the consistent performances of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' And considering the experiment results, TaCo can function as a strong baseline for the future work on the ZsLR benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='5 Ablation Studies To illustrate the effectiveness of each module of TaCo in the zero-shot setting, we conduct the following ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The results are listed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5 Detailedly, in part a), w/o type prefix and w/o reconstruction are related to the heuristic input reconstruction, where the former one ablates the type-oriented prefix, and the latter one maintains the simple concatenation of the question and option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Generally, the heuristic type prefix and input reconstruction bring more improvements on the metric of Test-Unseen with an average of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='95% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='26% respectively respectively on all six splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It shows that type-oriented prefix is more effective than input reconstruction in most cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', v2-v5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In part b), w/o graph reason ablates the whole module of graph construction and reasoning, and replaces the global node feature with the pooled output of the whole sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Since the reasoning process is the core of this task, it contributes most to the performance on both the metrics, with the average gains of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='58% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='23%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' While w/o global node simply replaces the global node feature with the pooled representation of all the other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It can be regarded as a small part of graph construction and reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' From the results, it works on the splits v2, v4 and v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' As the training set of these three splits are obviously larger, it can be concluded that global node is especially effective with more training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In part c), we ablate the type-aware contrastive learning module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' From the experiments, the modeling of reasoning types does great help to the unseen types of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In the unseen types of v1, v2 and v4, the margin loss brings significant improvements, with gains of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='81%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='17% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='70% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Overall, the three key modules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', type prefix, input reconstruction, type contrast) proposed to enhance the type JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 9 TABLE 4: Experimental results on 6 zero-shot splits of ReClor dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The percentage signs (%) of accuracy values are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The optimal and sub-optimal results are marked in bold and underline respectively (same for the following tables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Test-A and Test-U denote the abbreviations of the metrics Test-All and Test-Unseen respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Model v1 v2 v3 v4 v5 v6 Test-A Test-U Test-A Test-U Test-A Test-U Test-A Test-U Test-A Test-U Test-A Test-U BERT-Large 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='36 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='39 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='61 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='26 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='02 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='24 RoBERTa-Large 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='70 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='47 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='10 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='58 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='45 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='66 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='92 DAGN 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='37 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='70 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='56 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='73 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='51 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='63 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='15 LReasoner 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='49 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='76 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='12 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='02 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='93 Logiformer 43.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='85 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='10 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='88 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='44 TaCo 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='51 55.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='67 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='70 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='17 TABLE 5: Ablation Studies on zero-shot splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The drops on the performances are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The darker ones represent the greater decline in values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Model v1 v2 v3 v4 v5 v6 Test-A Test-U Test-A Test-U Test-A Test-U Test-A Test-U Test-A Test-U Test-A Test-U TaCo 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='51 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='79 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='26 54.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='01 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='08 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 53.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='54 w/o global node 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='96 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='70 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='78 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='51 53.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='64 c) Type-aware Contrastive Learning w/o type contrast 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='70 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='62 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='96 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='19 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='70 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='89 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='94 ∆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='23 perception all have positive effects on the model perfor- mances, especially on the unseen parts of the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='6 Parameter Analysis Further, we present the visualization of different selections for hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' All the hyper-parameters are selected in the full-data setting, since the zero-shot splits are diverse and hard to unify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The first one is the number of layers and the number of heads in the graph transformer network, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' To make a comprehensive comparison, we search both the head number and the layer number within {3,4,5,6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' As can be seen from the heatmap, the darker color represents the higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' TaCo reaches the optimal performance when the head number is 6 and the layer number is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In addition, we present the model performances un- der different margins γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The search scope of γ is within {8,10,12,14,16}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' And we select the optimal performance on the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The optimal value of γ is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 6 ANALYSIS OF MODEL GENERALIZATION In this section, we will discuss the generalization capability of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Since TaCo has achieved superior performances on the zero-shot setting, we further extend it to the full-data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Experiments are conducted on two mainstream logical reasoning dataset, ReClor and LogiQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 6, we present the comparison results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1 Generalization on Full-data Setting of ReClor Compared with all the current methods, TaCo achieves competitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In comparison with the SOTA model Logiformer, TaCo is better on the metric of Dev and Test-E with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='27% gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' And it is only slightly lower TABLE 6: Experimental results on ReClor with the full-data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Test-E and Test-H denote the easy part and the hard part of the ReClor test split respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Model ReClor Dev Test Test-E Test-H Random 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 Human Performance [13] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='10 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 BERT-Large [13] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 XLNet-Large [13] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='70 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50 RoBERTa-Large [13] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 DAGN [7] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='91 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='46 FocalReasoner [8] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='05 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='64 LReasoner [11] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50 MERIt [12] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='10 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 AdaLoGN [10] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='32 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='18 Logiformer [9] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='50 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='09 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='25 TaCo (Ours) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='36 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='11 than SOTA with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20% in the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We argue that in the full-data setting, the ideal distribution of types leads to the weakening of the type modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Combined with previous experiments on the zero-shot setting, though Logiformer does good at over-fitting the full-data setting, it loses some generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' From this perspective, TaCo also shows great superiority on the model generalization, performing consistently for both settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Further we conduct the ablation studies to analyze the effectiveness of the proposed modules in the full-data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' From the results, the graph reasoning contributes most to the performance on the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, we only witness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40% improvements for the consideration of type-aware margin loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Since in the full-data setting, the train and test splits share the same distributions, the effects of the reasoning types are reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Therefore, the slight gain of the contrastive learning is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In all, the generalization capability of JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 10 TABLE 7: Ablation Studies on the full-data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The drops on the performances are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The darker ones represent the greater decline in values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Model ReClor Valid ∆ Test ∆ Test-E Test-H TaCo (Ours) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='36 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='11 a) Heuristic Input Reconstruction w/o type prefix 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='45 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='25 w/o input reconstruction 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='09 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='25 b) Graph Construction & Reasoning w/o graph reasoning 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='55 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='39 w/o global node 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='82 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='25 c) Type-aware Contrastive Learning w/o type contrast 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='20 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='40 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='55 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='82 � � � � ������� � � � � ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� �� �� �� �� (a) The performance on the validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' � � � � ������� � � � � ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� �� �� �� �� (b) The performance on the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 5: The performances on the different margin γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' TaCo is well verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='2 Generalization on Other Dataset To discuss the generalization capability of TaCo, we further conduct some experiments on another logical reasoning dataset LogiQA [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Following the method utilized for ReClor, we also label each example of LogiQA with one of the 17 reasoning types, and consider the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We apply the above mentioned strategies to obtain three zero-shot splits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=', z1, z2 and z3) for LogiQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In comparison, we take some previous logical reasoning models into account, including two typical language model � �� �� �� �� γ �� �� �� �� �� ��������������������� (a) The performance on the validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' � �� �� �� �� γ �� �� �� �� �� �� �������������������� (b) The performance on the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 6: The performances on the different margin γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' TABLE 8: Experiments results on the zero-shot splits of the LogiQA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Model z1 z2 z3 Test-A Test-U Test-A Test-U Test-A Test-U BERT-L 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='26 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='64 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='80 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='06 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='26 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='14 RoBERTa-L 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='65 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='98 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='49 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='76 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='73 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='43 DAGN 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='41 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='78 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='48 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='12 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='79 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='00 Logiformer 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='33 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='41 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='18 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='29 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='56 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='43 TaCo (Ours) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='58 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='38 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='76 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='29 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='41 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='14 BERT-Large and RoBERTa-Large, the graph-based method DAGN and previous SOTA model Logiformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' We rerun the zero-shot setting for each baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' From the results shown in TABLE 8, TaCo shows its supe- riority in most cases, only except the Test-Unseen metric of z1 split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Specifically, compared with previous SOTA Logiformer, TaCo achieves an average gain of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='89% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='89% on the Test- All and Test-Unseen respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In general, experiments on the LogiQA dataset prove the good generalization capability of TaCo, which satisfies our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8, AUGUST 2015 11 (a) A successful case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' (b) A failure case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 7: Case Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 7 CASE STUDY In this section, we discuss the interpretability of TaCo for the perception of reasoning types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 7 shows a successful case and a failure case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For each case study, we present the visualization of the type perception via dimension reduction and make the comparison with the SOTA model Logiformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For the successful case, TaCo can well distance the ground truth type Implication with others and thus make the correct prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In this case, Logiformer which lacks the type perception fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' For the failure one, TaCo obviously struggles at distinguishing Necessary assumption from other reasoning types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It leads to the wrong prediction, which is in same with Logiformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It encourages us to explore the deeper modeling of types from the graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In all, the type-aware contrastive learning is helpful to the answer prediction and provides the interpretability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' 8 CONCLUSION AND FUTURE WORK To study the zero-shot capability of the logical reasoning models, we propose the first benchmark for the generalized zero-shot logical reasoning, named ZsLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' It includes six splits sampled with three strategies and two metrics to comprehen- sively evaluate the performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, we propose a model TaCo to enhance the reasoning type perception through the heuristic input reconstruction and the type-aware contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Also, we conduct extensive experiments on the zero-shot splits, full-data setting as well as other dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Superior results illustrate the effectiveness and generalization capability of the proposed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' In the future, we encourage more works to mind rea- soning manners 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Zeng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Hu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Hu, “A survey on machine reading comprehension—tasks, evaluation metrics and benchmark datasets,” Applied Sciences, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Context (Omitted) Options The nature of English B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' The origin of English played a role in shaping English literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Question Which one of the following can be most reasonably inferred from the information above?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content=' Type Perception Answer Prediction Implication A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1951 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='2169 Logiformer C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='5632 X D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0248 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0272 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='6349 ~ TaCo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1847 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfMAND/content/2301.02983v1.pdf'} +page_content='1532Context (Omitted) Options The present goal of 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sha256:4a96d12addff8254cdb1ab48a6f78818143b7ef15b0f793b05f48079bc08c8a0 +size 98720 diff --git a/h9FMT4oBgHgl3EQf4TEa/content/tmp_files/2301.12451v1.pdf.txt b/h9FMT4oBgHgl3EQf4TEa/content/tmp_files/2301.12451v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2be6bdc77da9497bf9a6c455fd55aa28e62dac73 --- /dev/null +++ b/h9FMT4oBgHgl3EQf4TEa/content/tmp_files/2301.12451v1.pdf.txt @@ -0,0 +1,2846 @@ +arXiv:2301.12451v1 [math.FA] 29 Jan 2023 +THE INVARIANT SUBSPACES OF PERIODIC FOURIER +MULTIPLIERS WITH APPLICATION TO ABSTRACT +EVOLUTION EQUATIONS +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +Abstract. By methods of harmonic analysis, we identify large classes of Ba- +nach spaces invariant of periodic Fourier multipliers with symbols satisfying the +classical Marcinkiewicz type conditions. Such classes include general (vector- +valued) Banach function spaces Φ and/or the scales of Besov and Triebel- +Lizorkin spaces defined on the basis of Φ. +We apply these results to the study of the well-posedness and maximal +regularity property of an abstract second-order integro-differential equation, +which models various types of elliptic and parabolic problems arising in dif- +ferent areas of applied mathematics. In particular, under suitable conditions +imposed on a convolutor c and the geometry of an underlying Banach space +X, we characterize the conditions on the operators A, B and P on X such +that the following periodic problem +∂P ∂u + B∂u + Au + c ∗ u = f +in D′(T; X) +is well-posed with respect to large classes of function spaces. The obtained +results extend the known theory on the maximal regularity of such problem. +1. Introduction +Fourier multipliers with operator-valued symbols have found many applications +in the theory of abstract evolution equations, in particular, in connection with +solvability (well-posedness) and regularity of integro-differential equations. A large +class of such equations can be modelled by the following abstract, degenerated +second-order problem with a convolution term: +(AP) +(Pu′)′ + Bu′ + Au + c ∗ u = f. +Here, A, B, P denote closed linear operators on a Banach space X and c is an +operator-valued function. +For particular forms of (AP), the studies of their well-posedness on diverse +vector-valued function spaces have been increased with occurring two seminal pa- +pers by Amann [2] and Weis [69], where operator-valued counterparts of classical +multiplier theorems for Besov and Lebesgue-Bochner spaces on R are provided. +Those results indicated a right form of multiplier conditions (see [31], [30]), which +have been further adapted to different situations; see, e.g. periodic multiplier results +in [7, 8, 67], which are relevant to this article. +2020 Mathematics Subject Classification. 42B37, 42A45, 45N05, 46N20, 43A15. +Key words and phrases. integro-differential equations, maximal regularity, well-posedness, pe- +riodic Fourier multipliers, Hardy-Littlewood maximal operator, Besov spaces, Triebel-Lizorkin +spaces, Rubio de Francia iteration algorithm. +1 + +2 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +In the literature one can extract two lines of research corresponding to such +studies: namely, when (AP) is considered in the euclidean setting, that is, on R +or R+, and in the periodic one, that is, on T := R/Z (i.e., when the periodic +conditions are imposed). +Each of these lines is represented by a long series of +papers; to mention a few representative results, see for the first one, e.g. [2, 69, +59, 6, 28, 11, 27, 5, 4, 51], and for the second one, e.g. +[7, 8, 47, 53, 23, 54, +57, 24, 49, 40, 56, 38, 20, 21, 22] (as well as the references therein). Such studies +correspond to the well-known research program formulated by Amman in [1, Section +3] and labelled as ’pairs of maximal regularity’. In both settings, the basic idea +for such studies is the same and relies on multiplier theorems. Roughly, by the +theory of vector-valued distributions, the well-posedness and regularity questions +for (AP), reduce to checking if corresponding Fourier multipliers with operator- +valued symbols are bounded in a space under consideration. Such Fourier multiplier +operators arise naturally via the representation formula for corresponding solution +operators associated to a given form of (AP). +In the euclidean setting, the so-called phenomenon of the extrapolation of Lp- +maximal regularity, which can be simply considered as a special variant of the +well-posedness with respect to various Banach function spaces, have been studied +recently; see, e.g. [59, 11, 27, 37, 29, 51]. Beyond a natural theoretical interest in +this phenomenon, the maximal regularity with respect to a more general function +space is an important tool for the study of associated non-linear problems; see, e.g. +[52, 60, 11]. +In the periodic case, the well-posedness and maximal regularity were addressed +mainly in the context of the classical Lebesgue, Besov, Triebel-Lizorkin spaces; see, +e.g. the corresponding series of the references mentioned above. The aim of this +article is to extend such results to a much wider context of general Banach function +spaces; see the main results of this paper, Theorems 6.5, 6.7, 6.9, and 6.12. In +particular, we clarify the phenomenon of the extrapolation of Lp-maximal regularity +for several periodic evolution equations modelled by (AP); see Theorems 6.7 and +6.12, as well as Section 7. These results, in particular, provide counterpart of the +euclidean line of research mentioned above for the periodic situation. In addition, +we provide a convenient framework for such studies, which reveals an underlying +structure, allows to simplify and unify technicalities mainly resulting from the fact +that we deal with higher order Marcinkiewicz’s conditions, and allows to handle +different questions (distributional or strong solvability, maximal regularity) in a +unified manner. It is achieved with the help of two auxiliary results Theorem 6.2 +and Lemma 6.4. On the other-hand, it allows to extend many results from the +related literature in several ways, by showing that assumptions usually made to +get those results in the Lp-setting are sufficient for a large class of Banach function +spaces; see Section 7. +To establish such results we make a revision of underlying multiplier results from +[7, 8, 25] applied in the context of Lp setting. Roughly, our main periodic multiplier +results, see Theorems 5.3 and 5.5, assert that the standard multiplier conditions, +which in the literature usually are imposed on the symbol of a Fourier multiplier +to get its boundedness on the classical (vector-valued) Lebesgue, Besov or Triebel- +Lizorkin spaces, are sufficient for its boundedness on much larger classes of spaces. +Such classes include general Banach function spaces Φ and/or the scales of Besov +and Triebel-Lizorkin spaces defined on the basis of Φ. In particular, these results + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +3 +extend [7, Theorem 1.3], [8, Theorem 4.5] and [25, Theorem 3.2]. Our proofs differ +from the proofs presented in those papers. We rely on direct maximal function +estimates; see the proofs of Lemma 2.2 and Theorem 5.3. +We conclude with a remark on the strategy of the proof of our abstract extrap- +olation result, Theorem 5.5. Since the theory of periodic distributions presents a +simplification in comparison to that on the real line, one could expect the same +in the context of multiplier theorems. In fact, we show that such simplification is +reflected mainly in the representation formulas for periodic Fourier multipliers; see +Section 3.1 for the further comments and Lemma 5.1. In particular we do not ad- +dress here problems which appear in the euclidean setting; see, e.g. [44, Problems +3.2 and 3.3] and [51, Section 4]. However, at some points (for instance, when the +interplay between the regularity of the symbol and its Fourier transform is crucial), +the euclidean setting presents some benefits in comparison to the periodic one. For +this reason, instead of trying to prove some periodic results in a complete analogy +to corresponding ones known in the euclidean setting, we deduce them from their +euclidean counterparts via transference techniques; see the proof of Theorem 5.5 +(cf. +also Lemma 2.1). +The tools for such transference methods are workout in +Section 4, which may be of independent interest; see operator-valued variants of +Jodeit’s type theorem, Theorems 4.1 and 4.4, as well as Lemmas 4.6 and 4.8. +The organization of the paper is well-reflected by the titles of the following +(sub)sections. +2. Auxiliary results +2.1. Function spaces. We refer the reader to the monograph by Bennett and +Sharpley [13] for the background on Banach function spaces. Here, we mention +only several facts we use in the sequel. +Let Ψ be a Banach function space over (G, dt), where G denotes R or T equipped +with the Lebesgue measure. It means that Ψ is a Banach space, which is an order +ideal of L0 := L0(G, dt), i.e. for every f ∈ L0 and g ∈ Ψ if |f| ≤ |g|, then f ∈ Ψ and +∥f∥Ψ ≤ ∥g∥Ψ. Here, L0 stands for the space of all complex measurable functions on +G (as usual, any two functions equal almost everywhere are identified). Moreover, +Ψ has Fatou’s property, and by the Lorentz-Luxemburg theorem [13, Theorem 2.7, +p.10], (Ψ′)′ = Ψ with equal norms. Here, Ψ′ denotes the (K¨othe) dual (or associated +space) of Ψ; see [13]. +We define the vector-valued variant of Banach function spaces Ψ as follows. Let +X be a Banach space with norm | · |X. Set +Ψ(G; X) := {f : G → X strongly measurable : +|f|X ∈ Ψ} +and ∥f∥Ψ(G;X) := ∥|f|X∥Ψ for f ∈ Ψ(X). Throughout, the symbol Φ is reserved +to denote a Banach function space over (T, dt). Note that if a function e0(τ) := 1 +(τ ∈ T) is in Φ, then by the ideal property of Φ we get that L∞(T; X) ⊂ Φ(T; X) =: +Φ(X). In particular, if P(T; X) denotes the set of all X-valued polynomials on T, +i.e. +P(X) := P(T; X) := +� +N +� +k=−N +ek ⊗ xk : N ∈ N, xk ∈ X +� +then P(X) ⊂ Φ(X). Here, (ek ⊗ x)(τ) := ek(τ)x, where ek(τ) := τ k (τ ∈ T, k ∈ Z, +x ∈ X). + +4 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +Moreover, we introduce a variant of vector-valued Besov and Triebel-Lizorkin +spaces corresponding to a Banach function space Φ. +Let D′(T; X) := L(D, X), where D := D(T) is a space of all complex-valued +infinitely differentiable functions on T equipped in the usual locally convex topology. +We refer to [63, Section 3] or [36] for the backgrounds on the scalar distributions on +T, and to [8, Section 2] for their vector-valued counterpart. For instance, relying +on [8, Proposition 2.1], it is readily seen that for each ψ ∈ C(R) with the compact +support, the operator ψ(∆) given by +ψ(∆)f := +� +k∈Z +ek ⊗ ψ(k) ˆf(k) +(f ∈ D′(T; X)) +is in L(D′(T; X)). +Let {ψj}j∈N0 be the resolution of the identity on R generated by a function +ψ ∈ C∞(R) such that ψ ≡ 1 on [−1, 1] and supp ψ ⊂ [−2, 2], i.e. +ψ0 := ψ, +ψj := ψ(2−j·) − ψ(2−j+1·) +for j ∈ N. +One can check that {ψj(∆)}j∈N0 is the resolution of the identity operator on +D′(T; X), i.e. for every f ∈ D′(T; X) +� +j≤N +ψj(∆)f = ψ(2−N∆)f → f +in D′(T; X) as N → ∞. +Let Φ be a Banach function space over (T, dt). For all s ∈ R and q ∈ [1, ∞] we +set (with usual modification when q = ∞): +Bs,q +Φ (T, X) := + + + + + +f ∈ D′(T; X) : ∥f∥Bs,q +Φ (T,X) := + + +∞ +� +j=0 +∥2sjψj(∆)f∥q +Φ(T;X) + + +1/q +< ∞ + + + + + +, +F s,q +Φ (T, X) := + + + + + +f ∈ D′(T; X) : ∥f∥F s,q +Φ +(T,X) := +������� + + +∞ +� +j=0 +|2sjψj(∆)f|q +X + + +1/q������� +Φ +< ∞ + + + + + +. +For G = R and a general Banach function spaces Ψ over (R, dt), the corre- +sponding generalized vector-valued Besov Bs,q +Ψ (R; X) and Triebel-Lizorkin spaces +F s,q +Ψ (R, X) were introduced in [51, Section 2]. In the case when G = T, the vector- +valued counterpart of the classical Besov spaces Bs,q +p (T), i.e. for Φ = Lp over (T, dt), +was introduced in [8]. In the both cases (G = T or G = R), under some additional +assumption on Ψ, one can show that the spaces Bs,q +Ψ (G; X) and F s,q +Ψ (G; X) share +most of the properties of their well-known scalar prototypes which correspond to +Ψ = Lp and X = C. For our further purposes, we only need a few basic properties +of the spaces Bs,q +Φ (T; X) and F s,q +Φ (T; X) corresponding to a general Banach func- +tion space Φ over (T, dt); see Proposition 2.3 below. For their proofs we need a +preliminary result on the boundedness of Fourier multipliers ψj(∆), j ∈ N0, (and +other ones) on the underlying space Φ(X). Then, the proof of Proposition 2.3 can +be carried out in an analogy to the non-periodic case when G = R as it has been +treated in [51]; see [51, Lemma 3.6] and [51, Lemma 5.3]. However, it should be +pointed out that in a comparison to the proofs of some results in the case G = R, +the proofs of their periodic counterparts admit an essential simplification, which +we indicate below; see also Subsection 3.1. + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +5 +As it could be already noted above, we omit ’T’ in the symbols of spaces over +(T, dt). Similarly, in the scalar case, i.e. when X = C, C is also omitted in the +corresponding symbols. For instance, Bs,q +Φ (X) stands for Bs,q +Φ (T; X) and C∞ +c (R) +denotes C∞ +c (R; C), etc. +2.2. Preliminary results on boundedness of periodic multipliers. For a +polynomially bounded sequence m : Z → L(X, Y ) we write ˇm to denote the corre- +sponding periodic distribution in D′(L(X, Y )), i.e. +ˇm := +� +k∈Z +ek ⊗ m(k). +Lemma 2.1. Let m : Z → L(X, Y ) be such that ˇm is in L∞(L(X, Y )). Then, for +every Banach function space Φ over (T, dt) such that L∞ ⊂ Φ ⊂ L1 the operator +m(∆) given by +m(∆)f := +� +k +ek ⊗ m(k) ˆf(k) +f ∈ D′(X) +is in L(Φ(X), Φ(Y )) with ∥m(∆)∥L(Φ(X),Φ(Y )) ≤ cΦ∥ ˇm∥L∞∥χT∥Φ, where cΦ denotes +the norm of embedding operator from Φ into L1. +Proof. A standard argument shows that Φ ֒→ L1, i.e. ∥g∥L1 ≤ cΦ∥g∥Φ (g ∈ Φ). +Since for every η ∈ L∞, g ∈ Φ and τ ∈ T we have +|(η ∗ g)(τ)| ≤ cΦ∥η∥L∞∥g∥Φ +we infer that +∥η ∗ g∥Φ ≤ cΦ∥η∥L∞∥χT∥Φ∥g∥Φ, +where χT is the characteristic function of T. Since for every f ∈ L1(X) we have +|(m(∆)f)(τ)|Y = |( ˇm ∗ f)(τ)|Y ≤ +� +∥ ˇm∥L(X,Y ) ∗ |f|X +� +(τ) +(τ ∈ T), +the proof is complete. +□ +In particular, Lemma 2.1 shows that each operator ψj(∆), j ∈ N0, is bounded +on Φ(X) if L∞ ⊂ Φ ⊂ L1. To show their uniform boundedness on Φ(X) we need +an additional assumption on the boundedness of the Hardy-Littlewood maximal +operator on Φ. +Recall that the Hardy-Littlewood maximal operators MT and MR are defined by +MTf(τ) := sup +ǫ>0 +1 +ǫ +� +Γ(τ,ǫ) +|f(ζ)| |dζ| +(τ ∈ T) +for f ∈ L1(T), where Γ(τ, ǫ) := T ∩ {z ∈ C : |z − τ| ≤ ǫ}, and +MRf(t) := sup +ǫ>0 +1 +2ǫ +� +[t−ǫ,t+ǫ] +|f(s)|ds +(t ∈ R) +for f ∈ L1 +loc(R). In the view of the standard identification between the function f +on T with its 2π-periodic extension �f on R, �f(t) := f(eit), t ∈ R, there exists a +constant c > 0 such that +c−1(MR �f)(t) ≤ (MTf)(eit) ≤ c(MR �f)(t) +(f ∈ L1(T), t ∈ R). +Note that the assumption that MT is bounded on a Banach function space Φ +implies that L∞ ⊂ Φ ⊂ L1. Indeed, if f ∈ Φ \ {0}, then there exists a constant +c > 0 and a measurable subset A of T such that |f| ≥ cχA, i.e. χA ∈ Φ. Hence, by + +6 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +the boundedness of MT on Φ, we get that MTχA ≥ |A| +2π . Consequently, by the ideal +property of Φ, we obtain that L∞ ⊂ Φ. +The following lemma provides a periodic counterpart of [66, Chapter 2, (17) p.57]. +For its proof we need a vector-valued variant of Fej´er’s theorem, which asserts that +for an arbitrary Banach space X and g ∈ L1(T; X), if Sl(g) := � +|k|≤l ek ⊗ ˆg(k) for +l ∈ N0, then +1 +N + 1 +N +� +l=0 +Sl(g) → g +as N → ∞ in L1(T; X). +Its proof follows the lines of the proof of its scalar prototype almost verbatim. +Lemma 2.2. (i) Let η ∈ Cc(R; L(X, Y )) be such that ∥F−1η(t)∥L(X,Y ) ≤ φ(t), +t ∈ R, for an even, radially decreasing, integrable function φ on R. Then, there +exists a constant c > 0 such that for every f ∈ L1(T; X) we have +(1) +|η(∆)f(τ)|Y ≤ c∥φ∥L1(R)(MT|f|X)(τ) +(τ ∈ T). +(ii) Let η ∈ C∞ +c (R) and set ηǫ := η(ǫ·), ǫ > 0. Then, for every Banach function +space Φ over (T, dt) such that MT is bounded on Φ, the operators ηǫ(∆), ǫ > 0, are +uniformly in L(Φ(X)). +Proof. (i) Let f ∈ L1(T; X). It is readily seen that for every t ∈ R the integral +� +R(F−1η)(s) �f(t − s)ds =: (F−1η ∗ �f)(t) is absolutely convergent. Suppose that +supp η ⊂ [−K, K] for some K ∈ Z. Note that for every t ∈ R and l > K we have +η(∆)f(eit) = +� +|k|≤l +eitk +� +R +e−iskF−1η(s)ds ˆf(k) += +� +R +F−1η(s) +� +|k|≤l +ei(t−s)k ˆf(k)ds. +Therefore, following the notation introduced in Fej´er’s theorem, for N > K and +t ∈ R we get +(2) +1 +N + 1 +N +� +l=K +Sl(η(∆)f)(eit) = +� +R +F−1η(s) +1 +N + 1 +N +� +l=K +Sl(gt)(eis)ds, +where gt(τ) := f(eit¯τ), τ ∈ T. By our assumption on F−1η it is straightforward to +show that the right-hand side of (2) converges to (F−1η ∗ �f)(t) as N → ∞. Since +the left-hand side is equal N−K+1 +N +η(∆)f(eit), we infer that for every t ∈ R +��η(∆)f(eit) +�� +Y = +���(F−1η ∗ �f)(t) +��� +Y ≤ (φ ∗ | �f|X)(t). +Further, note that for each ǫ > 0 there exists a function φǫ := �N +j=0 cjχBj, +where Bj denotes an interval with the center in 0 and cj > 0, such that 0 ≤ φǫ ≤ φ, +∥φ − φǫ∥L∞ < ǫ and ∥φ − φǫ∥L1 < ǫ . One can readily check that for every δ > 0 +there exists ǫ such that for every t ∈ R we have +|(φ ∗ | �f|X)(t)| ≤ ((φ − φǫ) ∗ | �f|X)(t) + (φǫ ∗ | �f|X)(t) ≤ δ + ∥φǫ∥L1(MR| �f|X)(t). +Since there exists c > 0, independent of f ∈ L1(T; X), such that (MR| �f|X)(t) ≤ +c(MT|f|X)(eit) for every t ∈ R, we get (1). +(ii) Note that (F−1ηǫ)(t) = +1 +ǫ(F−1η)( t +ǫ) for every ǫ > 0 and t ∈ R. +More- +over, there exists a constant C > 0 such that |(F−1η)(t)| ≤ +C +1+t2 , which yields + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +7 +|(F−1ηǫ)(t)| ≤ +ǫ−1C +1+(ǫ−1t)2 =: φǫ(t) for every t ∈ R and ǫ > 0. Since ∥φǫ∥L1 = Cπ for +every ǫ. Therefore, (1) gives the desired claim. +□ +2.3. Fundamental properties of generalized Besov and Triebel-Lizorkin +spaces. Here, we collect the fundamental properties of such spaces, which play a +role in our further studies; see Proposition 2.3 below. +We start with some preliminaries. Note that for every s ∈ R the space B−s,1 +Φ +(X∗) +embeds into (Bs,∞ +Φ +(X))∗. Indeed, this embedding is given by the following duality +pairing: for each f ∈ Bs,∞ +Φ +(X) and g ∈ B−s,1 +Φ′ +(X∗) we set +(3) +⟨g, f⟩ := +� +j,l∈N0 +� +T +⟨ψl(∆)g(t), ψj(∆)f(t)⟩X∗,Xdt. +Note that +⟨g, f⟩ := +� +r∈{±1,0} +� +l∈N0 +⟨ ˘ψj(∆)ψl(∆)g, χj(∆)f⟩Φ′(X∗),Φ(X), +where ˘ψj := ψj(−·) and χj := ψj−1 + ψj + ψj+1 (j ∈ N0) with ψ−1 ≡ 0 if j = 0. +Proposition 2.3. Let X be a Banach space and Φ be a Banach function space +over (T, dt). Then, the following assertions hold. +(i) If L∞ ⊂ Φ ⊂ L1, then for every +E ∈ {Bs,q +Φ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞)} +(4) +P(X) ⊂ E(X) ֒→ D′(X). +In particular, E(X) is a Banach space. +(ii) If MT is bounded on Φ, then for all s ∈ R, P(X) is a dense subset of +Bs,q +Φ (X) in the norm topology if q ∈ [1, ∞), and in the B−s,1 +Φ′ +(X∗)-topology +of Bs,∞ +Φ +(X) if q = ∞. +More precisely, for every f ∈ Bs,q +Φ (X) we have that +� +0≤j≤N +ψj(∆)f = ψ(2−N∆)f → f +as N → ∞. +in Bs,q +Φ (X) for each q < ∞, and in the other case, i.e. q = ∞, the conver- +gence holds in the σ(Bs,∞ +Φ +(X), B−s,1 +Φ′ +(X∗))-topology. +(ii’) If MT is bounded on Φ and its dual Φ′, then P(X) is a dense subset of +F s,q +Φ (X) and for ever f ∈ F s,q +Φ (X) we have that +� +0≤j≤N +ψj(∆)f = ψ(2−N∆)f → f +as N → ∞. +(iii) If MT is bounded on Φ, then for every distribution f ∈ D′(X), f belongs to +Bs,q +Φ (X) if and only if ∂f belongs to Bs−1,q +Φ +(X). Moreover, the function +(5) +Bs,q +Φ (X) ∋ f �→ ∥∂f∥Bs−1,q +Φ +(X) +is an equivalent norm on Bs,q +Φ (X). +Remark 2.4. (a) Compared to the case of the real line R (see [51, Lemma 3.6]), +note that in the periodic case there is a common dense subset, i.e. P(X), for all +E(X) with +E ∈ {Bs,q +Φ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞)}. + +8 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +(b) In contrast to [51, Lemma 3.6], in Proposition 2.3, in the case of Besov spaces +we do not assume that MT is bounded on the dual of Φ′. +By Lemma 2.2, the proof of Proposition 2.3, mimics the proof of the correspond- +ing statements on R, [51, Lemmas 3.6 and 5.4]. For the convenience of the reader +we provide some auxiliary observations which should be made. +The proof of Proposition 2.3. (i) We start with the case when E = Bs,q +Φ . +Since +P ⊂ L∞ ⊂ Φ, the left inclusion in (4) holds readily. For the second one, note that +for every f ∈ Bs,q +Φ (X) and φ ∈ D we have that +|(φj(∆)f) (φ)|X = +����� +� +k∈Z +� +T +ekφdt ψj(k) ˆf(k) +����� +X += 2π +����� +� +k∈Z +χj(k)ˆφ(−k)ψj(k) ˆf(k) +����� +X +≤ 2π +� +T +����� +� +k∈Z +ek ⊗ ψj(k) ˆf(k) +����� +X +����� +� +l∈Z +χj(l)ˆφ(−l)el +����� dt +≤ 2π +���ψj(∆) ˆf +��� +Φ(X) +���χj(∆)˘φ +��� +Φ′ +Here, ˘φ(eit) = φ(e−it), t ∈ [−π, π]. Since limN→∞ +�N +j=0 φj(∆)f = f in D′(X), we +obtain that +|f(φ)|X ≤ +� +j∈N0 +��2jsψj(∆)f +�� +Φ(X) ∥2−jsχj(∆)˘φ∥Φ′ +≤ + + � +j∈N0 +��2jsqψj(∆)f +��q +Φ(X) + + +1/q  + � +j∈N0 +∥2−jsχj(∆)˘φ∥q′ +Φ′ + + +1/q′ +≤ ∥f∥Bs,q +Φ (X) + + � +j∈N0 +∥2−jsχj(∆)ˇφ∥q′ +Φ′ + + +1/q′ +(with the usual modification when q = ∞). +Take α > |s| + 1 and set ρ(t) := +(1 + t2)−α, t ∈ R. Then, for every j ∈ N0 and τ ∈ T we have +2(|s|+1)j ���F−1(ρχj|Z)(τ) +��� = +����� +� +k∈Z +τ k 2(|s|+1)j +(1 + k2)α χj(k) +����� +≤ +� +2j−2 0 such that |s| < c|(ks − j)|, s2 < c(ks − j)2 +and |s|3 < c|(ks − j)|3 for j = 0, 1, 2 and |s| sufficiently large. +Therefore, if for all u ∈ [−1, 0] the function λ satisfied +(11) +λ′(u) + λ′(u + 1) + λ′(u + 2) + λ′(u + 3) = 0, +then m ∈ M3(Z; L(X, Y )) ⊂ M1(Z; L(X, Y )) would clearly yield the boundedness +of e and (·)e′. +Similarly, if for all u ∈ [−1, 0] the function λ satisfied +(12) +� +3λ′′(u) + 2λ′′(u + 1) + λ′′(u + 2) = 0, +−2λ′′(u) − λ′′(u + 1) + λ′′(u + 3) = 0, +then m ∈ M3(Z; L(X, Y )) ⊂ M2(Z; L(X, Y )) would imply the boundedness of (·)2e′′. +Finally, if for all u ∈ [−1, 0] the function λ satisfied +(13) + + + + + +3λ′′′(u) + λ′′′(u + 1) = 0, +−3λ′′′(u) + λ′′′(u + 2) = 0, +λ′′′(u) + λ′′′(u + 3) = 0, + +16 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +then m ∈ M3(Z; L(X, Y )) would give that (·)3e′′′ is bounded. +Notice that the conditions stated in (11), (12) and (13) are not mutually con- +tradictory. Indeed, one can check that (13) ⇒ (12) ⇒ (11). Because e must be an +extension of m, we have to additionally ensure that λ(0) = 1, λ(1) = 0, λ(2) = 0. +To construct a function λ, which obeys all of the above conditions, on the interval +[−1, 0] we take λ as any smooth function with +λ(−1) = 0, +λ(0) = 1, +λ′(−1) = 0, +λ′ +−(0) = 3/2, +λ′′ +−(0) = 1, +λ(j)(k) = 0 +for j ≥ 3 and k ∈ {−1, 0}. Then, on the interval [0, 3], we define λ in the following +way: +λ(u + 1) := −3λ(u) + (u + 1)2 +2 ++ 3 +2(u + 1) + 1, +λ(u + 2) := 3λ(u) − 2(u + 1)2 +2 +− 2(u + 1), +λ(u + 3) := −λ(u) + (u + 1)2 +2 ++ 1 +2(u + 1) +(u ∈ [0, 1]). +Now it is straightforward to check that for every u ∈ [−1, 0] we have +λ′(u+1) = −3λ′(u)+u+5 +2, +λ′(u+2) = 3λ′(u)−2u−4, +λ′(u+3) = −λ′(u)+u+3 +2, +λ′′(u + 1) = −3λ′′(u) + 1, +λ′′(u + 2) = 3λ′′(u) − 2, +λ′′(u + 3) = −λ′′(u) + 1, +λ′′′(u + 1) = −3λ′′′(u), +λ′′′(u + 2) = 3λ′′′(u), +λ′′′(u + 3) = −λ′′′(u). +Using the above expressions for λ′, λ′′ and λ′′′, it is easy to verify that such +function λ satisfies (11), (12) and (13). +The additional conditions imposed on +derivatives of λ at 0 ensure that λ is smooth. It completes the proof. +□ +Remark 4.5. The function λ which the existence is proven above for γ = 3 works +as well for γ = 1 and γ = 2. However, note that if we are interested merely in the +case when γ = 1 or γ = 2, then one can simplify the corresponding parts of the +above proof by considering λ with support in [−1, 1] or [−1, 2], respectively. +4.2. De Leeuw’s couples. Let Φ = Φ(T) and Ψ = Ψ(R) be two Banach function +spaces over (T, dt) and (R, dt), respectively. If for every m ∈ MΨ(R; X, Y ) such +that each k ∈ Z is a Lebesgue point of m its restriction to Z is in MΦ(Z; X, Y ), +then we call (Φ, Ψ) de Leeuw’s couple. +The classical de Leeuw theorem [33] (see also its operator-valued counterpart +[41, Theorem 5.93]) shows that the couples (Lp(T; X), Lp(R; X)), p ∈ [1, ∞), have +such property. Moreover, some further extensions of de Leeuw’s result done in the +literature provide other examples of such couples; see e.g. [15] and the references +therein. In particular, for our further purposes we need the following operator- +valued variant of a weighted extension of de Leeuw’s theorem due to Berkson and +Gillespie [14, Theorem 1.2]. +Lemma 4.6. Let p ∈ (1, ∞). Let �w be a 2π-periodic weight and w(eit) := �w(t) for +t ∈ R. Then, the pair (Lp +w(T), Lp +� +w(R)) is de Leeuw’s couple. +More precisely, for every Banach spaces X and Y , if m ∈ MLp +� +w(R; X, Y ) and +each point k ∈ Z is a Lebesgue point of m, then +∥m|Z(∆)∥L(Lp +w(T;X),Lp +w(T;Y )) ≤ ∥m(D)∥L(Lp +� +w(R;X),Lp +� +w(R;Y )). + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +17 +The proof can be obtained by an adaptation of the proof of its scalar counter- +part; see e.g. the presentation given in [3]. Below, we propose a slightly different +approach. +Proof. Let φ(t) := e− π +p t2 and ψ(t) := e− π +q t2 for t ∈ R and 1 +p + 1 +q = 1. As in [42, +Lemma 5.94] one can show that +� +[−π,π] +⟨m(∆)f(t), g(t)⟩Y,Y ∗dt = lim +ǫ→0+ ǫ +� +R +⟨m(D)(φ(ǫ·)f)(t), (φ(ǫ·)g)(t)⟩Y,Y ∗dt +for all trygonometric polynomials f : T → X, g : T → Y ∗. Furthermore, note that +ǫ +��� +� +R +⟨m(D)(φ(ǫ·)f)(t), (ψ(ǫ·)g)(t)⟩Y,Y ∗dt +��� +≤ ǫ∥m(D)(φ(ǫ·)f)∥Lp +� +w(R;Y )∥(ψ(ǫ·)g)∥Lq +� +w1−q(R;Y ∗) +≤ ∥m(D)∥ǫ +1 +p ∥φ(ǫ·)f)∥Lp +� +w(R;X)ǫ +1 +q ∥(ψ(ǫ·)g)∥Lq +� +w1−q(R;Y ∗) +≤ ∥m(D)∥ +� 1 +2π +� π +−π +2πǫ +� +n∈Z +|φ(ǫ(2πn + t))|p|f(t)|p +Xw(t)dt +� 1 +p +× +� 1 +2π +� π +−π +2πǫ +� +n∈Z +|ψ(ǫ(2πn + t))|q|g(t)|q +Y ∗w1−q(t)dt +� 1 +q +≤ ∥m(D)∥∥φ∥Lp(R)∥f∥Lp +w(X)∥ψ∥Lq(R)∥g∥Lq +w1−q(Y ∗) +≤ ∥m(D)∥∥f∥Lp +w(X)∥g∥Lq +w1−q(Y ∗). +Since Lq +w1−q(Y ∗) is norming for Lp +w(Y ) we have +∥m(∆)f∥Lp +w(Y ) = +sup +∥g∥Lq +w1−q (Y ∗)=1 +� +[−π,π] +⟨m(∆)f(t), g(t)⟩Y,Y ∗dt +≤ ∥m(D)∥∥f∥Lp +w(X). +It finishes the proof. +□ +Remark 4.7. (a) For our purposes (see, e.g. the proof of Theorem 5.5), for a +given Banach function space Φ over (T, dt) we are interested in constructing a +Banach function space Ψ over (R, dt) such that (Φ, Ψ) is a de Leeuw’s couple and, +in addition, Ψ inherits some analytic properties of Φ. For instance, we need to +know that the Hardy-Littlewood operator MR is bounded on Ψ if MT is bounded +on Φ. We do not know if such Ψ can be contracted for Banach function spaces +Φ, which are considered in the results of Section 5. Here, we only mention that +applying Wiener’s amalgam type construction we can define such a space Ψ under +additional density and rearrangement type assumptions on Φ. +It determines our strategy of the proofs of some results presented below (e.g. +Theorems 5.3 and 5.5), and does not allow to obtain some periodic variants of +results already known in the R-setting via a direct transference. +(b) However, by straightforward arguments (see also [16, Theorem 2.10]), one +can show that for every Muckenhoupt weight w ∈ Ap(T) its periodic extension �w +on R is in Ap(R). Therefore, by Muckenhoupt’s theorem and Lemma 4.6, for every +w ∈ Ap(T) and p ∈ (1, ∞) the couple (Lp +w(T), Lp +� +w(R)) is de Leeuw’s couple and MR +is bounded on Lp +� +w(R). To get desired results for general Φ’s we adopt the Rubio de +Francia iteration algorithm to the periodic setting. + +18 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +4.3. The extrapolation theorem. The following extrapolation theorem is a cru- +cial ingredient of results presented in the sequel. +Lemma 4.8. Let Φ be a Banach function space over (T, dt) such that the Hardy- +Littlewood operator MT is bounded on Φ and its dual Φ′. +(i) Let X be a Banach space. Then, for every p ∈ (1, ∞) +Φ(T; X) ⊂ +� +w∈Ap(T) +Lp +w(T; X) +(ii) Let X and Y be Banach spaces and p ∈ (1, ∞). Assume that {Tj}j∈J is +a family of linear operators Tj : P(T; X) → D′(T; Y ) such that for every +W ⊂ Ap(T) with supw∈W[w]Ap < ∞ +sup +w∈W +sup +j +∥Tj∥L(Lp +w(T;X),Lp +w(T;Y )) < ∞. +Then, each Tj extends to a linear operator Tj on � +w∈Ap(T) Lp +w(T; X) ⊂ +L1(T; X) and has the restriction to an operator in L(Φ(T; X), Φ(T; Y )). +Moreover, +sup +j∈J +∥Tj∥L(Φ(T;X),Φ(T;Y )) < ∞. +The proof follows the lines of the proof of [51, Theorem 3.1] almost verbatim. +Therefore, we leave it for the reader. +Remark 4.9. (a) Recall that the Hardy-Littlewood maximal operator MT is bounded +on Lp +w(T) for all p ∈ (1, ∞) and each Muckenhoupt weight w ∈ Ap(T); see, e.g. [16, +Theorem 5.2, Corollary 5.3]. +(b) By the reverse H¨older inequality for weights in Ap(T) one can show that for +every p ∈ (1, ∞) and w ∈ Ap(T) there exists q > p such that Lp +w(T) ֒→ Lq(T); +see the proof of [50, Theorem 4.1]. +In the context of a comment stated below +[50, Remark 4.4], note that for each p ∈ (1, ∞) one can construct a weight w on +[0, 2π), which is in the class Ap([0, 2π])(as it is defined in [50]), but the function +T ∋ τ �→ w(arg τ) is not in Ap(T). +By Lemma 4.8(i) we get that +(14) +� +Φ(T; X) = +� +p>1 +Lp(T; X), +where the first union is taken over all Banach function spaces Φ over (T, dt) such +that MT is bounded on Φ and Φ′. +We conclude this section with the following consequence of Lemmas 4.6 and 4.8. +Proposition 4.10. Let X be a Banach space with the UMD property. Then, for +every Banach function space Φ over (T, dt) such that the Hardy-Littlewood operator +MT is bounded on Φ and Φ′, the family {χI(∆) : I ⊂ R an interval} is uniformly +bounded in L(Φ(X)). +Proof. Recall that the Hilbert transform HR = −i sgn(D) is bounded on L2(R; X) +and its kernel satisfies the Calder´on-Zygmund conditions; see, e.g. +[42, Theo- +rem 5.1, p. 374]. In particular, for each W ⊂ A2(R) with supw∈W[w]A2 < ∞, +supw∈W ∥HR∥L(L2(R;X)) < ∞; see, e.g. [62] or [45]. By Lemma 4.6 we infer that +the periodic Hilbert transform HT := −i sgn(∆) has the analogues boundedness + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +19 +property as HR does. Therefore, by Lemma 4.8 we get that HT is in L(Φ(X)). +Since for every a, b ∈ Z with a < b we have +χ{a,...,b}(k) = 1 +2 sgn(k − a) − 1 +2 sgn(k − b) + χ{a}(k) + χ{b}(k) +(k ∈ Z) +and ∥ sgn(·−n)(∆)∥L(Φ(X)) = ∥HT∥L(Φ(X)) for n ∈ Z, we get the desired claim. +□ +5. The boundedness results for periodic Fourier multipliers +This section provides the boundedness results for periodic Fourier multipliers on +spaces introduced in Section 2. We treat the case of Besov and Triebel-Lizorkin +spaces separately to the case of Banach function spaces Φ. +In the both subsections, our purpose is to show that the typical multiplier condi- +tions, sufficient for the boundedness on Bs,q +Lp -spaces/F s,q +Lp -spaces/Lp-spaces, ensure +the boundedness on Bs,q +Φ -spaces/F s,q +Φ -spaces/Φ-spaces corresponding to general Ba- +nach functions spaces Φ (see Theorems 5.3 and 5.5 below). +5.1. Multipliers on generalized Besov and Triebel-Lizorkin spaces. In the +case of Besov spaces, it is readily seen from the definition of their norms ∥ · ∥Bs,q +Φ , +that the problem whether a sequence m is a multiplier on Bs,q +Φ (X) reduces to +showing that its dyadic parts ψjm(∆), j ∈ N0, are uniformly bounded on the +underlying Banach function space Φ(X). Note that, in the contrast to the real line +case, no extension procedures are needed in the periodic setting; see Subsection 3.1. +Moreover, the boundedness of each dyadic part mψj(∆) of m(∆) is automatic (see +Lemma 2.1), and the problem reduces to the uniform boundedness of their norms. +For this reason we postpone the study of the boundedness of general multipliers on +Banach function spaces Φ to the next section. +The following lemma makes the above observation rigorous. +Lemma 5.1. Let X and Y be Banach spaces. Let Φ be a Banach function space +over (T, dt). Then the following statements hold. +Let m : Z → L(X, Y ) be such that +(15) +µ := sup +j∈N0 +∥(ψjm)(∆)∥L(Φ(X),Φ(Y )) < ∞. +Then, for every s ∈ R and q ∈ [1, ∞], m ∈ MBs,q +Φ (T; X, Y ). +More precisely, the multiplier m(∆) restricts to an operator in L(Bs,q +Φ (X), L(Bs,q +Φ (Y )) +and for every f ∈ Bs,q +Φ (X) +(16) +m(∆)f = +∞ +� +j=0 +(ψjm)(∆)f = lim +N→∞ +� +k∈Z +ek ⊗ ψ(2−Nk)m(k) ˆf(k) +with the convergence in Bs,q +Φ (Y ) if q < ∞, and in the B−s,1 +Φ′ +(Y ∗)-topology if q = ∞. +Moreover, if q = ∞, then the restriction of m(∆) to Bs,∞ +Φ +(X) is σ(Bs,∞ +Φ +(X), B−s,1 +Φ′ +(X∗))- +to-σ(Bs,∞ +Φ +(Y ), B−s,1 +Φ′ +(Y ∗))-continuous. +Proof. Since ψj(∆)m(∆)f = (χjm)(∆)ψj(∆)f for every f ∈ D′(X) and j ∈ N0, +where χj := ψj−1 + ψj + ψj+1 with ψ−1 ≡ 0, the condition (15) readily gives that +m ∈ MBs,q +Φ (T; X, Y ) for all s ∈ R and q ∈ [1, ∞]. The point is to show the claimed +representation formula for m(∆) and its continuity when q = ∞. +For q < ∞, since {ψj(∆)}j∈N0 is the resolution of the identity operator on D′(X) +and, by Lemma 2.3, P(X) is a dense subset of Bs,q +Φ (X), it is sufficient to show that + +20 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +the operators � +0≤l≤N(ψlm)(∆), N ∈ N, are uniformly in L(Bs,q +Φ (X), Bs,q +Φ (Y )). +Let f ∈ Bs,q +Φ (X). Then, +������ +� +0≤l≤N +(ψlm)(∆)f +������ +Bs,q +Φ (Y ) += + + + + +� +j∈N0 +������� +2js +� +0≤|l−j|≤1 +0≤l≤N +(ψlm)(∆)ψj(∆)f +������� +q +Φ(Y ) + + + + +1 +q +≤ 3µ ∥f∥Bs,q +Φ (X). +Let q = ∞. We start with the continuity statement. First note that, for all j ∈ N0, +the adjoint operator to the operator (ψjm)(∆) ∈ L(Φ(X), Φ(Y )) on functions g ∈ +Φ′(Y ∗) ⊂ [Φ(Y )]∗ (see (3)) acts as the multiplier m(−·)∗(∆). Here, m(s)∗ for s ∈ R +stands for the adjoint of m(s) ∈ L(X, Y ). Let f ∈ Bs,∞ +Φ +(X) and g ∈ B−s,1 +Φ′ +(Y ∗). +Since for j, l ∈ N0 +⟨ψj(∆)g, ψl(∆)[m(∆)f]⟩Φ′(Y ∗),Φ(Y ) = ⟨ψj(∆)[m(−·)∗(∆)g], ψl(∆)f⟩Φ′(X∗),Φ(X), +it suffices to show that m(−·)∗(∆)g is in B−s,1 +Φ′ +(X∗). For, since Φ(X) ⊂ [Φ′(X∗)]∗ +is a norming subspace of Φ′(X∗), for each j ∈ N0 there exists hj ∈ Φ(X) such that +∥ψj(∆)[m(−·)∗(∆)g]∥Φ′(X∗) ≤ +��⟨ψj(∆)[m(−·)∗(∆)g], hj⟩Φ′(X∗),Φ(X) +�� + 2sj−j. +However, +|⟨ψj(∆)[m(−·)∗(∆)g], hj⟩Φ′(X∗),Φ(X)| = +��⟨ψj(∆)g, (χjm)(∆)hj⟩Φ′(X∗),Φ(X) +�� +≤ 3µ∥ψj(∆)g∥Φ′(Y ∗), +which gives the desired claim. The convergence of the series in (16) in the B−s,1 +Φ′ +(Y ∗)- +topology follows from similar arguments to those presented above. We omit it. +□ +In the context of Subsection 3.1, it is interesting to compare the proof of the +above boundedness principle for periodic Fourier multipliers with the proof of the +corresponding result for R, see [51, Theorem 3.8]; cf. also [44, Problems 3.2 and 3.3]. +We complete Lemma 5.1 with the following supplementary observation on the +convergence of the series in (16) under additional assumptions on the geometry of +the underlying spaces X and Y . +Corollary 5.2. Let X and Y be Banach spaces and suppose that Y has the UMD +property. Let Φ be a Banach function space on (T; dt) such that MT is bounded on +Φ and Φ′. Then, for every q ∈ [1, ∞), s ∈ R, m ∈ MBs,q +Φ (T; X, Y ) and f ∈ Bs,q +Φ (X) +(17) +m(∆)f = lim +N→∞ +� +|k|≤N +ek ⊗ m(k) ˆf(k) +with the convergence in Bs,q +Φ (Y ). If, in addition, Y is a Banach function space +and m(∆)f ∈ Φ(Y ) (for instance, when s > 0), then the above convergence holds +pointwise almost everywhere on T. +Proof. By Lemma 5.1 and Proposition 4.10, we infer that χ[−N,N](∆), N ∈ N, are +uniformly in L(Bs,q +Φ (Y )). Since (17) holds for every f ∈ P(X) and P(X) is dense +in Bs,q +Φ (X) (see Lemma 2.3), by Lemma 5.1, we get (17) for every f ∈ Bs,q +Φ (X) +For the additional statement, note that by (14), m(∆)f ∈ Lp(Y ) for some p > 1. +By Rubio de Francia’s vector-valued counterpart of Carleson’s theorem (see [61]), +we get the pointwise convergence in (17). It completes the proof. +□ + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +21 +Theorem 5.3. Let X and Y be Banach spaces. Let Φ denote a Banach function +space over (T, dt) such that the Hardy-Littlewood operator MT is bounded on Φ. +Then the following assertions hold. +(i) For every E := Bs,q +Φ +with s ∈ R and q ∈ [1, ∞], we have that +M2(Z; L(X, Y )) ⊂ ME(T; X, Y ). +(ii) Let X and Y have the UMD property and, in addition, MT is bounded on +Φ′. Then, for every E := Bs,q +Φ +with s ∈ R and q ∈ [1, ∞], we have that +V ar(Z; L(X, Y )) ⊂ ME(T; X, Y ). +(iii) If, in addition, MT is bounded on Φ′, then the above statements (i) and (ii) +hold for every E = F s,q +Φ +with s ∈ R and q ∈ (1, ∞). +Proof. (i) Let {ψj}j∈N0 be the resolution of the identity on R. +By Lemma 4.4 +there exists an extension �m of m on R such that �m ∈ M2(R; L(X, Y )). Therefore, +to get the uniform boundedness of the operators ψjm(∆) = (ψj �m)(∆), j ∈ N, in +L(Φ(X), Φ(Y )), by Lemma 2.2, it is sufficient to show that there exists an even, +radially decreasing functions φj, j ∈ N, on R such that supj∈N ∥φj∥L1 < ∞ and +∥F−1(ψj �m)(t)∥L(X,Y ) ≤ φj(t) for all t ∈ R. The fact that (ψ0m)(∆) = ψ0 �m(∆) is +in L(Φ(T; X), Φ(T; Y )) follows directly from, e.g. Lemma 2.1. But, the existence +of such majorants it is exactly what the proof of [51, Proposition 4.2(i)] shows +(φj(t) = C[ � +m]M2 2j +1+22jt2 +(t ∈ R)). Therefore, by Lemma 2.2 and Lemma 5.1 we get the +desired claim. +(ii) First note that for every j ≥ 1 and f ∈ D′(X) we can write +χ[2j−1,2j+1](∆)m(∆)ψj(∆)f += +2j+1 +� +l=2j−1 +el ⊗ m(2j+1)ψj(l) ˆf(l) ++ +2j+1−1 +� +k=2j−1 +k +� +l=2j−1 +el ⊗ [m(k) − m(k + 1)]ψj(l) ˆf(l) += m(2j+1)χ[2j−1,2j+1](∆)ψj(∆)f ++ +2j+1−1 +� +k=2j−1 +[m(k) − m(k + 1)]χ[2j−1,k](∆)ψj(∆)f +Therefore, +��χ[2j−1,2j+1](∆)ψj(∆)m(∆)f +�� +Φ(Y ) +≤ +��m(2j+1) +�� +L(X,Y ) +��χ[2j−1,2j+1](∆)ψj(∆)f +�� +Φ(X) ++ +sup +2j−1≤k≤2j+1−1 +∥χ[2j−1,k](∆)ψj(∆)f∥Φ(X) +��� +2j+1−1 +� +k=2j−1 +[m(k) − m(k + 1)] +��� +L(X,Y ). +Similarly, the analogous estimate holds for ∥χ[−2j+1,−2j−1](∆)m(∆)ψj(∆)f∥Φ(Y ). +Since supp ψj ⊂ {2j−1 ≤ |t| ≤ 2j+1} (j ≥ 1), and m ∈ V ar(Z; L(X, Y )), by + +22 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +Proposition 4.10, there exists a constant µ > 0 such that for all f ∈ Φ(X) and +j ≥ 1 we have +∥ψj(∆)m(∆)f∥Φ(Y ) ≤ µ[m]V ar∥ψj(∆)f∥Φ(Y ). +Therefore, Lemma 5.1 completes the proof. +(iii) The proof of this part follows arguments presented in the proof of Proposition +2.3. We sketch some details. First, since F s,q +Ψ (X) = Bs,q +Ψ (X) for every Ψ = Lq +w +(q ∈ (1, ∞), w ∈ Ap(T)), m(∆) restricts to an operator in L(F s,q +Ψ (X), F s,q +Ψ (Y )) for +such Ψ′s. Its norm is bounded by +µw,q := sup +j∈N0 +∥(ψjm)(∆)∥L(Ψ(X),Ψ(Y )), +supw∈W µw,q < ∞ for each W ⊂ Aq(T) with supw∈W[w]Aq < ∞. Furthermore, for +every f ∈ F s,q +Ψ (X) and j ∈ N0 +(18) +∥ψj(∆)m(∆)f∥Ψ(Y ) ≤ 3µw,q∥ψj(∆)f∥Ψ(X). +Fix Φ and f ∈ P(X). Let +Gf := + + � +j∈N0 +|2sjψj(∆)f(·)|q +X + + +1/q +. +and similarly for G(m(∆)f). Of course, Gf and G(m(∆)f) are in Φ; see, e.g. (4). +Let h ∈ Φ′, h ̸= 0, and set +w := wGf,h,q := R(Gf)1−qR′h. +Then Gf ∈ Lq +w, i.e. f ∈ F s,q +Lq +w(T; X) and m(∆)f ∈ F s,q +Lq +w(T; X). By (18) and a +similar argument as in the proof of Proposition 2.3(ii) give +3µw,q∥R∥L(Φ)∥Gf∥Φ∥R′∥L(Φ′)∥h∥Φ′ ≥ 3µw,q∥RGf∥Φ∥R′h∥Φ′ +≥ 3µw,q +�� +R +R(Gf)R′h dt +� 1 +q �� +T +R(Gf)R′h dt +� 1 +q′ +≥ +�� +T +G(m(∆)f)qw dt +� 1 +q �� +T +R(Gf)R′hdt +� 1 +q′ +≥ +� +T +G(m(∆)f)h dt +Since +sup +� +[wGf,h,q]Aq : f ∈ P(X), h ∈ Φ′� +< ∞, +µ := sup {3µw,q : w = wGf,h,q with f ∈ P(X), h ∈ Φ′} < ∞. +Therefore, for every f ∈ P(X) +∥m(∆)f∥F s,q +Φ +(T;Y ) ≤ µ∥R∥L(Φ)∥R′∥L(Φ′)∥f∥F s,q +Φ +(X). +Since P(X) is dense in F s,q +Φ (T; X) (see Lemma 2.3), m(∆) restricts to an operator +in L(F s,q +Φ (X), F s,q +Φ (Y )). +□ +Remark 5.4. Note that [8, Theorem 4.2] shows that the boundedness of MT on Φ′ +cannot be drop in the part (ii) in general. Indeed, V ar(Z; L(X)) ⊈ MBs,∞ +Φ +(T; X, X) +for Φ = L∞ and s ∈ (0, 1), when X is not isomorphic to a Hilbert space. + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +23 +5.2. Multipliers on general Banach function spaces. The main aim of this +section is to provide a result on the extrapolation of the boundedness of periodic +Fourier multipliers; see Theorem 5.5. +More precisely, we identify the classes of +multipliers m for which a priori knowledge that m is in MLp(T; X, Y ) for some +p ∈ (1, ∞) implies that m ∈ MΦ(T; X, Y ) for a large class of Banach function +spaces Φ. In the next section, we apply this result to prove the phenomenon of the +extrapolation of the Lp-maximal regularity for a large class of abstract evolution +equations; see Theorems 6.7, 6.9 and 6.12. +Theorem 5.5. Let X and Y be Banach spaces. Let Φ be a Banach function space +over (T, dt) such that MT is bounded on Φ and its dual Φ′. Then the following +assertions hold. +(i) Assume that m ∈ M3(Z; L(X, Y )) and m ∈ MLp(T; X, Y ) for some p ∈ +(1, ∞). Then, m ∈ ME(T; X, Y ) for every +E ∈ {Φ, Bs,q +Φ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞)} . +(ii) Assume that m ∈ M1 +R(Z; L(X, Y )) and, in addition, X and Y have the +UMD property. Then, the conclusion of (i) holds. +Proof. (i) The case E = Bs,q +Φ +and E = F s,q +Φ +follows from Theorem 5.3. Therefore, +let E = Φ. By Theorem 4.4 there exists an extension �m := e(Λ, m) of m on R +such that �m ∈ � +M3(R; L(X, Y )). Moreover, Theorem 4.1 yields �m ∈ MLp(R; X, Y ). +Consequently, �m(D) is a Calder´on-Zygmund operator; see [66, Proposition 4.4.2, +p.254], or [51, Lemma 4.1]. In particular, by [62, Theorem 1.6], for every q ∈ (1, ∞) +and for every � +W ⊂ Aq(R) with sup � +w∈� +W[ �w]Aq(R) < ∞ we have that +sup +� +w∈� +W +∥ �m(D)∥L(Lq +� +w(X),Lq +� +w(Y )) < ∞. +Let W ⊂ Aq(T) be such that supw∈W[w]Aq(T) < ∞. By �w we denote the periodic +extension of w on R. Set � +W := { �w : w ∈ W}. Then simple argumentation shows +that � +W ⊂ Aq(R) and there exists a constant C > 0 such that +sup +� +w∈� +W +[ �w]Aq(R) ≤ C sup +w∈W +[w]Aq(T) < ∞. +By Lemma 4.6 we get that +sup +w∈W +∥m(∆)∥L(Lq +w(X),Lq +w(Y )) ≤ sup +� +w∈� +W +∥ �m(D)∥L(Lq +� +w(X),Lq +� +w(Y )) < ∞. +Therefore, by Lemma 4.8 we conclude that �m|Z = m is in MΦ(T; X, Y ). +(ii) By Theorem 4.4 we find λ ∈ C∞ +c (R) such that �m := e(Λ, m) (for Λ := λ(·)IY ) +extends m and is in M1(R; L(X, Y )). By Kahane’s contraction principle (see [52, +Proposition 2.5]), we easily get that �m satisfies, in fact, the M1 +R-condition. Now +[37, Theorem 3.5.(a)] shows that for every W ⊂ Aq(R) with supw∈W[w]Aq(R) < ∞ +sup +w∈W +∥ �m(D)∥L(Lp +w(R;X),Lp +w(R;Y )) < ∞. +Therefore, Lemma 4.6, Lemma 4.8 and similar reasoning as in (i) give that m is in +MΦ(T; X, Y ). The case of the Besov spaces follows now directly from Lemma 5.1. +For the Triebel-Lizorkin case, note that the functions ψj �m satisfy the M1 +R-condition +uniformly in j ∈ N0. +Therefore, in this case, the proof mimics the arguments +presented already in the proof of Theorem 5.3(iii). +□ + +24 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +Remark 5.6. (a) In the context of applications, see Theorem 6.5(c), Theorem 5.5(i) +can be read as an extrapolation of the Lp-maximal regularity property. Indeed, as +Theorem 6.2 shows, for Fourier multipliers m related to some evolution equations +(see Subsection 7.2), the fact that m ∈ MLp(R; X, Y ) implies that m satisfies +Marcinkiewicz’s condition (Mγ) of an arbitrary order γ ∈ N. +(b) The assumptions of Theorem 5.5 should be confronted with [9, Theorem 1], +which says that for every Banach space X, which is not isomorphic to a Hilbert +space, there is a sequence m : Z → L(X) satisfying the Mγ-condition for every +γ ∈ N, but m is not in MLp(Z; X, X) (p ∈ (1, ∞)). +6. Applications +In this section we apply the results developed above to study the solvability and +maximal regularity of an abstract second-order integro-differential equation, see +(AEE) below. We start with some preliminaries. +Let A, B, P be closed operators on a Banach space X. +By DA, DB, DP we +denote their domains equipped with the corresponding graph norms. Let A, B and +P denote the evaluations of operators A, B and P on D′(DA), D′(DB) and D′(DP ), +respectively. More precisely, A ∈ L(D′(DA), D′(X)) is given by (Au)(φ) = A(u(φ)) +for every u ∈ D′(DA) and φ ∈ D. The operators B and P are defined in the similar +manner. Moreover, let c ∈ D′(L(Z, X)), where Z is a Banach space continuously +embedded in X. +Let us consider the following abstract degenerated, second-order problem with +the convolution term: +(AEE) +∂P∂u + B∂u + Au + c ∗ u = f +(in D′(X)) +where f ∈ D′(X) is a given X-valued distribution. The convolution term c ∗ u we +interpret as the Fourier multiplier, i.e. c ∗ u := ˆc(∆)u. For a given f ∈ D′(X), +a distribution u ∈ D′(X) is called the distributional solution of (AEE) if u ∈ +D′(DA) ∩ D′(Z), ∂u ∈ D′(DB) ∩ D′(DP ) and (AEE) holds in D′(X). +Set +(19) +Y := DA ∩ DB ∩ DP ∩ Z +with the norm +|y|Y := max(|y|A, |y|B, |y|P , |y|Z) +(y ∈ Y ). +Our first result shows how the structure of (AEE) affects the relation between the +regularities of the symbols of Fourier multipliers, which are involved in the study of +solvability of (AEE), which we address below. To make this result applicable to the +different situations (see Section 7), we need an abstract joint multiplier condition +on two symbols. For γ ∈ N we say that a sequence d : Z → L(Z, X) satisfies the +Mγ-condition with respect to a sequence a : Z → L(X, Z), if +(Mγ(a)) +[d]Mγ(a) := +max +l=0,...,γ sup +k∈Z +��kl(∆ld)(k)a(k + l) +�� +L(X) < ∞. +We write d ∈ Mγ(a), when the above holds. +Remark 6.1. Of course, if d ∈ Mγ(Z; L(Z, X)) and a ∈ l∞(Z; L(X, Z)), then +d ∈ Mγ(a). However, under further information on the boundedness of a one can +provide more suitable conditions on d to show that d ∈ Mγ(a). For example, if, + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +25 +in addition, we have that (·)a ∈ l∞(L(X, Z)), then (·)−1d ∈ Mγ(Z; L(Z, X)) yields +d ∈ Mγ(a). To see it, notice that +sup +l=0,...,γ +sup +k∈Z +∥kl−1(∆ld)(k)∥L(Z,X) < ∞ +if and only if +[(·)−1d]Mγ(Z;L(Z,X)) < ∞. +Furthermore, note that d ∈ Mγ−1(Z; L(Z, X)) implies (·)−1d ∈ Mγ(Z; L(Z, X)), +but the converse does not hold in general. For instance, for γ = 2 and the sequence +d(k) := k (k ∈ Z) we have that (·)−1d ∈ M2 but d does not satisfy the M1-condition. +We apply these observations below. +We say that a sequence d : Z → L(Z, X) satisfies the variational Marcinkiewicz +condition with respect to a sequence a : Z → L(X, Z), and write d ∈ V ar(a), if +(Var(a)) +[d]V ar(a) := [d]M0(a) + sup +j≥0 +� +2j≤|k|<2j+1 +∥∆d(k)a(k + 1)∥L(X) < ∞ +We say that a sequence d : Z → L(Z, X) satisfies the Mγ +R-condition with respect +to a sequence a : Z → L(X, Z) (and write d ∈ Mγ +R(a)), if for each l = 0, 1, ..., γ +the set {kl(∆ld)(k)a(k + l)}k∈Z is R-bounded in L(X). +Theorem 6.2. Let A, B, P, c and Y be as stated above. Assume that for every +k ∈ Z \ {0} the operator +b(k) := −k2P + ikB + A + ˆc(k) ∈ L(Y, X) +is invertible. Let +a(k) := b(k)−1, a0(k) := kBa(k), a1(k) := k2Pa(k), a2(k) := ka(k) +(k ∈ Z\{0}) +and a(0) = a0(0) = a1(0) = a2(0) = 0 ∈ L(X). Then, the following assertions +hold. +(i) Assume that a ∈ l∞(L(X, Z)) and a0, a1 ∈ l∞(L(X)). +Then, for ev- +ery γ ∈ {1, 2, 3}, if ˆc ∈ Mγ(a) (respectively, ˆc ∈ V ar(a)), then a ∈ +Mγ(Z; L(X, Y )), a0, a1 ∈ Mγ(Z; L(X)) (respectively, a ∈ V ar(Z; L(X, Y ), +a0, a1 ∈ V ar(Z; L(X))). +In addition, if the sequence a2 ∈ l∞(L(X)), then a2 ∈ Mγ(Z; L(X)) (re- +spectively, a2 ∈ V ar(Z; L(X)). +(ii) The statement (i) holds in its R-bound reformulation, that is, if, in addition, +the sequences a, a0 and a1 are R-bounded, then for every γ ∈ {1, 2, 3} the +fact that ˆc satisfies the Mγ +R-condition with respect to a sequence a, implies +that the sequences a, a0 and a1 satisfy the Mγ +R-condition. +If, in addition, a2 is R-bounded then it satisfies the Mγ +R-condition. +Proof. First we show that a ∈ l∞(L(X, Y )). +Indeed, note that each condition +imposed on ˆc implies that ˆc(·)a(·) ∈ l∞(L(X)). Since +IX := b(k)a(k) = −k2Pa(k) + ikBa(k) + Aa(k) + ˆc(k)a(k), +where IX denotes the identity operator on X, we get Aa(·) ∈ l∞(L(X)). Moreover, +Z ֒→ X implies that a ∈ l∞(L(X)), which gives our claim that a ∈ l∞(L(X, Y )). +(i) First we prove the statement for a and γ ∈ {1, 2, 3}. For γ = 1, note that by +Leibniz’ rule for difference operators we have +(20) +(∆a)(k) = −a(k)(∆b)(k)a(k + 1) +(k ∈ Z), + +26 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +and one easily gets that +(21) +(∆b)(k) = −(2k + 1)P + iB + (∆ˆc)(k) +(k ∈ Z). +Thus, by our assumptions on a0, a1 and ˆc, +(22) +(k(∆b)(k)a(k + 1))k∈Z ∈ l∞(L(X)). +Consequently, since a ∈ l∞(L(X, Y )), +(k(∆a)(k))k∈Z ∈ l∞(L(X, Y )). +For γ = 2, since the M2-condition implies the M1-condition, it is sufficient to show +that +(k2(∆2a)(k))k∈Z ∈ l∞(L(X, Y )). +For, note that for all k ∈ Z we have +(23) +(∆2a)(k) = −2(∆a)(k)(∆b)(k + 1)a(k + 2) − a(k)(∆2b)(k)a(k + 2). +Therefore, by the step for γ = 1 (see (22)), it is enough to check that +(24) +(k2(∆2b)(k)a(k + 2))k∈Z ∈ l∞(L(X)), +But this again follows directly from our assumptions, since +(∆2b)(k) = −2P + (∆2ˆc)(k) +(k ∈ Z). +For γ = 3, similarly it suffices to show that +� +k3(∆3a)(k) +� +k∈Z ∈ l∞(L(X, Y )). +For, note that for all k ∈ Z we have +(∆3a)(k) = −3(∆2a)(k)(∆b)(k + 2)a(k + 3) − a(k)(∆3b)(k)a(k + 3) +(25) +− 3(∆a)(k)(∆2b)(k + 1)a(k + 3). +By the steps for γ = 1, 2 (see (22) and (24)), it is sufficient to show the boundedness +of +� +k3(∆3b)(k)a(k + 3) +� +k∈Z ⊂ L(X). +However, since ∆3b = ∆3ˆc, it follows directly from the assumption on ˆc. +This +finishes the proof of the statement about a. +Now we turn to the sequence a0. For γ = 1, note that +(26) +(∆a0)(k) = iBa(k + 1) + ikB(∆a)(k). +Thus, combining (20) and (22) with the boundedness of a0, we get +(k(∆a0)(k))k∈Z ∈ l∞(L(X)). +For γ = 2, note that +(∆2a0)(k) = 2iB(∆a)(k + 1) + ikB(∆2a)(k) +(k ∈ Z). +Therefore, the boundedness of +� +k2(∆2a0)(k) +� +k∈Z ⊂ L(X) +follows from (23), (24), and the assumption on a0. +Finally, for γ = 3, note that +(∆3a0)(k) = 3iB(∆2a) + ikB(∆3a)(k) +(k ∈ Z). +Hence, the boundedness of +� +k3(∆3a0)(k) +� +k∈Z ⊂ L(X) + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +27 +is implied by the formula (25) and our assumptions on a0 and ˆc. It completes the +proof of the statement about a0. The fact that a1 ∈ Mγ(Z; L(X)) follows from +very similar arguments. Therefore, we omit it. +Now we assume that ˆc ∈ V ar(Z; a). Then, by (20) and (21), for every j ∈ N we +have that +� +2j≤|k|<2j+1 +∥∆a(k)∥L(X,Y ) ≤ ∥a∥l∞(L(X,Y )) +� +� +2j≤|k|<2j+1 +1 +|k|∥k(2k + 1)Pa(k + 1)∥L(X) ++ +� +2j≤|k|<2j+1 +1 +|k|∥kBa(k + 1)∥L(X) ++ +� +2j≤|k|<2j+1 +∥(∆ˆc)(k)a(k + 1)∥L(X) +� +. +Hence, by the boundedness of a0, a1 and the condition imposed on ˆc, we get that +a ∈ V ar(L(X, Y )). To show that a0 ∈ V ar(Z; L(X)), using the formulas (20),(21) +and (26), for every j ∈ N we get that +� +2j≤|k|<2j+1 +∥∆a0(k)∥L(X,Y ) += +� +2j≤|k|<2j+1 +∥Ba(k + 1) + kB(∆a)(k)∥L(X,Y ) +≤ +� +2j≤|k|<2j+1 +1 +|k|∥kBa(k + 1)∥L(X) ++ ∥a0∥l∞(L(X,Y )) +� +� +2j≤|k|<2j+1 +1 +|k|∥k(2k + 1)Pa(k + 1)∥L(X) ++ +� +2j≤|k|<2j+1 +1 +|k|∥kBa(k + 1)∥L(X) ++ +� +2j≤|k|<2j+1 +∥(∆ˆc)(k)a(k + 1)∥L(X) +� +. +Therefore, the claim about a0 follows from assumptions imposed on a0, a1 and ˆc. +In a similar manner we prove that a1 ∈ V ar(Z; L(X)). The proof of the additional +statement about a2 mimics that for a0 (formally, note that a2 = a0, when B = I). +(ii) The proof of the R-bounded version follows from the same arguments as +presented in (i) and the fact that for any Banach spaces X, Y, Z, if τ, σ ⊂ L(X, Y ) +and ρ ⊂ L(Y, Z) are R-bounded, then the families τ+σ and τ◦ρ are R-bounded. +□ +Remark 6.3. By the open mapping theorem, the assumption on the operators +b(k) made in Theorem 6.2, i.e. the invertibility of b(k) considered as an operator in +L(Y, X), is equivalent to say that b(k) considered as an operator on X is bijective +and its inverse is in L(X). Indeed, the norm | · |Y is stronger then the graph norm +of b(k). +We call a distributional solution of (AEE), the strong solution, if u ∈ W 1,1(X) +with u(t) ∈ DA, u′(t) ∈ DB ∩ DP for a.e. t ∈ R, and +Au, Bu′, c ∗ u ∈ L1(X), and Pu′ ∈ W 1,1(X). + +28 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +Recall that W 1,1(X) ⊂ C(X). Moreover, note that the existence of a strong +solution of (AEE) requires that f ∈ L1(X), and then (AEE) reads as +(Pu′(t))′ + Bu′(t) + Au(t) + (c ∗ u)(t) = f(t) +for a.e. t ∈ R +with u(0) = u(2π) and (Pu′)(0) = (Pu′)(2π). The ’prime’ in the symbols (Pu′(t))′ +and Bu′(t) refers to the classical derivative, which exists almost everywhere on R in +the topology of X. In the context of Theorem 6.2, note that the multiplier sequence +a corresponds to the solution operator of (AEE), f �→ a(∆)f, the sequences a0 and +a1 correspond to the summands B∂u and ∂P∂u in (AEE), and a2 relates to the +strong differentiability of a(∆)f. +In the following lemma we abstract some facts which allow to adapt multiplier +results from the previous section to the study of the solvability of (AEE). +Lemma 6.4. (i) Assume that A, B and P are closed, linear operators on a Banach +spaces X and c ∈ D′(L(Z, X)), where Z is a Banach space with Z ֒→ X. Then, +the distributional solution u of (AEE) is a strong one if and only if +u, ∂u, P∂u, ∂P∂u, B∂u, Au, c ∗ u ∈ L1(X). +(ii) Assume that for every function f = ek ⊗ x, where k ∈ Z and x ∈ X, the +problem (AEE) has a unique distributional solution. Then, for every k ∈ Z the +operator b(k) = −k2P + ikB + A + ˆc(k) is bijective with the bounded inverse. +Proof. (i) The necessity is readily seen. Combining Lebesgue’s differentiation theo- +rem and closedness of operators A, B and P it is straightforward to show that this +condition is sufficient. For instance, if u is a distributional solution of (AEE) such +that u ∈ W 1,1(X) and the distribution P∂u is represented by v ∈ L1(X), then for +every φ ∈ D +� +T +vφdt = (P∂u)(φ) = P +�� +T +u′φdt +� +, +and both integrals are convergent in X. The Lebesgue differentiation theorem and +closedness of P give that for a.e. t ∈ [0, 2π], u′(t) ∈ DP and Pu′(t) = v(t). Since +v ∈ L1(X), we get that Pu′ ∈ L1(X). (ii) For the surjectivity, first note that if +u ∈ D′(X) is a distributional solution of (AEE) then ˆu(k) ∈ DA ∩ Z for all k ∈ Z +and ˆu(k) ∈ DB ∩ DP for all k ∈ Z \ {0}. Indeed, note that ∂u ∈ D′(DB) if and +only if u − ˆu(0) ∈ D′(DB), and similarly for P. Therefore, if u is the corresponding +solution of (AEE) for f = ek ⊗ x, where x ∈ X and k ̸= 0, then ˆu(k) ∈ Y , and in +the case when k = 0, ˆu(0) ∈ DA ∩ Z. Testing the both sides of (AEE) on e−k ∈ D +(k ∈ Z) we get that b(k)ˆu(k) = x, which yields the suriectivity of b(k). +Suppose now that for some k ∈ Z, b(k)y = 0 for some y in the domain of b(k). +Then, the function u := ek ⊗ y satisfies +∂P∂u + B∂u + Au + c ∗ u = 0. +Therefore, the postulated uniqueness yields u ≡ 0, that is, y = 0. Consequently, +the operators b(k), k ∈ Z, are bijective. +Finally, since b(k) ∈ L(Y, X) for k ̸= 0, and b(0) ∈ L(DA ∩ Z, X), and Y, DA ∩ +Z ֒→ X, the boundedness of the inverse of b(k) on X follows from the open mapping +theorem. It finishes the proof of (ii). +□ +For E ∈ {Φ, Bs,q +Φ , F s,q +Φ +: Φ a Banach function space over (T, dt)}, we say that +the problem (AEE) has E-maximal regularity, if for every f ∈ E(X) there exists a + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +29 +unique distributional solution u of (AEE) such that +u − ˆu(0) ∈ E(Y ) +and +∂P∂u, B∂u, Au, c ∗ u ∈ E(X). +Moreover, for E ⊂ L1, we say that (AEE) is E-well-posed, if for every f ∈ E(X) +the problem (AEE) has a unique strong solution such that +(27) +u, u′, Pu′, (Pu′)′, Bu′, Au, c ∗ u ∈ E(X). +Note that by Lemma 6.4(i), if E ⊂ L1 and (AEE) has E-maximal regularity, +then (AEE) is E-well-posed if and only if +u, ∂u, P∂u ∈ E(X). +The following result is the main result of this section. In the points (a) and (b) we +address the questions of the maximal regularity and well-posedness of the problem +(AEE) under different assumptions on the geometry of the underlying Banach space +X, as well as multiplier conditions imposed on corresponding multiplier symbols. +The point (c) clarifies the phenomenon of extrapolation of Lp-maximal regularity +and Lp-well-posedness for such problem. +For the simplicity of its formulation, let LM denote the family of all Banach func- +tion spaces over (T, dt) on which the Hardy-Littlewood operator MT is bounded, i.e. +LM := {Φ a Banach space over (T, dt) : MT is bounded on Φ}. +Theorem 6.5. Let X and Z be Banach spaces such that Z ֒→ X. Let A, B and +P be closed, linear operators on a Banach space X, c ∈ D′(T; L(Z, X)) and Y has +the meaning specified in (19). For every k ∈ Z let +b(k) := −k2P + ikB + A + ˆc(k). +(a) Assume that for every k ∈ Z the operator b(k) is bijective and a ∈ l∞(L(X, Z)) +and a0, a1 ∈ l∞(L(X)), where +a(k) := b(k)−1, +a0(k) := ikBa(k), +a1(k) := −k2Pa(k) +(k ∈ Z \ {0}). +and a(0) = a0(0) = a1(0) = 0. Then, the following assertions hold. +(a1) If ˆc ∈ M2(Z; a), then for every +E ∈ +� +Bs,q +Ψ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′, Ψ ∈ LM +� +. +the problem (AEE) has the E-maximal regularity. In addition, if the se- +quence (ka(k))k∈Z is bounded in L(X) and E ⊂ L1, then (AEE) is E-well- +posed. +(a2) If ˆc ∈ V ar(Z; a) and, in addition, X has UMD-property, then the conclusion +of (a1) holds for each +E ∈ +� +Bs,q +Φ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′ ∈ LM +� +. +(b) Assume that for every k ∈ Z the operator b(k) is bijective, the sequences a, +and a0, a1 are R-bounded in L(X, Z) and L(X), respectively, and X has the UMD +property. +If ˆc ∈ M1 +R(Z; a), then for every +E ∈ +� +Φ, Bs,q +Φ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′ ∈ LM +� +the problem (AEE) has E-maximal regularity. + +30 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +In addition, if (ka(k))k∈Z is R-bounded in L(X), then for every E ⊂ L1, the +problem (AEE) is E-well-posed. +In particular, the last statement holds for each +E ∈ +� +Φ, Bs,q +Φ , F s,r +Φ +: s > 0, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′ ∈ LM +� +. +(c) Assume that the problem (AEE) has Lp-maximal regularity for some p ∈ +(1, ∞) (respectively, (AEE) is Lp-well-posed). +If ˆc ∈ M3(Z; a), then for every +E ∈ +� +Φ, Bs,q +Ψ , F s,q +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′, Ψ ∈ LM +� +it has E-maximal regularity (respectively, in addition, if E ⊂ L1, it is E-well-posed). +Proof. (a1) Combining Theorem 6.2 with Theorem 5.3(i) and (iii) we infer that +a ∈ ME(T; X, Y ) and a0, a1, Aa(·) ∈ ME(T; X, X). In particular, for every f ∈ +E(X), if we put u := a(∆)f + b(0)−1 ˆf(0), then u ∈ D′(DA) ∩ D′(Z) and ∂u ∈ +D′(Y ) ⊂ D′(DB) ∩ D′(DP ). It is easy to check that u is a distributional solution +of (AEE). Since the Fourier coefficients of distributions are uniquely determined, +the injectivity of the operators b(k), k ∈ Z, gives that u is the unique solution of +(AEE). +Moreover, note that +∂P∂u = a1(∆)f, B∂u = a0(∆)f, Au = [Aa(·)](∆)f ∈ E(X) +and, since f ∈ E(X), +c ∗ u = ˆc(∆)a(∆)f = ∂P∂u + B∂u + Au − f ∈ E(X). +It proves the E-maximal regularity of (AEE). +For the additional statement, again by Theorem 6.2 and Theorem 5.3(i), we get +that a2(k) := ika(k) and a3(k) := ikPa(k), k ∈ Z, are in ME(T; X, X). Since +Y ֒→ X and DA ∩ Z ֒→ X, we have that a ∈ ME(T; X, X). Hence, if f ∈ E(X), +then for the corresponding solution u = a(∆)f + b(0)−1 ˆf(0) of (AEE) we get that +u, ∂u = a2(∆)f, P∂u = a3(∆)f ∈ E(X). +If, in addition, E ⊂ L1, Lemma 6.4(i) shows that u is a strong solution and (27), +that is, the E-well-posedness of (AEE). It completes the proof of (a1). +Relying on Theorem 6.2, the parts (ii) and (iii) of Theorem 5.3, and Lemma +6.4(i), the proof of the statement (a2) follows the similar arguments presented for +the proof of (a1). Therefore, we omit them. +For the statement (b), note that Theorem 5.5(ii) reduces the proof of (b) to the +arguments provided in the proof of (a1). +Finally, for the proof of (c), by Lemma 6.4(ii), the Lp-maximal regularity of +(AEE) implies that for each k ∈ Z, b(k) is bijective with a bounded inverse. That +is, a(k) = b(k)−1 ∈ L(X, Y ) for all k ∈ Z \ {0} and a(0) = b(0)−1 ∈ L(X, DA ∩ Z) +(see Remark 6.3(i)). It is straighforward to check that the corresponding solution +operator for (AEE) is given by the Fourier multiplier a(∆). +We show that a ∈ l∞(L(X, Y )) and a0, a1 ∈ l∞(L(X)). +The Lp-maximal +regularity implies that the maps +Lp(X) ∋ f �→ a(∆)f ∈ Lp(Y ) and Lp(X) ∋ f �→ B∂a(∆)f, ∂P∂a(∆)f ∈ Lp(X), +are well-defined and (by the uniqueness) linear. By the closed graph theorem, it is +straightforward to see that these maps are bounded. Moreover, since a(∆)(ek⊗x) = + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +31 +ek ⊗ b(k)−1x for every x ∈ X and k ∈ Z, we infer that for k ̸= 0 +|b−1(k)x|Y = ∥a(∆)(ek ⊗ x)∥Lp(Y ) ≤ ∥a(∆)∥L(Lp(X),Lp(Y ))|x|X +and similarly +|kBb(k)−1x|X ≤ ∥B∂a(∆)∥L(Lp(X))|x|X, |k2Pb(k)−1x|X ≤ ∥∂P∂a(∆)∥L(Lp(X))|x|X. +It gives our claim. Combining it with the assumption on ˆc, by Theorem 6.2 and +Theorem 5.5(i), we are in the position to use the same arguments as in (a1) to get +the desired assertion on the E-maximal regularity of (AEE). For the additional +statement, similarly as above, the Lp-well-posedness of (AEE) implies that the +sequences a2(k) = ika(k) and a3(k) = ikPa(k) (k ∈ Z) are in ME(T; X, X). Hence, +the proof follows the corresponding lines of the proof of (a1). This completes the +proof of (c). +□ +Remark 6.6. Since for each M ∈ {Mγ, Mγ +R, V ar}, if (kBa(k))k∈Z, (k2Pa(k))k∈Z +are in M(Z, L(X)) then (Ba(k))k∈Z, (kPa(k))k∈Z, (Pa(k))k∈Z ∈ M(Z; L(X)), the +proof of Theorem 6.5 shows that the E-maximal regularity of (AEE) concluded +in each of its statements (a), (b), (c) (no additional assumptions on (ka(k))k∈Z is +required) gives that the problem +∂2Pu + ∂Bu + Au + c ∗ u = f +is E-well-posed whenever E ⊂ L1. That is, for each f ∈ E(X) there exists a unique +distributional solution u such that u − ˆu(0) ∈ E(Y ), Pu ∈ W 2,1(X) ⊂ C1(X), +Bu ∈ W 1,1(X) ⊂ C(X) and +(Pu)′′(t) + (Bu)′ + Au(t) + c ∗ u(t) = f(t) +a.e. t ∈ [0, 2π]. +In this context note that if ∂2Pu, ∂Bu ∈ Bs,q +Φ (X) ⊂ Φ(X) for some s > 0, then by +Proposition 2.3(iii), we immediately get Pu ∈ W 2,1(X) and Bu ∈ W 1,1(X). +As was already mentioned in Remark 6.1 one can impose separate conditions +on a and ˆc which imply their joint condition ˆc ∈ Mγ(a) (or others). It leads to +the following characterisation of the maximal regularity and well-posedness of the +problem (AEE). Here, we assume that Z = X; see Remark 6.11 below. We start +with the characterization and extrapolation of the maximal regularity of (AEE). +Theorem 6.7. Let A, B and P be closed, linear operators on a Banach space X. +Let c ∈ D′(L(X)). +(a) Assume that ˆc ∈ M2(Z; L(X)). Then, the following assertions are equivalent. +(i) The problem (AEE) has E-maximal regularity for every E such that +E ∈ +� +Bs,q +Ψ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′, Ψ ∈ LM +� +. +(ii) The problem (AEE) has Bs,q +Lp -maximal regularity for some p ∈ (1, ∞) and +q ∈ [1, ∞]. +(iii) For every k ∈ Z the operator b(k) := −k2P + ikB + A + ˆc(k) is bijective +and the sequences +� +b(k)−1� +k∈Z , +(kBb(k)−1)k∈Z, +(k2Pb(k)−1)k∈Z +are bounded in L(X). +(b) Assume that X has the UMD property and ˆc ∈ M1 +R(Z; L(X)). +Then, the +following assertions are equivalent. + +32 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +(i) The problem (AEE) has E-maximal regularity for every E such that +E ∈ +� +Φ, Bs,q +Ψ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′, Ψ ∈ LM +� +. +(ii) The problem (AEE) has Lp-maximal regularity for some p ∈ (1, ∞). +(iii) For every k ∈ Z the operator b(k) := −k2P + ikB + A + ˆc(k) is bijective +and the sequences +(b(k)−1)k∈Z, +(kBb(k)−1)k∈Z, +(k2Pb(k)−1)k∈Z +are R-bounded in L(X) . +Remark 6.8. (a) It is easily seen that for (AEE) with B = I ∈ L(X) one can +relax the assumption on ˆc in the statement (a) and (b) of Theorem 6.7 to (·)−1ˆc ∈ +M2(Z; L(X)) in (a), and to (·)−1ˆc ∈ M1 +R(Z; L(X)) in (b), respectively. +Furthermore, in the case when P = I ∈ L(X), one can further weaken the as- +sumption on ˆc, namely, to (·)−2ˆc ∈ M2(Z; L(X)) in (a), and to (·)−2ˆc ∈ M1 +R(Z; L(X)) +in (b), respectively; cf. also Remark 6.1. +(b) The assertion (ii) in Theorem 6.7(a) can be replaced with the following one +(ii’) The problem (AEE) has E-maximal regularity for some E such that +E ∈ +� +Bs,q +Ψ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′, Ψ ∈ LM +� +. +Indeed, one can easily check that the maximal regularity of (AEE) with respect +to every such E implies the boundedness of the sequences stated (iii). It is not clear +if E-maximal regularity of (AEE) for general E yields the R-boundedness of these +sequences. +(c) In a similar manner to that in Theorem 6.7 one can formulate the statement +corresponding to (a2) of Theorem 6.5. Its proof follows the same arguments. We +leave it to the interested reader. +Proof of Theorem 6.7. (a) Of course (i)⇒(ii). For (ii)⇒(iii), since ek ⊗x ∈ Bs,q +Lp (X) +for every k ∈ Z and x ∈ X (see, e.g. Proposition 2.3(i)), by Lemma 6.4(ii) we get +that for all k ∈ Z the operator b(k) is bijective with inverse in L(X, Y ) when k ̸= 0, +and in L(X, Z) when k = 0. Analogously as in the proof of Theorem 6.5(c), by the +closed graph theorem, we infer that the maps +Bs,q +Lp (X) ∋ f �→ a(∆)f ∈ Bs,q +Lp (Z) and +Bs,q +Lp (X) ∋ f �→ B∂a(∆)f, ∂P∂a(∆)f ∈ Bs,q +Lp (X) +are bounded, where a(k) := b−1(k), k ∈ Z. Since for every k ∈ Z and x ∈ X +∥ek ⊗ x∥Bs,q +Lp (X) = +� � +j≥0 +2sjqψj(k)q� 1 +q |x|X, +we get the boundedness of desired sequences in (iii). The proof of (iii)⇒(i) follows +directly from Theorem 6.5(a1); see Remark 6.1. +The proof of (b) mimics the same arguments, we only recall here that each +sequence in Mp(Z, X, Y ) (for arbitrary Banach spaces X and Y ) is necessarily +R-bounded; see [31] or [7]. +□ +Now we formulate analogous result for the well-posedness of (AEE). Its proof +follows the lines of the proof of Theorem 6.7 with straightforward modifications. +Therefore, we omit it. + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +33 +Theorem 6.9. Let A, B and P be closed, linear operators on a Banach space X. +Let c ∈ D′(L(X)). +(a) Assume that (k−1ˆc(k))k∈Z ∈ M2(Z; L(X)). Then, the following assertions are +equivalent. +(i) The problem (AEE) is E-well-posed for every E ⊂ L1 such that +E ∈ +� +Bs,q +Ψ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′, Ψ ∈ LM +� +(ii) The problem (AEE) is Bs,q +Lp -well-posed for some p ∈ (1, ∞) and q ∈ [1, ∞]. +(iii) For every k ∈ Z the operator b(k) := −k2P + ikB + A + ˆc(k) is bijective +and the sequences +� +kb(k)−1� +k∈Z , (kBb(k)−1)k∈Z, (k2Pb(k)−1)k∈Z +are bounded in L(X). +(b) Assume that X has the UMD property and (k−1ˆc(k))k∈Z ∈ M1 +R(Z; L(X)). +Then, the following assertions are equivalent. +(i) The problem (AEE) is E-well-posed for every E such that +E ∈ +� +Φ, Bs,q +Ψ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′, Ψ ∈ LM +� +. +(ii) The problem (AEE) is Lp-well-posed for some p ∈ (1, ∞). +(iii) For every k ∈ Z the operator b(k) := −k2P + ikB + A + ˆc(k) is bijective +and the sequences +(kb(k)−1)k∈Z, +(kBb(k)−1)k∈Z, +(k2Pb(k)−1)k∈Z +are R-bounded in L(X). +Remark 6.10. The similar statements to those stated in Remark 6.8 hold in the +context of Theorem 6.9. In particular, for (AEE) with P = I ∈ L(X) in (AEE), +one can relax the assumption on ˆc in the statements (a) and (b) of Theorem 6.9 to +(·)−2ˆc ∈ M2(Z; L(X)) in (a) and to (·)−2ˆc ∈ M1 +R(Z; L(X)) in (b), respectively. +It is worth noticing that according to Remark 6.1, if ˆc ∈ M1(Z; L(X)), then +(·)−1ˆc ∈ M2(Z; L(X)). Thus, for c such that ˆc ∈ M1(Z; L(X)) the equivalence +in Theorem 6.9(a) also holds. This is a strictly stronger condition, but in some +situations it might be easier to verify. +Remark 6.11. If we modify the notion of the well-posedness of (AEE) (which +sometimes can be motivated by a special form of the convolution therm; see, e.g. +(28)) by replacing the condition u ∈ W 1,1(X) by a stronger one, e.g. u ∈ W 1,1(Z), +where Z is a Banach space such that Z ֒→ X, then the above characterizations, The- +orems 6.7 and 6.9 (where Z = X) can be easily adjusted to such modified setting. +The proofs of such modifications have the same pattern as those of Theorems 6.7 +and 6.9. Here, we do not provide such reformulation of the above characterization +results, which corresponds to such stronger notion of well-posedness of (AEE). +For our further purposes we present a counterpart of Theorems 6.7 and 6.9 for +a convolutor c ∈ D′(L(Z, X)) with and arbitrary Z ֒→ X, which we apply in the +next section. +Theorem 6.12. Let A, B and P be closed, linear operators on a Banach space X. +Let c ∈ D′(L(Z, X)), where Z is a Banach space continuously embedded in X. +(a) Assume that ˆc ∈ M2(Z; L(Z, X)). Then, the following assertions are equiv- +alent. + +34 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +(i) The problem (AEE) has E-maximal regularity (respectively, is E-well-posed) +for every E such that +E ∈ +� +Bs,q +Ψ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′, Ψ ∈ LM +� +(respectively, in addition, E ⊂ L1). +(ii) The problem (AEE) has Bs,q +Lp -maximal regularity (respectively, is Bs,q +Lp -well- +posed) for some p ∈ (1, ∞) and q ∈ [1, ∞]. +(iii) For every k ∈ Z the operator b(k) := −k2P + ikB + A + ˆc(k) is bijective +and the sequences +� +b(k)−1� +k∈Z ⊂ L(X, Z), (kBb(k)−1)k∈Z ⊂ L(X), (k2Pb(k)−1)k∈Z ⊂ L(X) +are bounded (respectively, in addition, +� +kb(k)−1� +k∈Z ⊂ L(X) is bounded). +(b) Assume that X has the UMD property and ˆc ∈ M1 +R(Z; L(Z, X)). Then, the +following assertions are equivalent. +(i) The problem (AEE) has E-maximal regularity (respectively, is E-well-posed) +for every E such that +E ∈ +� +Φ, Bs,q +Ψ , F s,r +Φ +: s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞), Φ, Φ′, Ψ ∈ LM +� +. +(respectively, in addition, E ⊂ L1). +(ii) The problem (AEE) has Lp-maximal regularity (respectively, is E-well-posed) +for some p ∈ (1, ∞). +(iii) For every k ∈ Z the operator b(k) := −k2P + ikB + A + ˆc(k) is bijective +and the sequences +(b(k)−1)k∈Z ⊂ L(X, Z), +(kBb(k)−1)k∈Z ⊂ L(X), +(k2Pb(k)−1)k∈Z ⊂ L(X) +are R-bounded (respectively, in addition, +� +kb(k)−1� +k∈Z ⊂ L(X) is R-bounded). +7. Particular forms of (AEE) +In this section we specialize our general results from the previous section to +particular forms of the abstract problem (AEE), which have been studied in the +literature. In particular, Theorems 6.7 and 6.9 extend many results from a long +series of articles, where such characterisations have been studied progressively; [32, +58, 7, 8, 47, 53, 23, 54, 57, 24, 49, 40, 56, 38, 20, 21, 22] and the references therein. +7.1. Integro-differential equations. Here, we consider (AEE) for special classes +of convolutors c, which arise in the (abstract) reformulation of the integro-differential +equations describing physical processes in materials with fading memory. We start +with equations, where a convolutor c is given by the so-called finite delay operators, +that is, c is of the form +(28) +c ∗ u := Hu· + Gu′ +·, +where H, G ∈ L(Lp(T; X), X) for some p ∈ (1, ∞), ut(s) := u(t + s) and, if, for +instance, u ∈ W 1,p(T; X), u′ +t(s) = u′(t + s) (s ∈ [0, 2π]). Note that if for all k ∈ Z +and x ∈ X we set +Hkx := H(ek ⊗ x) +and +Gkx := G(ek ⊗ x), + +THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS +35 +then � +Hu·(k) = Hkˆu(k) and � +Gu·(k) = Gkˆu(k) for all u ∈ Lp(X). Consequently, the +sequences h := (Hk)k∈Z and g := (Gk)k∈Z are in Mp(Z; X, X). It shows that one +can extend the meaning of (28) to all u ∈ D′(X) putting +c ∗ u = h(∆)u + ∂g(∆)u = h(∆)u + g(∆)(∂u), +that is, ˆc(k) = h(k) + ikg(k), k ∈ Z. +This abstract reformulation of (28) leads to the following characterisation of the +maximal regularity and well-posedness for this class of equations. +Corollary 7.1. Let A, B and P be closed, linear operators on a Banach space X. +Let (h(k))k∈Z, (g(k))k∈Z ⊂ L(X). +(1) Assume that (·)−αh ∈ M2(Z; L(X)) and (·)1−αg ∈ M2(Z; L(X)) for some +α ∈ {0, 1, 2}. Let ˆc := h + i(·)g. Then, the following statements are true. +(i) If α = 0, then the equivalence (i) ⇔ (ii) ⇔ (iii) of Theorem 6.7(a) holds +for (AEE) with ˆc. +(ii) If α = 1, then the conclusion of Theorem 6.9(a) holds, and if, in addition, +B = I, then the conclusion of Theorem 6.7(a) holds too. +(iii) If α = 2 and, in addition, P = I, then the conclusion of Theorem 6.7(a) +and Theorem 6.9(a) hold. +(2) Let X have the UMD property. Assume that (·)−αh ∈ M1 +R(Z; L(X)) and +(·)1−αg ∈ M1 +R(Z; L(X)) for some α ∈ {0, 1, 2}. Then the analogous statements to +those of the point (1) hold, that is, the statements which are obtained by replacing +the assertion (a) of Theorems 6.7 and 6.9 with the corresponding (b) therein. +Remark 7.2. (a) One can readily seen, that Corollary 7.1 extends several related +results from the literature, where the convolutors c are given by finite delay opera- +tors; see, e.g. [22], [21] and also references therein. +(b) Moreover, Theorem 7.1(2) covers [38, Theorem 3.4]. Indeed, recall that Fu +and Li in [38] studied the problem (AEE) with P = I and the convolutor c given +by the infinite delay operators, that is, +c ∗ u = Hu· + Gu′ +·, +where H, G : B → X are bounded, linear operators, B is a space of X-valued +functions on R− := (−∞, 0], which is axiomatically defined in [38], and u· is given +as before by ut(s) := u(t + s) (s ∈ R−). Since ek, k ∈ Z, considered as functions +on R−, belong to Cb(R−) ⊂ B, one can define sequences h and g similarly as +above, i.e. h(k)x := H(ek(·)x) and g(k)x := (ek(·)x), k ∈ Z. Then, c ∈ D′(X) +with ˆc(k) = h(k) + ikg(k), k ∈ Z. +The axioms of B easily give that h and g +are R-bounded in L(X); see e.g. the proof of [38, Lemma 3.2]. Finally, one can +easily check that the condition (ii) of [38, Theorem 3.4] implies the assumptions of +Corollary 7.1(2) for α = 2 (with P = I). +Analogously, one can check that Theorem 7.1(a) extends [38, Theorems 4.4 and 4.7]; +see also references therein, as well corresponding results in [48, 23, 24, 54, 40]. +To illustrate Theorem 6.12, we remark on the integro-differential equations with +infinite delays, which have been studied in [47]. Namely, in the problem studied in +[47] (i.e., (AEE) with P = 0 and B = I) the infinite delay operator is given by +c ∗ u := d(·)A ∗ u, + +36 +SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI +where d ∈ L1 +loc(R+) ⊂ L1 +loc(R) is such that the Fourier transform of d exists at +points k ∈ Z and the sequence (Fd(k))k∈Z satisfies some further assumptions cor- +responding to a γ-regularity (γ = 1, 2); see [47, Theorems 2.12 and 3.9]). It is +readily seen that, these assumptions imply ˆc ∈ Mγ +R(Z; L(Z, X)) with Z := DA +(γ = 1, 2). Therefore, Theorem 6.12 might also be seen as an extension of those +results. +7.2. Differential equations. We conclude with some remarks on (AEE) with +c ≡ 0, that is, +(29) +∂P∂u + B∂u + Au = f +(in D′(X)). +The particular form of (29) with P = 0 and B = I was, in a sense, a prototype for +further periodic extensions done in the literature; see Arendt and Bu [7, Theorem +2.3 and Corollary 2.4] for the corresponding result on the Lp-well-posedness and +its p-independence. +For the studies of the other forms of (29) in connection to +their well-posedness in the context of the classical Lebesgue-Bochner, Besov, and +Triebel-Lizorkin spaces (i.e., corresponding to Φ = Lp) see for instance [7, Section +2], [8, 55, 19], as well as the references provided for the integro-differential equations. +Theorem 6.9 (for c ≡ 0), in particular, gives an extension of those results and also +provides periodic variant of extrapolation results known in the euclidean setting; +see, e.g. [60, 12, 27, 29, 37, 51] and references therein. We leave the formulation +of this result to the interested reader. 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Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory in- +tegrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, +1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. +[67] ˇZ. ˇStrkalj and L. Weis, On operator-valued Fourier multiplier theorems, Trans. Amer. Math. +Soc. 359 (2007), 3529-3547. +[68] H.Triebel, Spaces of Besov-Hardy-Sobolev Type, Teubner-Texte Math. 15, Leipzig: Teubner +1978. +[69] L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Math. Ann. +319 (2001), 735-758. +Sebastian Kr´ol, Faculty of Mathematics and Computer Science, Adam Mickiewicz +University in Pozna´n, ul. Uniwersytetu Pozna´nskiego 4, 61-614 Pozna´n, Poland +Email address: sebastian.krol@amu.edu.pl +Jaros�law Sarnowski, Faculty of Mathematics and Computer Science, Nicolaus Coper- +nicus University in Toru´n, ul. Chopina 12/18, 87-100 Toru´n, Poland +Email address: jsarnowski@doktorant.umk.pl + diff --git a/h9FMT4oBgHgl3EQf4TEa/content/tmp_files/load_file.txt b/h9FMT4oBgHgl3EQf4TEa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..af83877d755f80c679684f8f7c1d673527534897 --- /dev/null +++ b/h9FMT4oBgHgl3EQf4TEa/content/tmp_files/load_file.txt @@ -0,0 +1,1822 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf,len=1821 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='12451v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='FA] 29 Jan 2023 THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS WITH APPLICATION TO ABSTRACT EVOLUTION EQUATIONS SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' By methods of harmonic analysis, we identify large classes of Ba- nach spaces invariant of periodic Fourier multipliers with symbols satisfying the classical Marcinkiewicz type conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Such classes include general (vector- valued) Banach function spaces Φ and/or the scales of Besov and Triebel- Lizorkin spaces defined on the basis of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' We apply these results to the study of the well-posedness and maximal regularity property of an abstract second-order integro-differential equation, which models various types of elliptic and parabolic problems arising in dif- ferent areas of applied mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In particular, under suitable conditions imposed on a convolutor c and the geometry of an underlying Banach space X, we characterize the conditions on the operators A, B and P on X such that the following periodic problem ∂P ∂u + B∂u + Au + c ∗ u = f in D′(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) is well-posed with respect to large classes of function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' The obtained results extend the known theory on the maximal regularity of such problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Introduction Fourier multipliers with operator-valued symbols have found many applications in the theory of abstract evolution equations, in particular, in connection with solvability (well-posedness) and regularity of integro-differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' A large class of such equations can be modelled by the following abstract, degenerated second-order problem with a convolution term: (AP) (Pu′)′ + Bu′ + Au + c ∗ u = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Here, A, B, P denote closed linear operators on a Banach space X and c is an operator-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For particular forms of (AP), the studies of their well-posedness on diverse vector-valued function spaces have been increased with occurring two seminal pa- pers by Amann [2] and Weis [69], where operator-valued counterparts of classical multiplier theorems for Besov and Lebesgue-Bochner spaces on R are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Those results indicated a right form of multiplier conditions (see [31], [30]), which have been further adapted to different situations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' periodic multiplier results in [7, 8, 67], which are relevant to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' 42B37, 42A45, 45N05, 46N20, 43A15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' integro-differential equations, maximal regularity, well-posedness, pe- riodic Fourier multipliers, Hardy-Littlewood maximal operator, Besov spaces, Triebel-Lizorkin spaces, Rubio de Francia iteration algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' 1 2 SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI In the literature one can extract two lines of research corresponding to such studies: namely, when (AP) is considered in the euclidean setting, that is, on R or R+, and in the periodic one, that is, on T := R/Z (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=', when the periodic conditions are imposed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Each of these lines is represented by a long series of papers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' to mention a few representative results, see for the first one, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' [2, 69, 59, 6, 28, 11, 27, 5, 4, 51], and for the second one, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' [7, 8, 47, 53, 23, 54, 57, 24, 49, 40, 56, 38, 20, 21, 22] (as well as the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Such studies correspond to the well-known research program formulated by Amman in [1, Section 3] and labelled as ’pairs of maximal regularity’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In both settings, the basic idea for such studies is the same and relies on multiplier theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Roughly, by the theory of vector-valued distributions, the well-posedness and regularity questions for (AP), reduce to checking if corresponding Fourier multipliers with operator- valued symbols are bounded in a space under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Such Fourier multiplier operators arise naturally via the representation formula for corresponding solution operators associated to a given form of (AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In the euclidean setting, the so-called phenomenon of the extrapolation of Lp- maximal regularity, which can be simply considered as a special variant of the well-posedness with respect to various Banach function spaces, have been studied recently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' [59, 11, 27, 37, 29, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Beyond a natural theoretical interest in this phenomenon, the maximal regularity with respect to a more general function space is an important tool for the study of associated non-linear problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' [52, 60, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In the periodic case, the well-posedness and maximal regularity were addressed mainly in the context of the classical Lebesgue, Besov, Triebel-Lizorkin spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' the corresponding series of the references mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' The aim of this article is to extend such results to a much wider context of general Banach function spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see the main results of this paper, Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='7, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='9, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In particular, we clarify the phenomenon of the extrapolation of Lp-maximal regularity for several periodic evolution equations modelled by (AP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='12, as well as Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' These results, in particular, provide counterpart of the euclidean line of research mentioned above for the periodic situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In addition, we provide a convenient framework for such studies, which reveals an underlying structure, allows to simplify and unify technicalities mainly resulting from the fact that we deal with higher order Marcinkiewicz’s conditions, and allows to handle different questions (distributional or strong solvability, maximal regularity) in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' It is achieved with the help of two auxiliary results Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='2 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' On the other-hand, it allows to extend many results from the related literature in several ways, by showing that assumptions usually made to get those results in the Lp-setting are sufficient for a large class of Banach function spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' To establish such results we make a revision of underlying multiplier results from [7, 8, 25] applied in the context of Lp setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Roughly, our main periodic multiplier results, see Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='5, assert that the standard multiplier conditions, which in the literature usually are imposed on the symbol of a Fourier multiplier to get its boundedness on the classical (vector-valued) Lebesgue, Besov or Triebel- Lizorkin spaces, are sufficient for its boundedness on much larger classes of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Such classes include general Banach function spaces Φ and/or the scales of Besov and Triebel-Lizorkin spaces defined on the basis of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In particular, these results THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS 3 extend [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3], [8, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='5] and [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Our proofs differ from the proofs presented in those papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' We rely on direct maximal function estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see the proofs of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' We conclude with a remark on the strategy of the proof of our abstract extrap- olation result, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Since the theory of periodic distributions presents a simplification in comparison to that on the real line, one could expect the same in the context of multiplier theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In fact, we show that such simplification is reflected mainly in the representation formulas for periodic Fourier multipliers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='1 for the further comments and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In particular we do not ad- dress here problems which appear in the euclidean setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' [44, Problems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3] and [51, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' However, at some points (for instance, when the interplay between the regularity of the symbol and its Fourier transform is crucial), the euclidean setting presents some benefits in comparison to the periodic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For this reason, instead of trying to prove some periodic results in a complete analogy to corresponding ones known in the euclidean setting, we deduce them from their euclidean counterparts via transference techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='5 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' also Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' The tools for such transference methods are workout in Section 4, which may be of independent interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see operator-valued variants of Jodeit’s type theorem, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='4, as well as Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' The organization of the paper is well-reflected by the titles of the following (sub)sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Auxiliary results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' We refer the reader to the monograph by Bennett and Sharpley [13] for the background on Banach function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Here, we mention only several facts we use in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Let Ψ be a Banach function space over (G, dt), where G denotes R or T equipped with the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' It means that Ψ is a Banach space, which is an order ideal of L0 := L0(G, dt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' for every f ∈ L0 and g ∈ Ψ if |f| ≤ |g|, then f ∈ Ψ and ∥f∥Ψ ≤ ∥g∥Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Here, L0 stands for the space of all complex measurable functions on G (as usual, any two functions equal almost everywhere are identified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Moreover, Ψ has Fatou’s property, and by the Lorentz-Luxemburg theorem [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='10], (Ψ′)′ = Ψ with equal norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Here, Ψ′ denotes the (K¨othe) dual (or associated space) of Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' We define the vector-valued variant of Banach function spaces Ψ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Let X be a Banach space with norm | · |X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Set Ψ(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) := {f : G → X strongly measurable : |f|X ∈ Ψ} and ∥f∥Ψ(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='X) := ∥|f|X∥Ψ for f ∈ Ψ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Throughout, the symbol Φ is reserved to denote a Banach function space over (T, dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Note that if a function e0(τ) := 1 (τ ∈ T) is in Φ, then by the ideal property of Φ we get that L∞(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) ⊂ Φ(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) =: Φ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In particular, if P(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) denotes the set of all X-valued polynomials on T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' P(X) := P(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) := � N � k=−N ek ⊗ xk : N ∈ N, xk ∈ X � then P(X) ⊂ Φ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Here, (ek ⊗ x)(τ) := ek(τ)x, where ek(τ) := τ k (τ ∈ T, k ∈ Z, x ∈ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' 4 SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI Moreover, we introduce a variant of vector-valued Besov and Triebel-Lizorkin spaces corresponding to a Banach function space Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Let D′(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) := L(D, X), where D := D(T) is a space of all complex-valued infinitely differentiable functions on T equipped in the usual locally convex topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' We refer to [63, Section 3] or [36] for the backgrounds on the scalar distributions on T, and to [8, Section 2] for their vector-valued counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For instance, relying on [8, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='1], it is readily seen that for each ψ ∈ C(R) with the compact support, the operator ψ(∆) given by ψ(∆)f := � k∈Z ek ⊗ ψ(k) ˆf(k) (f ∈ D′(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X)) is in L(D′(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Let {ψj}j∈N0 be the resolution of the identity on R generated by a function ψ ∈ C∞(R) such that ψ ≡ 1 on [−1, 1] and supp ψ ⊂ [−2, 2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' ψ0 := ψ, ψj := ψ(2−j·) − ψ(2−j+1·) for j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' One can check that {ψj(∆)}j∈N0 is the resolution of the identity operator on D′(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' for every f ∈ D′(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) � j≤N ψj(∆)f = ψ(2−N∆)f → f in D′(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Let Φ be a Banach function space over (T, dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For all s ∈ R and q ∈ [1, ∞] we set (with usual modification when q = ∞): Bs,q Φ (T, X) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 f ∈ D′(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) : ∥f∥Bs,q Φ (T,X) := \uf8eb \uf8ed ∞ � j=0 ∥2sjψj(∆)f∥q Φ(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='X) \uf8f6 \uf8f8 1/q < ∞ \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe , F s,q Φ (T, X) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 f ∈ D′(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) : ∥f∥F s,q Φ (T,X) := ������� \uf8eb \uf8ed ∞ � j=0 |2sjψj(∆)f|q X \uf8f6 \uf8f8 1/q������� Φ < ∞ \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For G = R and a general Banach function spaces Ψ over (R, dt), the corre- sponding generalized vector-valued Besov Bs,q Ψ (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) and Triebel-Lizorkin spaces F s,q Ψ (R, X) were introduced in [51, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In the case when G = T, the vector- valued counterpart of the classical Besov spaces Bs,q p (T), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' for Φ = Lp over (T, dt), was introduced in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In the both cases (G = T or G = R), under some additional assumption on Ψ, one can show that the spaces Bs,q Ψ (G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) and F s,q Ψ (G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) share most of the properties of their well-known scalar prototypes which correspond to Ψ = Lp and X = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For our further purposes, we only need a few basic properties of the spaces Bs,q Φ (T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) and F s,q Φ (T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) corresponding to a general Banach func- tion space Φ over (T, dt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For their proofs we need a preliminary result on the boundedness of Fourier multipliers ψj(∆), j ∈ N0, (and other ones) on the underlying space Φ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Then, the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3 can be carried out in an analogy to the non-periodic case when G = R as it has been treated in [51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see [51, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='6] and [51, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' However, it should be pointed out that in a comparison to the proofs of some results in the case G = R, the proofs of their periodic counterparts admit an essential simplification, which we indicate below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see also Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS 5 As it could be already noted above, we omit ’T’ in the symbols of spaces over (T, dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Similarly, in the scalar case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' when X = C, C is also omitted in the corresponding symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For instance, Bs,q Φ (X) stands for Bs,q Φ (T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) and C∞ c (R) denotes C∞ c (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' C), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Preliminary results on boundedness of periodic multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For a polynomially bounded sequence m : Z → L(X, Y ) we write ˇm to denote the corre- sponding periodic distribution in D′(L(X, Y )), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' ˇm := � k∈Z ek ⊗ m(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Let m : Z → L(X, Y ) be such that ˇm is in L∞(L(X, Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Then, for every Banach function space Φ over (T, dt) such that L∞ ⊂ Φ ⊂ L1 the operator m(∆) given by m(∆)f := � k ek ⊗ m(k) ˆf(k) f ∈ D′(X) is in L(Φ(X), Φ(Y )) with ∥m(∆)∥L(Φ(X),Φ(Y )) ≤ cΦ∥ ˇm∥L∞∥χT∥Φ, where cΦ denotes the norm of embedding operator from Φ into L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' A standard argument shows that Φ ֒→ L1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' ∥g∥L1 ≤ cΦ∥g∥Φ (g ∈ Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Since for every η ∈ L∞, g ∈ Φ and τ ∈ T we have |(η ∗ g)(τ)| ≤ cΦ∥η∥L∞∥g∥Φ we infer that ∥η ∗ g∥Φ ≤ cΦ∥η∥L∞∥χT∥Φ∥g∥Φ, where χT is the characteristic function of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Since for every f ∈ L1(X) we have |(m(∆)f)(τ)|Y = |( ˇm ∗ f)(τ)|Y ≤ � ∥ ˇm∥L(X,Y ) ∗ |f|X � (τ) (τ ∈ T), the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' □ In particular, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='1 shows that each operator ψj(∆), j ∈ N0, is bounded on Φ(X) if L∞ ⊂ Φ ⊂ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' To show their uniform boundedness on Φ(X) we need an additional assumption on the boundedness of the Hardy-Littlewood maximal operator on Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Recall that the Hardy-Littlewood maximal operators MT and MR are defined by MTf(τ) := sup ǫ>0 1 ǫ � Γ(τ,ǫ) |f(ζ)| |dζ| (τ ∈ T) for f ∈ L1(T), where Γ(τ, ǫ) := T ∩ {z ∈ C : |z − τ| ≤ ǫ}, and MRf(t) := sup ǫ>0 1 2ǫ � [t−ǫ,t+ǫ] |f(s)|ds (t ∈ R) for f ∈ L1 loc(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In the view of the standard identification between the function f on T with its 2π-periodic extension �f on R, �f(t) := f(eit), t ∈ R, there exists a constant c > 0 such that c−1(MR �f)(t) ≤ (MTf)(eit) ≤ c(MR �f)(t) (f ∈ L1(T), t ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Note that the assumption that MT is bounded on a Banach function space Φ implies that L∞ ⊂ Φ ⊂ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Indeed, if f ∈ Φ \\ {0}, then there exists a constant c > 0 and a measurable subset A of T such that |f| ≥ cχA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' χA ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Hence, by 6 SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI the boundedness of MT on Φ, we get that MTχA ≥ |A| 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Consequently, by the ideal property of Φ, we obtain that L∞ ⊂ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' The following lemma provides a periodic counterpart of [66, Chapter 2, (17) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For its proof we need a vector-valued variant of Fej´er’s theorem, which asserts that for an arbitrary Banach space X and g ∈ L1(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X), if Sl(g) := � |k|≤l ek ⊗ ˆg(k) for l ∈ N0, then 1 N + 1 N � l=0 Sl(g) → g as N → ∞ in L1(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Its proof follows the lines of the proof of its scalar prototype almost verbatim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (i) Let η ∈ Cc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' L(X, Y )) be such that ∥F−1η(t)∥L(X,Y ) ≤ φ(t), t ∈ R, for an even, radially decreasing, integrable function φ on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Then, there exists a constant c > 0 such that for every f ∈ L1(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X) we have (1) |η(∆)f(τ)|Y ≤ c∥φ∥L1(R)(MT|f|X)(τ) (τ ∈ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (ii) Let η ∈ C∞ c (R) and set ηǫ := η(ǫ·), ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Then, for every Banach function space Φ over (T, dt) such that MT is bounded on Φ, the operators ηǫ(∆), ǫ > 0, are uniformly in L(Φ(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (i) Let f ∈ L1(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' It is readily seen that for every t ∈ R the integral � R(F−1η)(s) �f(t − s)ds =: (F−1η ∗ �f)(t) is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Suppose that supp η ⊂ [−K, K] for some K ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Note that for every t ∈ R and l > K we have η(∆)f(eit) = � |k|≤l eitk � R e−iskF−1η(s)ds ˆf(k) = � R F−1η(s) � |k|≤l ei(t−s)k ˆf(k)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Therefore, following the notation introduced in Fej´er’s theorem, for N > K and t ∈ R we get (2) 1 N + 1 N � l=K Sl(η(∆)f)(eit) = � R F−1η(s) 1 N + 1 N � l=K Sl(gt)(eis)ds, where gt(τ) := f(eit¯τ), τ ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' By our assumption on F−1η it is straightforward to show that the right-hand side of (2) converges to (F−1η ∗ �f)(t) as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Since the left-hand side is equal N−K+1 N η(∆)f(eit), we infer that for every t ∈ R ��η(∆)f(eit) �� Y = ���(F−1η ∗ �f)(t) ��� Y ≤ (φ ∗ | �f|X)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Further, note that for each ǫ > 0 there exists a function φǫ := �N j=0 cjχBj, where Bj denotes an interval with the center in 0 and cj > 0, such that 0 ≤ φǫ ≤ φ, ∥φ − φǫ∥L∞ < ǫ and ∥φ − φǫ∥L1 < ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' One can readily check that for every δ > 0 there exists ǫ such that for every t ∈ R we have |(φ ∗ | �f|X)(t)| ≤ ((φ − φǫ) ∗ | �f|X)(t) + (φǫ ∗ | �f|X)(t) ≤ δ + ∥φǫ∥L1(MR| �f|X)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Since there exists c > 0, independent of f ∈ L1(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' X), such that (MR| �f|X)(t) ≤ c(MT|f|X)(eit) for every t ∈ R, we get (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (ii) Note that (F−1ηǫ)(t) = 1 ǫ(F−1η)( t ǫ) for every ǫ > 0 and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' More- over, there exists a constant C > 0 such that |(F−1η)(t)| ≤ C 1+t2 , which yields THE INVARIANT SUBSPACES OF PERIODIC FOURIER MULTIPLIERS 7 |(F−1ηǫ)(t)| ≤ ǫ−1C 1+(ǫ−1t)2 =: φǫ(t) for every t ∈ R and ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Since ∥φǫ∥L1 = Cπ for every ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Therefore, (1) gives the desired claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Fundamental properties of generalized Besov and Triebel-Lizorkin spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Here, we collect the fundamental properties of such spaces, which play a role in our further studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' We start with some preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Note that for every s ∈ R the space B−s,1 Φ (X∗) embeds into (Bs,∞ Φ (X))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Indeed, this embedding is given by the following duality pairing: for each f ∈ Bs,∞ Φ (X) and g ∈ B−s,1 Φ′ (X∗) we set (3) ⟨g, f⟩ := � j,l∈N0 � T ⟨ψl(∆)g(t), ψj(∆)f(t)⟩X∗,Xdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Note that ⟨g, f⟩ := � r∈{±1,0} � l∈N0 ⟨ ˘ψj(∆)ψl(∆)g, χj(∆)f⟩Φ′(X∗),Φ(X), where ˘ψj := ψj(−·) and χj := ψj−1 + ψj + ψj+1 (j ∈ N0) with ψ−1 ≡ 0 if j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Let X be a Banach space and Φ be a Banach function space over (T, dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Then, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (i) If L∞ ⊂ Φ ⊂ L1, then for every E ∈ {Bs,q Φ , F s,r Φ : s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞)} (4) P(X) ⊂ E(X) ֒→ D′(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' In particular, E(X) is a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (ii) If MT is bounded on Φ, then for all s ∈ R, P(X) is a dense subset of Bs,q Φ (X) in the norm topology if q ∈ [1, ∞), and in the B−s,1 Φ′ (X∗)-topology of Bs,∞ Φ (X) if q = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' More precisely, for every f ∈ Bs,q Φ (X) we have that � 0≤j≤N ψj(∆)f = ψ(2−N∆)f → f as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' in Bs,q Φ (X) for each q < ∞, and in the other case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' q = ∞, the conver- gence holds in the σ(Bs,∞ Φ (X), B−s,1 Φ′ (X∗))-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (ii’) If MT is bounded on Φ and its dual Φ′, then P(X) is a dense subset of F s,q Φ (X) and for ever f ∈ F s,q Φ (X) we have that � 0≤j≤N ψj(∆)f = ψ(2−N∆)f → f as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (iii) If MT is bounded on Φ, then for every distribution f ∈ D′(X), f belongs to Bs,q Φ (X) if and only if ∂f belongs to Bs−1,q Φ (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Moreover, the function (5) Bs,q Φ (X) ∋ f �→ ∥∂f∥Bs−1,q Φ (X) is an equivalent norm on Bs,q Φ (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (a) Compared to the case of the real line R (see [51, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='6]), note that in the periodic case there is a common dense subset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' P(X), for all E(X) with E ∈ {Bs,q Φ , F s,r Φ : s ∈ R, q ∈ [1, ∞], r ∈ (1, ∞)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' 8 SEBASTIAN KR ´OL & JAROS�LAW SARNOWSKI (b) In contrast to [51, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='6], in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3, in the case of Besov spaces we do not assume that MT is bounded on the dual of Φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='2, the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3, mimics the proof of the correspond- ing statements on R, [51, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For the convenience of the reader we provide some auxiliary observations which should be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' (i) We start with the case when E = Bs,q Φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Since P ⊂ L∞ ⊂ Φ, the left inclusion in (4) holds readily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' For the second one, note that for every f ∈ Bs,q Φ (X) and φ ∈ D we have that |(φj(∆)f) (φ)|X = ����� � k∈Z � T ekφdt ψj(k) ˆf(k) ����� X = 2π ����� � k∈Z χj(k)ˆφ(−k)ψj(k) ˆf(k) ����� X ≤ 2π � T ����� � k∈Z ek ⊗ ψj(k) ˆf(k) ����� X ����� � l∈Z χj(l)ˆφ(−l)el ����� dt ≤ 2π ���ψj(∆) ˆf ��� Φ(X) ���χj(∆)˘φ ��� Φ′ Here, ˘φ(eit) = φ(e−it), t ∈ [−π, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Since limN→∞ �N j=0 φj(∆)f = f in D′(X), we obtain that |f(φ)|X ≤ � j∈N0 ��2jsψj(∆)f �� Φ(X) ∥2−jsχj(∆)˘φ∥Φ′ ≤ \uf8eb \uf8ed � j∈N0 ��2jsqψj(∆)f ��q Φ(X) \uf8f6 \uf8f8 1/q \uf8eb \uf8ed � j∈N0 ∥2−jsχj(∆)˘φ∥q′ Φ′ \uf8f6 \uf8f8 1/q′ ≤ ∥f∥Bs,q Φ (X) \uf8eb \uf8ed � j∈N0 ∥2−jsχj(∆)ˇφ∥q′ Φ′ \uf8f6 \uf8f8 1/q′ (with the usual modification when q = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Take α > |s| + 1 and set ρ(t) := (1 + t2)−α, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FMT4oBgHgl3EQf4TEa/content/2301.12451v1.pdf'} +page_content=' Then, for every j ∈ N0 and τ ∈ T we have 2(|s|+1)j ���F−1(ρχj|Z)(τ) ��� = ����� � k∈Z τ k 2(|s|+1)j (1 + k2)α χj(k) ����� ≤ � 2j−20 for all i, j such that ( +� +1≤i≤t−1 +mi)+ +1 ≤ i < j ≤ ( +� +1≤i≤t−1 +mi) + mt, 1 ≤ t ≤ k. +From the construction of T (λ), we know that Tm1 will be a Young tableau +with a single column of entries being (λm1, ..., λ1) from the first row to the last +row. So c1(Tm1) = m1. In the process of constructing Young tableaux from Tm1 to +Tm1+m2, we may add some boxes to the first column. So we have c1(Tm1+m2) ≥ m1 +and c1(Tm1+m2) + c2(Tm1+m2) = m1 + m2. Finally, we will have c1(Tn) ≥ m1, +c1(Tn) + c2(Tn) ≥ m1 + m2,..., +� +1≤j≤k−1 +cj(T (n)) ≥ +� +1≤j≤k−1 +mj, +� +1≤j≤k +cj(T (n)) = +� +1≤j≤k +mj. We use cj to denote cj(T (n)) = cj(T (λ)). So from lemma 3.1 we can +see that +� +1≤j≤k +c2 +j will take the minimal value +� +1≤j≤k +m2 +j if and only if cj = mj for +all 1 ≤ j ≤ k. +This proves our lemma since a(λ) = 1 +2( � +1≤j≤k +c2 +j − n). +□ +When p is a standard decreasing parabolic subalgebra of size (m1, m2, ..., mk), +from this lemma we can see that an integral simple module L(λ) in Op is socular +if and only if λ is a maximal standard weight of size (m1, m2, ..., mk). +Suppose d = [n1, n2, ..., nk] is a partion of n, we denote the corresponding de- +creasing partition by ¯d = [m1, m2, ..., mk]. Now for a given parabolic subalgebra p + +SOCULAR SIMPLE MODULES OF sl(n, C) +7 +of size (n1, n2, ..., nk), we use qp to denote the corresponding decreasing parabolic +subalgebra of size (m1, m2, ..., mk). +Lemma 3.3. Let g = sl(n, C). +Let p be a parabolic subalgebra of size (n1, n2) +(n1 < n2) with qp being the corresponding decreasing parabolic subalgebra of size +(n2, n1). +An integral weight λ ∈ Op is socular if and only if there exists some +integral socular weight µ ∈ Oqp such that λ +K∼= µ. +Proof. When λ is socular in Op, from Bai-Xie [BX19] we know that c1(T (λ)) = +n2 and c2(T (λ)) = n1. We denote the elements in the first column of T (λ) by +(t1, t2, ..., tn2) and the elements in the second column of T (λ) by (s1, s2, ..., sn1). +We let µ + ρ = (tn2, tn2−1, ..., t1, sn1, ..., s1). So this µ is an integral socular weight +in Oqp and T (µ) = T (λ). So by lemma 2.5, we have λ +K∼= µ. The other direction is +obvious. +□ +Remarks 3.4. In general, we have c1(T (λ)) ≥ n2 and c2(T (λ)) ≤ n1. +Example 3.5. Let g = sl(5, C) and p is a parabolic subalgebra of size (n1, n2) = +(2, 3). Suppose λ + ρ = (3, 1, 4, 2, −10), which is an integral socular weight in Op. +Then we have (3, 1, 4, 2, −10) +K∼= (3, 4, 1, 2, −10) +K∼= (3, 4, 1, −10, 2) +K∼= (3, 1, 4, −10, 2) +K∼= +(3, 1, −10, 4, 2) := µ + ρ. This µ is an integral socular weight in Oqp, where qp is +the corresponding decreasing parabolic subalgebra of size (3, 2). +Let g = sl(n, C) and p be a parabolic subalgebra of size (n1, n2, ..., nk). When +we restrict λ + ρ to its some k0 parts (np1, ..., npk0 ) for some 1 < k0 < k, we get a +new weight and denote it by µ = (µ1, ..., µN), where N = np1 + ...+ npk0. Then the +simple module L(µ) may not be a module of sl(N, C). We denote c = 1 +N ( +� +1≤j≤N +µj). +Let ˜µ = µ − c(1, ..., 1). Then L(˜µ) is a simple module of sl(N, C). We will call ˜µ +the restriction of λ to its k0 parts (np1, ..., npk0 ). Usually we still use µ to denote +this restriction. +Lemma 3.6. Let g = sl(n, C). +Let p be a standard decreasing parabolic subal- +gebra of size (m1, m2, ..., mk). For an integral socular weight λ in Op, we write +λ + ρ = (λ1, .., λn). If for some 1 < i0 < k, we have mi0 > mi0+1, then λ is +Knuth equivalent to some integral socular weight µ ∈ Oqp, where qp is a parabolic +subalgebra of size (m1, m2, ..., mi0−1, mi0+1, mi0, mi0+2, ..., mk). +Proof. We restrict our λ+ρ to its i0-th part and (i0+1)-th part. We denote p = mi0 +and q = mi0+1. So p > q. We use t to denote the sequence in these two parts, and +denote it by t = (t1, ..., tp, tp+1, ..., tp+q). We put {t1, .., tp−q} and {tp+1, ..., tp+q} +together, and denote the corresponding decreasing sequence by (s1, ..., sp). We let +τ = (tp−q+1, ..., tp, s1, , , , , sp). From the above lemma, we know t +K∼= τ. We use τ to +replace the i0-th part and (i0 + 1)-th part of λ, and denote the new weight by µ. +Then we have λ +K∼= µ. +□ +Lemma 3.7. Let g = sl(n, C). Let p be a standard decreasing parabolic subalge- +bra of size (m1, m2, ..., mk). Let (n1, ..., nk) be any partition of n corresponding +to (m1, m2, ..., mk). +Suppose λ is an integral socular weight in Op. +Then λ is +Knuth equivalent to some integral socular weight µ ∈ Oqp, where qp is a parabolic +subalgebra of size (n1, n2, ..., nk). + +8 +ZHANQIANG BAI, WEI XIAO*, AND XUN XIE +Proof. We have cj(T (λ)) = mj for 1 ≤ j ≤ k since λ is socular in Op. If nk = mk, +we do nothing and consider nk−1 = mk−1. +Suppose j0 is the first index such +that nj0 ̸= mj0. Then we must have nj0 > mj0 since (m1, m2, ..., mk) is a de- +creasing sequence. +We use kj0 to denote nj0 − mj0. +There exists a maximal +index i0 < j0 such that mi0 = nj0 > mj0. +From the previous lemma 3.6, we +can move mi0 − mi0+1 elements from the i0-th part of λ + ρ to the (i0 + 1)-th +part of λ + ρ. +This new weight µ(1) is Knuth equivalent to our λ. +This µ(1) +is an integral socular weight in Oqp, where qp is a parabolic subalgebra of size +(m1, m2, ..., mi0−1, mi0+1, mi0, mi0+2, mi0+3, ..., mk). Now the (i0 + 1)-th part and +(i0 + 2)-th part of µ(1) still satisfy the condition of our lemma 3.6, so we can +move mi0 − mi0+2 elements from the (i0 + 1)-th part of µ(1) to the (i0 + 2)- +th part of µ(1). The second new weight µ(2) is still Knuth equivalent to our λ. +We continue this process and finally we can move mi0 − mj0 elements from the +(j0 − 1)-th part to the (j0)-th part of the previous new weight. We denote this new +weight by µ(j0−i0). It is an integral socular weight in Oqp, where qp is a parabolic +subalgebra of size (m1, m2, ..., mi0−1, mi0+1, mi0+2, ..., mj0−1, mi0, mj0+1, ..., mk) = +(m1, m2, ..., mi0−1, mi0+1, mi0+2, ..., mj0−1, nj0, nj0+1, ..., nk). We continue this pro- +cess and finally we can get an integral socular weight µ ∈ Oqp, where qp is a para- +bolic subalgebra of size (n1, n2, ..., nk). This µ is Knuth equivalent to our λ. +□ +Lemma 3.8. Let g = sl(n, C). Let p be a parabolic subalgebra of size (n1, n2, ..., nk) +with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, ..., mk). +An integral weight λ ∈ Op is socular if and only if there exists some integral socular +weight µ ∈ Oqp such that λ +K∼= µ. +Proof. Suppose λ is an integral socular weight in Op. In the process of constructing +the Young tableau, Tn1 is a a Young tableau with a single column and we have +c1(Tn1) = n1. Then Tn1+n2 is a a Young tableau with at most two columns and +we have c1(Tn1+n2) ≥ max(n1, n2), and c1(Tn1+n2) + c2(Tn1+n2) = n1 + n2. We +continue this process and finally we have T (λ) = Tn = T(n1+...+nk), which is a +Young tableau with at most k columns. We denote ci = ci(Tn) for 1 ≤ i ≤ k. Then +we have +c1 ≥ m1 = max{n1, ..., nk}, +c1 + c2 ≥ m1 + max({n1, ..., nk} \ m1) = m1 + m2, +..., +c1 + ... + ck−1 ≥ ( +� +1≤j≤k−2 +mj) + max({n1, ..., nk} \ {m1, ..., mk−2}) = +� +1≤j≤k−1 +mj, +and +c1 + ... + ck = m1 + ... + mk = n. +Since λ is socular, so by lemma 3.1 and 3.2 we must have cj = mj for all +1 ≤ j ≤ k. +We let µ be the integral weight whose i-th part consists of all the elements in +the i-th column of T (λ). Then this µ is a socular weight in Oqp such that λ +K∼= µ. +□ +Proof of Theorem 1.1. From lemma 3.8, we know that the integral socular +weight λ ∈ Op is Knuth equivalent to some integral socular weight µ ∈ Oqp, where + +SOCULAR SIMPLE MODULES OF sl(n, C) +9 +qp is the corresponding decreasing parabolic subalgebra. Then from lemma 2.5, we +know λ +Y∼= µ. This completes the proof of our main theorem. +□ +From lemma 3.7 and 3.8, we have the following corollary. +Corollary 3.9. Let g = sl(n, C). Let p be a parabolic subalgebra of size (n1, n2, ..., nk) +with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, ..., mk). +Then there exists a bijection between the integral socular weights in Op and integral +socular weights µ ∈ Oqp. +From the proof of our main theorem, we have the following corollary. +Corollary 3.10. Let g = sl(n, C). Let p be a parabolic subalgebra of size (n1, n2, ..., nk) +with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, ..., mk). +Any integral weight λ ∈ Op is socular if and only if for any 1 < k0 ≤ k, when we +restrict our λ to its any k0 parts, the restricted weight is still socular in the corre- +sponding parabolic category. +Proof. Suppose λ ∈ Op is an integral socular weight. We denote ci = ci(Tn) for +1 ≤ i ≤ k. Then from the proof of lemma 3.8, we have: c1(Tn1+n2) = max(n1, n2), +c2(Tn1+n2) = min(n1, n2); c1(Tn1+n2+n3) = max(n1, n2, n3), c2(Tn1+n2+n3) = +max({n1, n2, n3} \ max(n1, n2, n3)), c3(Tn1+n2+n3) = min(n1, n2, n3); ...; c1 = +m1, c2 = m2, ..., ck = mk. +For (n1, ..., nj), we denote the corresponding decreasing sequence by (mj +1, mj +2, .., mj +j). +This process of constructing Young tableau T (λ) means that for any 1 < j ≤ k, +when we restrict our λ to its first j parts, the restricted weight is still socular in the +corresponding parabolic category Oqp, where qp is the corresponding decreasing +parabolic subalgebra of size (mj +1, mj +2, ..., mj +j). +For an integral socular weight λ ∈ Op, we know the Knuth relation doesn’t +change the corresponding Young tableau. So we can use the Knuth relation to +change some i-th part and (i + 1)-th part of λ. Suppose the new weight is denoted +by µ(i). Then it must be an integral socular weight in Oqp, where qp is the cor- +responding parabolic subalgebra of size (n1, ..., ni−1, ni+1, ni, ni+2, ..., nk). So for +any 1 < j ≤ k, when we restrict our λ to its any successive j parts, the restricted +weight is still socular in the corresponding parabolic category. +Now for any 1 < k0 ≤ k, we restrict our λ to its any k0 parts. We denote them +by (np1, np2, ..., npk0 ). From the Knuth relation, we can change λ to a new weight +η(1) whose size is +(n1, .., np1, n′ +p1+1, ..., n′ +p2−1, np2, ..., n′ +pk0 −1, npk0 , ..., nk), +where (n′ +p1+1, ..., n′ +p2−1) is the corresponding decreasing sequence of the sequence +(np1+1, ..., np2−1) and so on. +Then we restrict η(1) to its (np1, n′ +p1+1, ..., n′ +p2−1, np2, ..., n′ +pk0 −1, npk0 ) parts and +denote this new weight by η(11). In the construction of the Young tableau T (η(11)), +we can see that the restriction of η(11) to its (np1, np2, ..., npk0 ) parts is an integral +socular weight in Oqp, where qp is the corresponding parabolic subalgebra of size +(np1, np2, ..., npk0 ). +Thus we have proved our conclusion. +□ + +10 +ZHANQIANG BAI, WEI XIAO*, AND XUN XIE +Example 3.11. Let g = sl(10, C) and p be a parabolic subalgebra of size +(n1, n2, n3, n4) = (2, 4, 3, 1). +Suppose λ + ρ = (5, −48, 7, 6, 4, 2, 9, 8, 3, 4), which is an integral socular weight +in Op. +Then we can check that the following restrictions are socular weights +in the corresponding parabolic category: +(5, −48, 9, 8, 3, 4), (5, −48, 7, 6, 4, 2, 4), +(7, 6, 4, 2, 9, 8, 3), (5, −48, 9, 8, 3). +Since the Knuth relation doesn’t change the Young tableau for a given integral +socular weight, we will have the following corollary. +Corollary 3.12. Let g = sl(n, C). Let p be a parabolic subalgebra of size (n1, n2, ..., nk) +with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, ..., mk). +An integral weight λ ∈ Op is socular if and only if for each 1 ≤ i ≤ k, we can choose +mi parallel increasing subsequences of length i from the longer i parts of λ + ρ. +4. The non-integral case +Let g = sl(n, C). Let p be a parabolic subalgebra of size (n1, n2, ..., nk) with qp +being the corresponding decreasing parabolic subalgebra of size (m1, m2, ..., mk). +When a weight λ ∈ Op is non-integral, from Bai-Xie [BX19], we can associate +some Young tableaux (more than one Young tableau) to λ. For any λ ∈ h∗, we +write λ + ρ = (λ1, · · · , λn). Let P(λ) be a set of some Young tableaux as follows. +Let λY : λi1, λi2, . . . , λir be a maximal subsequence of λ1, λ2, . . . , λn such that λik, +1 ≤ k ≤ r are congruent to each other by Z. Then let T (λY ) be the Young tableau +obtained from λY by using RS-insertion algorithm, which is a Young tableau in +P(λ). +Now we put these Young tableaux together and make them into one bigger Young +tableau ¯T(λ) by inserting the columns of other Young tableaux into the first Young +tableau such that ci( ¯T(λ)) is decreasing for 1 ≤ i ≤ k. In other words, the Young +tableau ¯T (λ) = +c⊔ +T (λY )∈P (λ) T (λY ). Here T1 +c⊔T2 denotes the Young tableau whose +multiset of nonzero column lengths equals the union of the two Young tableaux T1 +and T2. +Then from our main theorem and corollary 3.10,we have the following theorem +and corollary. +Theorem 4.1. Let g = sl(n, C). Let p be a parabolic subalgebra of size (n1, n2, ..., nk) +with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, ..., mk). +A non-integral weight λ ∈ Op is socular if and only if λ is Young equivalent to some +socular weight µ corresponding to a standard decreasing parabolic subalgebra of size +(m1, m2, ..., mk). Equivalently, ci( ¯T(λ)) = mi for 1 ≤ i ≤ k. +Corollary 4.2. Let g = sl(n, C). Let p be a parabolic subalgebra of size (n1, n2, ..., nk) +with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, ..., mk). +A weight λ ∈ Op is socular if and only if we divide λ + ρ into several subsequence +such that each subsequence is an integral weight and the restriction of λ to each +subsequence is a socular weight in the corresponding parabolic category. +References +[BXX] Z. Q. Bai, W. Xiao and X. Xie, Gelfand-Kirillov dimensions and associated varieties of +highest weight modules, to appear in IMRN, https://doi.org/10.1093/imrn/rnac081. 3 + +SOCULAR SIMPLE MODULES OF sl(n, C) +11 +[BX19] Z. Q. Bai and X. Xie, Gelfand-Kirillov dimensions of highest weight Harish-Chandra +modules for SU(p,q), Int. Math. Res. Not., 17, 4392-4418 (2019). 2, 3, 4, 7, 10 +[B11] +J. Brundan, Mœglin’s theorem and Goldie rank polynomials in Cartan type A, Compositio +Math. 147 (2011), 1741-1771. 1 +[BK08] J. Brundan and A. Kleshchev, Schur-Weyl duality for higher levels. Selecta Math. (N.S.) +14 (2008), no. 1, 1-57 1 +[BLW17] J. Brundan, I. Losev and B. Webster, Tensor product categorifications and the super +Kazhdan-Lusztig conjecture. Int. Math. Res. Not. IMRN 2017, no. 20, 6329-6410. 1 +[I85] +R. S. Irving, Projective modules in the category OS: self-duality, Trans. Amer. Math. +Soc. 291 (1985), no. 2, 701-732. 1 +[MC08] MV. Mazorchuk and C. Stroppel, Projective-injective modules, Serre functors and sym- +metric algebras. J. Reine Angew. Math. 616 (2008), 131-165. 1 +[S01] +B.E. Sagan, The Symmetric Group. Representations, Combinatorial Algorithms, and +Symmetric Functions, second edition, Graduate Texts in Mathematics, vol. 203, Springer- +Verlag, New York, 2001. 3, 4 +[V78] +D. A. Vogan, Jr., Gelfand-Kirillov dimension for Harish-Chandra modules, Invent.Math. +48 (1978), 75-98. 2 +[X15] +W. Xiao, Leading weight vectors and homomorphisms between generalized Verma mod- +ules. J. Algebra 430 (2015), 62-93. 1 +(Bai) School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. +China +Email address: zqbai@suda.edu.cn +(Xiao) College of Mathematics and statistics, Shenzhen Key Laboratory of Advanced +Machine Learning and Applications, Shenzhen University, Shenzhen, 518060, Guangdong, +P. R. China +Email address: xiaow@szu.edu.cn +(Xie) School of Mathematics and Statistics, Beijing Institute of Technology, Beijing +100081, China +Email address: Xieg7@163.com + diff --git a/htE5T4oBgHgl3EQfFg4V/content/tmp_files/load_file.txt b/htE5T4oBgHgl3EQfFg4V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae387e9a9b6f0a6e89f3c75cd23ec32d769cfb46 --- /dev/null +++ b/htE5T4oBgHgl3EQfFg4V/content/tmp_files/load_file.txt @@ -0,0 +1,749 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf,len=748 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='05422v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='RT] 13 Jan 2023 AN EXPLICIT CHARACTERIZATION OF SOCULAR SIMPLE MODULES OF sl(n, C) ZHANQIANG BAI, WEI XIAO*, AND XUN XIE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g be a simple complex Lie algebra with a Cartan subalgebra h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We fix a standard parabolic subalgebra p ⊃ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The socular simple modules play an important role in the parabolic versions of category Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From Irving’s work, we know that these modules are just those modules with largest possible Gelfand-Kirillov dimension in Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' In this article, we will give an explicit characterization for these modules of sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Our characterization is given in the information of the corresponding highest weight and Young tableau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Introduction Let g be a finite-dimensional complex semisimple Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Fix a Cartan subalgebra h and denote by Φ the root system associated with (g, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Choose a positive root system Φ+ ⊂ Φ and a simple system ∆ = {αi|1 ≤ i ≤ n} ⊂ Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let ρ be the half sum of roots in Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = ¯n ⊕ h ⊕ n be the Cartan decomposition of g with nilpotent radical n and its dual space ¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Moreover, b = h ⊕ n is the Borel subalgebra corresponding to Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Choose a subset I ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then ∆I = {αi|i ∈ I} generates a subsystem ΦI ⊂ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let pI be the standard parabolic subalgebra corresponding to I with Levi decomposition pI = lI ⊕uI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We frequently drop the subscript I if there is no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let F(λ) be a finite-dimensional irreducible l-module with highest weight λ ∈ h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' It can also be viewed as a p-module with trivial u-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The generalized Verma modules MI(λ) is defined by MI(λ) := U(g) ⊗U(p) F(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The simple quotient of MI(λ) is denoted by L(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We use Op to denote the corre- sponding parabolic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' In [I85], Irving called L(µ) or µ socular if L(µ) is a summand of the socle of a generalized Verma module MI(λ) in the category Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From an observation of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Vogan, Irving showed that these socular simple modules L(µ) are just those modules with largest possible Gelfand-Kirillov dimension in Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' These modules play an important role in the study of parabolic category Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Many mathematicians use these modules to study the properties of category O or other type algebras, for example, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' [B11, BK08, BLW17, MC08, X15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 17B10, 17B20, 22E47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Highest weight module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Gelfand-Kirillov dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Young tableau;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Parabolic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The first author is supported by NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 12171344 and National Key R&D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 2018YFA0701700 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 2018YFA0701701), the second author is supported by NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 11701381 and the third author is supported by NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 11801031 and 12171030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 1 2 ZHANQIANG BAI, WEI XIAO*, AND XUN XIE Recently, Bai and Xie [BX19] have found a practical combinatorial method to compute the Gelfand-Kirillov dimension of any simple highest weight module when g is of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' In this paper, we will use their method to give an explicit charac- terization for these socular simple modules of sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now we fix g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We choose Φ+ = {ei − ej|1 ≤ i < j ≤ n} and a simple system ∆ = {αi := ei − ei+1|1 ≤ i ≤ n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We choose a subset I ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' There will exist some positive integers n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk and n0 = 0 with n1 + n2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' + nk = n such that I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n1 − 1, n1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n1 + n2 − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', ( � 1≤i≤k−1 ni)+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', ( � 1≤i≤k−1 ni)+nk −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Note that when some nt = 1, there will be no elements between ( � 1≤i≤t−1 ni) + 1 and ( � 1≤i≤t−1 ni) + nt − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This I will generate a subsystem ΦI ⊂ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let pI be the standard parabolic subalgebra corresponding to I with Levi decomposition pI = lI ⊕ uI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So there will be a bijection between the partitions of n and parabolic subalgebras of g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When n1 ≥ n2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' ≥ nk, we call pI a standard decreasing parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From Bai-Xie [BX19], we know that for each integral weight λ, by using Robinson- Schensted insertion algorithm, there is a Young tableau T (λ) corresponding to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' λ and µ are called Young equivalent, written as λ Y∼= µ, if T (λ) = T (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For an integral standard weight λ of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk), if the number ci of entries in the i-th column of T (λ) is mi for all 1 ≤ i ≤ k, we will call λ a maximal standard weight of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Our theorem is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose the parabolic subalgebra p corresponds to a partition d = [n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk] of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We use (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk) to denote the corresponding decreasing partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' A simple module L(λ) (λ is integral) in Op is socular if and only if λ is Young equivalent to some maximal standard weight µ corresponding to a standard decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Two increasing sequences (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='., xp) and (y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='., yp) are called parallel if xi < yi for each 1 ≤ i ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From this theorem, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose the parabolic subalgebra p corresponds to a partition d = [n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk] of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' An integral weight λ is socular if and only if for each 1 ≤ i ≤ k, we can choose mi parallel increasing subsequences of length i from λ + ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose λ + ρ = (7, 4, 9, 8, 2, −52, 10, 9, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The corresponding partition of n is d = [2, 4, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The decreasing partition is ¯d = [4, 3, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For i = 1, m1 = 4, we can choose (9), (8), (2), (−52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For i = 2, m2 = 3, we can choose (9, 10), (8, 9), (2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For i = 3, m3 = 2, we can choose (7, 9, 10), (4, 8, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From Bai-Xie [BX19], we know that this simple module L(λ) has the maximal Gelfand- Kirillov dimension 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So this λ is socular in Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Preliminaries Before we prove our main theorems, we first recall some useful results about Gelfand-Kirillov dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The details can be found in [V78] and [BX19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let M be a U(g)-module generated by a finite-dimensional subspace M0 (that is, M is finitely generated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let Un(g) be the standard filtration of U(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Set SOCULAR SIMPLE MODULES OF sl(n, C) 3 Mn = Un(g) · M0 and gr(M) = ∞ � n=0 grnM, where grnM = Mn/Mn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Thus gr(M) is a graded module of gr(U(g)) ≃ S(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='1 ([BXX, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The Gelfand-Kirillov dimension of M is defined by GKdim M = lim n→∞ log dim(Un(g)M0) log n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' It is easy to see that the above definition is independent of the choice of M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Denote ϕM,M0(n) = dim(Un(g)M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' By a theorem of Hilbert and Serre, there exists a unique polynomial ˜ϕM,M0(n) such that ϕM,M0(n) = ˜ϕM,M0(n) for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The leading term of ˜ϕM,M0(n) is c(M) (dM)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='ndM, where c(M) is an integer, called Bernstein degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The integer dM is the Gelfand-Kirillov dimension of M, that is, dM = GKdim(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now we fix g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We choose a subset I ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n − 1} and the corresponding partition of n is d = [n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let ¯d = [m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk] be the corresponding decreasing partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This I will generate a subsystem ΦI ⊂ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let pI be the standard parabolic subalgebra corresponding to I with Levi decomposition pI = lI ⊕ uI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let F(λ) be a finite-dimensional irreducible lI-module with highest weight λ ∈ h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The generalized Verma modules MI(λ) = U(g) ⊗U(pI) F(λ) has maximal possible Gelfand-Kirillov dimension in Op (we will omit I if there is no confusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' That is, GKdim(M(λ)) = dim(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From the construction of l, we know dim(l+) = 1 2 � 1≤j≤k nj(nj − 1) = 1 2( � 1≤j≤k n2 j − n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So dim(u) = n(n−1) 2 − dim(l+) = 1 2(n2 − � 1≤j≤k n2 j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote this number by dm(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now our problem is to find out all simple modules L(λ) in Op with maximal Gelfand-Kirillov dimension dm(p) = 1 2(n2 − � 1≤j≤k n2 j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C) with a Cartan subalgebra h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The Weyl group of g is Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From Sagan [S01], we know that there is a bijection between the symmetric group Sn and the Young tableaux through the famous Robinson-Schensted algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We use T (σ) to denote the corresponding Young tableau for any σ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Two permutations π, σ ∈ Sn are said to be Young equivalent, written as π Y∼= σ, if T (π) = T (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From Bai-Xie [BX19], we know that for any integral weight λ ∈ h∗, there is a Young tableau corresponding to it through a similar method with the R-S algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We recall this method from Bai-Xie [BX19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For an integral weight λ ∈ h∗, we write λ + ρ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', λn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We associate to λ a Young tableau T (λ) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let T0 be an empty Young tableau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Assume that we have constructed Young tableau Tk associated to (λ1, · · · , λk), 0 ≤ k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then Tk+1 is obtained by adding λk+1 to Tk as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' First add λk+1 to the first row of Tk by replacing the leftmost entry x in the first row which is strictly bigger than λk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' (If there is no such an entry x, we just add a box with entry λk+1 to the right side of the first row, and end this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=') Then add x to the next row as the same way of adding λk+1 to the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then we put T (λ) = Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 4 ZHANQIANG BAI, WEI XIAO*, AND XUN XIE For the Young tableau T (λ), we define a(λ) := � i≥1 ci(ci−1) 2 where ci is the number of entries in the i-th column of T (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='2 (Bai-Xie [BX19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For any integral weight λ ∈ h∗, we have GKdimL(λ) = n(n − 1) 2 − a(λ) = 1 2(n2 − � 1≤i≤k c2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When p is a parabolic subalgebra of sl(n, C) with the corresponding partition of n being d = [n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk], the maximal Gelfand-Kirillov dimension of simple modules in Op will be dm(p) = 1 2(n2 − � 1≤i≤k n2 i ) = 1 2(n2 − � 1≤i≤k c2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So a simple module L(λ) in Op is socular if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='3) � 1≤i≤k c2 i = � 1≤i≤k n2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let ¯d = [m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk] be the corresponding decreasing partition associated with the given partition d = [n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then we have � 1≤i≤k m2 i = � 1≤i≤k n2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now we recall the famous Knuth relation about Young tableaux [S01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose x < y < z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then π, σ ∈ Sn differ by a Knuth relation of the first kind, written π 1∼= σ if π = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='yxz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='xn and σ = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='yzx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='xn or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' They differ by a Knuth relation of the second kind, written π 2∼= σ if π = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='xzy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='xn and σ = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='zxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='xn or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We let K∼= denote the equivalence relation generated by 1∼= and 2∼=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This relation is called the Knuth relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From Bai-Xie [BX19], we know that for any integral weight λ ∈ h∗, there is a unique σλ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Two integral weight λ, µ ∈ h∗ are called Knuth equivalent, written as λ K∼= µ if σλ K∼= σµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose λ, µ ∈ h∗ are two integral weight, then λ K∼= µ ⇔ λ Y∼= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From Sagan [S01], we know σλ K∼= σµ ⇔ σλ Y∼= σµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From Bai-Xie [BX19, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='5], we know λ Y∼= µ ⇔ σλ Y∼= σµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So we have λ K∼= µ ⇔ λ Y∼= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Proof of the main theorem In this section, we will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Firstly, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' SOCULAR SIMPLE MODULES OF sl(n, C) 5 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose {m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk} is a sequence of decreasing positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We have a function f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) = � 1≤j≤k x2 j, where (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) ∈ D with D = {(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) ∈ Rk |x1 ≥ m1, x1 + x2 ≥ m1 + m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', � 1≤j≤k−1 xj ≥ � 1≤j≤k−1 mj, � 1≤j≤k xj = � 1≤j≤k mj, x1 ≥ x2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' ≥ xk ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then f will take the minimal value � 1≤j≤k m2 j if and only if xj = mj for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When f is a function of two variables, the conclusion is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We assume the conclusion is true for all functions containing less that or equal to k−1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now we assume f is a function of k variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote m = � 1≤j≤k mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' If there is no restriction, the function f will take the minimal value at the point P0 = ( m k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', m k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now the problem is equivalent to finding out all points on the given plane Π : � 1≤j≤k xj = m with some restriction condition such that the distance from the origin to these points will take the minimal value d = � � 1≤j≤k m2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From the condition x1 ≥ m1, x1 + x2 ≥ m1 + m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', � 1≤j≤k−1 xj ≥ � 1≤j≤k−1 mj, x1 ≥ x2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' ≥ xk ≥ 0, we know the domain D ⊆ Π of our function f is a bounded connected closed subset in the first quadrant and P0 /∈ D (unless all the integers mi are equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So f will take its minimal value d2 at the boundary ∂D of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When x1 = m1, we have f(m1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) = m2 1 + � 2≤j≤k x2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote f2(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) = � 2≤j≤k x2 j, where (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) ∈ D2 = {(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) ∈ Rk−1|x2 ≥ m2, x2 + x3 ≥ m2 + m3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', � 2≤j≤k−1 xj ≥ � 2≤j≤k−1 mj, � 2≤j≤k xj = � 2≤j≤k mj, x2 ≥ x3 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' ≥ xk ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So f2 is a function of k − 1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' By our induction, f2 will take its minimal value if and only if xj = mj for all 2 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So f will take its minimal value at the boundary x1 = m1 if and only if xj = mj for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When x1 + x2 = m1 + m2, we have f(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) = (x2 1 + x2 2) + � 3≤j≤k x2 j := g2(x1, x2) + f3(x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This is a sum of two functions which satisfy our induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then we know g2 will take its minimal value if and only if x1 = m1, x2 = m2 and f3 will take its minimal value if and only if xj = mj for all 3 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So f will take its minimal value at the boundary x1 + x2 = m1 + m2 if and only if xj = mj for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 6 ZHANQIANG BAI, WEI XIAO*, AND XUN XIE We continue this process and finally when � 1≤j≤k xj = m = � 1≤j≤k mj, we have f(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) = x2 k+ � 1≤j≤k−1 x2 j = (m− � 1≤j≤k−1 xj)2+ � 1≤j≤k−1 x2 j := fk(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk−1), where (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk−1) ∈ Dk = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk−1) ∈ Rk−1|x1 ≥ m1, x1 + x2 ≥ m1 + m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', � 1≤j≤k−1 xj ≥ � 1≤j≤k−1 mj, x1 ≥ x2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' ≥ xk−1 ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' If there is no restriction, the function fk will take the minimal value at the point Q0 = ( m k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', m k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' But Q0 /∈ Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So the function fk will take the minimal value at the boundary ∂Dk of Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When x1 = m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', � 1≤j≤k−2 xj = � 1≤j≤k−2 mj, the arguments are the same with the case of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When � 1≤j≤k−1 xj = � 1≤j≤k−1 mj, we have f(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', xk) = ( � 1≤j≤k−1 xj) + m2 k := gk−1 + m2 k, where gk−1 is a function of k − 1 variables which satisfies our induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Thus we have proved our lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let pI be a standard decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For an integral standard weight λ of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk) in h∗, the number a(λ) will take the minimal value 1 2( � 1≤j≤k m2 j − n) if and only if λ is a maximal standard weight of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', the number cj of entries in the j-th column of T (λ) is mj for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We write λ + ρ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', λn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Since λ is an integral standard weight of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk) in h∗, we will have λi−λj ∈ Z>0 for all i, j such that ( � 1≤i≤t−1 mi)+ 1 ≤ i < j ≤ ( � 1≤i≤t−1 mi) + mt, 1 ≤ t ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From the construction of T (λ), we know that Tm1 will be a Young tableau with a single column of entries being (λm1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', λ1) from the first row to the last row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So c1(Tm1) = m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' In the process of constructing Young tableaux from Tm1 to Tm1+m2, we may add some boxes to the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So we have c1(Tm1+m2) ≥ m1 and c1(Tm1+m2) + c2(Tm1+m2) = m1 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Finally, we will have c1(Tn) ≥ m1, c1(Tn) + c2(Tn) ≥ m1 + m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', � 1≤j≤k−1 cj(T (n)) ≥ � 1≤j≤k−1 mj, � 1≤j≤k cj(T (n)) = � 1≤j≤k mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We use cj to denote cj(T (n)) = cj(T (λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So from lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='1 we can see that � 1≤j≤k c2 j will take the minimal value � 1≤j≤k m2 j if and only if cj = mj for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This proves our lemma since a(λ) = 1 2( � 1≤j≤k c2 j − n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' □ When p is a standard decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk), from this lemma we can see that an integral simple module L(λ) in Op is socular if and only if λ is a maximal standard weight of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose d = [n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk] is a partion of n, we denote the corresponding de- creasing partition by ¯d = [m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now for a given parabolic subalgebra p SOCULAR SIMPLE MODULES OF sl(n, C) 7 of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk), we use qp to denote the corresponding decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a parabolic subalgebra of size (n1, n2) (n1 < n2) with qp being the corresponding decreasing parabolic subalgebra of size (n2, n1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' An integral weight λ ∈ Op is socular if and only if there exists some integral socular weight µ ∈ Oqp such that λ K∼= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When λ is socular in Op, from Bai-Xie [BX19] we know that c1(T (λ)) = n2 and c2(T (λ)) = n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote the elements in the first column of T (λ) by (t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', tn2) and the elements in the second column of T (λ) by (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', sn1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We let µ + ρ = (tn2, tn2−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', t1, sn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So this µ is an integral socular weight in Oqp and T (µ) = T (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So by lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='5, we have λ K∼= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The other direction is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' □ Remarks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' In general, we have c1(T (λ)) ≥ n2 and c2(T (λ)) ≤ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(5, C) and p is a parabolic subalgebra of size (n1, n2) = (2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose λ + ρ = (3, 1, 4, 2, −10), which is an integral socular weight in Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then we have (3, 1, 4, 2, −10) K∼= (3, 4, 1, 2, −10) K∼= (3, 4, 1, −10, 2) K∼= (3, 1, 4, −10, 2) K∼= (3, 1, −10, 4, 2) := µ + ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This µ is an integral socular weight in Oqp, where qp is the corresponding decreasing parabolic subalgebra of size (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C) and p be a parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When we restrict λ + ρ to its some k0 parts (np1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', npk0 ) for some 1 < k0 < k, we get a new weight and denote it by µ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', µN), where N = np1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='+ npk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then the simple module L(µ) may not be a module of sl(N, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote c = 1 N ( � 1≤j≤N µj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let ˜µ = µ − c(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then L(˜µ) is a simple module of sl(N, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We will call ˜µ the restriction of λ to its k0 parts (np1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', npk0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Usually we still use µ to denote this restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a standard decreasing parabolic subal- gebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For an integral socular weight λ in Op, we write λ + ρ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='., λn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' If for some 1 < i0 < k, we have mi0 > mi0+1, then λ is Knuth equivalent to some integral socular weight µ ∈ Oqp, where qp is a parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mi0−1, mi0+1, mi0, mi0+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We restrict our λ+ρ to its i0-th part and (i0+1)-th part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote p = mi0 and q = mi0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So p > q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We use t to denote the sequence in these two parts, and denote it by t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', tp, tp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', tp+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We put {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='., tp−q} and {tp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', tp+q} together, and denote the corresponding decreasing sequence by (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', sp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We let τ = (tp−q+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', tp, s1, , , , , sp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From the above lemma, we know t K∼= τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We use τ to replace the i0-th part and (i0 + 1)-th part of λ, and denote the new weight by µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then we have λ K∼= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a standard decreasing parabolic subalge- bra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk) be any partition of n corresponding to (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose λ is an integral socular weight in Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then λ is Knuth equivalent to some integral socular weight µ ∈ Oqp, where qp is a parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 8 ZHANQIANG BAI, WEI XIAO*, AND XUN XIE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We have cj(T (λ)) = mj for 1 ≤ j ≤ k since λ is socular in Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' If nk = mk, we do nothing and consider nk−1 = mk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose j0 is the first index such that nj0 ̸= mj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then we must have nj0 > mj0 since (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk) is a de- creasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We use kj0 to denote nj0 − mj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' There exists a maximal index i0 < j0 such that mi0 = nj0 > mj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From the previous lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='6, we can move mi0 − mi0+1 elements from the i0-th part of λ + ρ to the (i0 + 1)-th part of λ + ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This new weight µ(1) is Knuth equivalent to our λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This µ(1) is an integral socular weight in Oqp, where qp is a parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mi0−1, mi0+1, mi0, mi0+2, mi0+3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now the (i0 + 1)-th part and (i0 + 2)-th part of µ(1) still satisfy the condition of our lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='6, so we can move mi0 − mi0+2 elements from the (i0 + 1)-th part of µ(1) to the (i0 + 2)- th part of µ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The second new weight µ(2) is still Knuth equivalent to our λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We continue this process and finally we can move mi0 − mj0 elements from the (j0 − 1)-th part to the (j0)-th part of the previous new weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote this new weight by µ(j0−i0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' It is an integral socular weight in Oqp, where qp is a parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mi0−1, mi0+1, mi0+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mj0−1, mi0, mj0+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk) = (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mi0−1, mi0+1, mi0+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mj0−1, nj0, nj0+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We continue this pro- cess and finally we can get an integral socular weight µ ∈ Oqp, where qp is a para- bolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This µ is Knuth equivalent to our λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk) with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' An integral weight λ ∈ Op is socular if and only if there exists some integral socular weight µ ∈ Oqp such that λ K∼= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose λ is an integral socular weight in Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' In the process of constructing the Young tableau, Tn1 is a a Young tableau with a single column and we have c1(Tn1) = n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then Tn1+n2 is a a Young tableau with at most two columns and we have c1(Tn1+n2) ≥ max(n1, n2), and c1(Tn1+n2) + c2(Tn1+n2) = n1 + n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We continue this process and finally we have T (λ) = Tn = T(n1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='+nk), which is a Young tableau with at most k columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote ci = ci(Tn) for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then we have c1 ≥ m1 = max{n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk}, c1 + c2 ≥ m1 + max({n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk} \\ m1) = m1 + m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', c1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' + ck−1 ≥ ( � 1≤j≤k−2 mj) + max({n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk} \\ {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk−2}) = � 1≤j≤k−1 mj, and c1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' + ck = m1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' + mk = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Since λ is socular, so by lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='2 we must have cj = mj for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We let µ be the integral weight whose i-th part consists of all the elements in the i-th column of T (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then this µ is a socular weight in Oqp such that λ K∼= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='8, we know that the integral socular weight λ ∈ Op is Knuth equivalent to some integral socular weight µ ∈ Oqp, where SOCULAR SIMPLE MODULES OF sl(n, C) 9 qp is the corresponding decreasing parabolic subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then from lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='5, we know λ Y∼= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This completes the proof of our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' □ From lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='8, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk) with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then there exists a bijection between the integral socular weights in Op and integral socular weights µ ∈ Oqp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From the proof of our main theorem, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk) with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Any integral weight λ ∈ Op is socular if and only if for any 1 < k0 ≤ k, when we restrict our λ to its any k0 parts, the restricted weight is still socular in the corre- sponding parabolic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose λ ∈ Op is an integral socular weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote ci = ci(Tn) for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then from the proof of lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='8, we have: c1(Tn1+n2) = max(n1, n2), c2(Tn1+n2) = min(n1, n2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' c1(Tn1+n2+n3) = max(n1, n2, n3), c2(Tn1+n2+n3) = max({n1, n2, n3} \\ max(n1, n2, n3)), c3(Tn1+n2+n3) = min(n1, n2, n3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' c1 = m1, c2 = m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', ck = mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nj), we denote the corresponding decreasing sequence by (mj 1, mj 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='., mj j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' This process of constructing Young tableau T (λ) means that for any 1 < j ≤ k, when we restrict our λ to its first j parts, the restricted weight is still socular in the corresponding parabolic category Oqp, where qp is the corresponding decreasing parabolic subalgebra of size (mj 1, mj 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mj j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For an integral socular weight λ ∈ Op, we know the Knuth relation doesn’t change the corresponding Young tableau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So we can use the Knuth relation to change some i-th part and (i + 1)-th part of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose the new weight is denoted by µ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then it must be an integral socular weight in Oqp, where qp is the cor- responding parabolic subalgebra of size (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', ni−1, ni+1, ni, ni+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' So for any 1 < j ≤ k, when we restrict our λ to its any successive j parts, the restricted weight is still socular in the corresponding parabolic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now for any 1 < k0 ≤ k, we restrict our λ to its any k0 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' We denote them by (np1, np2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', npk0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' From the Knuth relation, we can change λ to a new weight η(1) whose size is (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='., np1, n′ p1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n′ p2−1, np2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n′ pk0 −1, npk0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk), where (n′ p1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n′ p2−1) is the corresponding decreasing sequence of the sequence (np1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', np2−1) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then we restrict η(1) to its (np1, n′ p1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n′ p2−1, np2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', n′ pk0 −1, npk0 ) parts and denote this new weight by η(11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' In the construction of the Young tableau T (η(11)), we can see that the restriction of η(11) to its (np1, np2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', npk0 ) parts is an integral socular weight in Oqp, where qp is the corresponding parabolic subalgebra of size (np1, np2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', npk0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Thus we have proved our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' □ 10 ZHANQIANG BAI, WEI XIAO*, AND XUN XIE Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(10, C) and p be a parabolic subalgebra of size (n1, n2, n3, n4) = (2, 4, 3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Suppose λ + ρ = (5, −48, 7, 6, 4, 2, 9, 8, 3, 4), which is an integral socular weight in Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then we can check that the following restrictions are socular weights in the corresponding parabolic category: (5, −48, 9, 8, 3, 4), (5, −48, 7, 6, 4, 2, 4), (7, 6, 4, 2, 9, 8, 3), (5, −48, 9, 8, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Since the Knuth relation doesn’t change the Young tableau for a given integral socular weight, we will have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk) with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' An integral weight λ ∈ Op is socular if and only if for each 1 ≤ i ≤ k, we can choose mi parallel increasing subsequences of length i from the longer i parts of λ + ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' The non-integral case Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk) with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' When a weight λ ∈ Op is non-integral, from Bai-Xie [BX19], we can associate some Young tableaux (more than one Young tableau) to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' For any λ ∈ h∗, we write λ + ρ = (λ1, · · · , λn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let P(λ) be a set of some Young tableaux as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let λY : λi1, λi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' , λir be a maximal subsequence of λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' , λn such that λik, 1 ≤ k ≤ r are congruent to each other by Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then let T (λY ) be the Young tableau obtained from λY by using RS-insertion algorithm, which is a Young tableau in P(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Now we put these Young tableaux together and make them into one bigger Young tableau ¯T(λ) by inserting the columns of other Young tableaux into the first Young tableau such that ci( ¯T(λ)) is decreasing for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' In other words, the Young tableau ¯T (λ) = c⊔ T (λY )∈P (λ) T (λY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Here T1 c⊔T2 denotes the Young tableau whose multiset of nonzero column lengths equals the union of the two Young tableaux T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Then from our main theorem and corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='10,we have the following theorem and corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk) with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' A non-integral weight λ ∈ Op is socular if and only if λ is Young equivalent to some socular weight µ corresponding to a standard decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Equivalently, ci( ¯T(λ)) = mi for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let g = sl(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Let p be a parabolic subalgebra of size (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', nk) with qp being the corresponding decreasing parabolic subalgebra of size (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=', mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' A weight λ ∈ Op is socular if and only if we divide λ + ρ into several subsequence such that each subsequence is an integral weight and the restriction of λ to each subsequence is a socular weight in the corresponding parabolic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' References [BXX] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Bai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Xiao and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' Xie, Gelfand-Kirillov dimensions and associated varieties of highest weight modules, to appear in IMRN, 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215006, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' China Email address: zqbai@suda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='cn (Xiao) College of Mathematics and statistics, Shenzhen Key Laboratory of Advanced Machine Learning and Applications, Shenzhen University, Shenzhen, 518060, Guangdong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content=' China Email address: xiaow@szu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='cn (Xie) School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China Email address: Xieg7@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE5T4oBgHgl3EQfFg4V/content/2301.05422v1.pdf'} diff --git a/i9E1T4oBgHgl3EQffwT8/content/tmp_files/2301.03223v1.pdf.txt b/i9E1T4oBgHgl3EQffwT8/content/tmp_files/2301.03223v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ce30266cf4733bb0f84f545ebc25cd4a5ee2c02 --- /dev/null +++ b/i9E1T4oBgHgl3EQffwT8/content/tmp_files/2301.03223v1.pdf.txt @@ -0,0 +1,2786 @@ +Mean-field approximation of the Hubbard model expressed in a many-body basis. +Antoine Honet and Luc Henrard +Department of Physics and Namur Institute of Structured Materials, +University of Namur, Rue de Bruxelles 51, 5000 Namur, Belgium +Vincent Meunier +Department of Engineering Science and Mechanics, +The Pennsylvania State University, University Park, PA, USA +(Dated: January 10, 2023) +The effective independent-particle (mean-field) approximation of the Hubbard Hamiltonian is +described in a many-body basis to develop a formal comparison with the exact diagonalization of +the full Hubbard model, using small atomic chain as test systems. This allows for the development +of an intuitive understanding of the shortcomings of the mean-field approximation and of how +critical correlation effects are missed in this popular approach. The description in the many-body +basis highlights a potential ambiguity related to the definition of the density of states. Specifically, +satellite peaks are shown to emerge in the mean-field approximation, in departure from the common +belief that they characterize correlation effects. The scheme emphasizes the importance of correlation +and how different many-body corrections can improve the mean-field description. The pedagogical +treatment is expected to make it possible for researchers to acquire an improved understanding of +many-body effects as found in various areas related to electronic properties of molecules and solids, +which is highly relevant to current efforts in quantum information and quantum computing. +Keywords: Hubbard model, mean-field approximation, exact diagonalization, many-electron basis +I. +INTRODUCTION +The Hubbard model [1] is a popular and simple model +to describe electron correlation in solids, molecules, and +nanoparticles. The exact description of correlation is a +tremendous task in the field of electronics, and it is of +paramount importance for the accurate description of +magnetism, optical properties, electron transport, and +plasmonics [2–5]. +In spite of its apparent simplicity, finding exact so- +lutions of the Hubbard model is a formidable effort in +general. +Starting from the Hubbard Hamiltonian, the +simplest conceptual way to solve it is the exact diag- +onalization (ED) method [6–8], that can however only +be performed analytically or numerically for very small +systems [6–14]. This method hinges on the diagonaliza- +tion of the Hubbard Hamiltonian expressed in a many- +electron basis and yields the eigen-energies and eigen- +vectors of the Hamiltonian, expressed as linear combina- +tions of the (many-body) basis states. The major issue +with this method is that the number of basis states grows +exponentially with the number of electrons. For example, +at half-filling, the many-body basis for the single-orbital +Hubbard model of a two-site system has dimension 6, di- +mension 20 for three-site system, and already dimension +924 for a six-site system. The dimension of the Hilbert +space for a ten-site system reaches 184, 756, leading to +a Hamiltonian matrix with more that 3.4 × 1010 ele- +ments [7]. This illustrates how the numerical resolution +of the method is rapidly limited, even for modest size +systems. For this reason, researchers have realized the +need for approximation methods to render the compu- +tational treatment of the Hubbard model at a tractable +computational cost. +The mean-field approximation (MF) is one of the sim- +plest approximations for the Hubbard model. It consists +in replacing the two-body interaction term of the Hub- +bard model as an interaction between one electron and a +mean-field due to the other electrons. As a result, a given +electron no longer interacts directly with other electrons +but rather with a field. This makes it possible to write +the Hamiltonian in a single-electron basis and to consider +electrons as particles that are effectively free. The main +advantage of MF is that the corresponding dimension of +the Hilbert space is reduced to 2Nel, where Nel is the +number of electrons and the factor 2 accounts for the +spin degree of freedom. The MF approximation is often +used to describe electronic systems [2, 3, 15] and we will +refer to the formulation of the MF approximation in the +single-electron basis as MF-U for mean-field usual. +One major drawback of the MF-U approximation is +that it misses all the correlation between electrons. Sev- +eral techniques have been developed to move beyond +MF-U to include (at least a part of the) correlation, +such as the Green’s function many-body approxima- +tion (GFMBA), which consists of a sum of Feynman +diagrams [16, 17]. +This family of approximations in- +cludes the second-order Born approximation, the GW +approximation, and the T-matrix approximation. There +exists several other approaches such as the dynamical +mean-field theory and the quantum Monte Carlo ap- +proaches. [15, 18]. Furthermore, other methods are based +on the MF-U computation augmented by a symmetry +restoration procedure [19, 20]. +What’s more, machine +learning based self-energy construction have also been +investigated more recently [21–23]. +Up to now, the discrepancies between MF-U and ED +wave functions have not been fully described or under- +arXiv:2301.03223v1 [cond-mat.str-el] 9 Jan 2023 + +2 +stood in a general framework despite their importance +for the development of intuitive and accurate corrections +to the approximation. +We believe that this lack of a +deeper understanding is partly due to the fact that ED +methods must be expressed in the many-electron basis +whereas the MF-U is, by design, usually implemented +in the single-electron basis. It is therefore challenging to +compare the two methods since this change of basis is not +a simple unitary transformation but a complete change +of paradigm. +The objective of this paper is to propose an in-depth +comparison between MF and ED by formulating the MF +approximation in the many-body basis framework. This +approximation will be referred to as the MF-MB in the +rest of the paper. The goal of this formulation is not to +reduce the computational cost of the method but to build +a MF approximation in the same basis as the ED, and +thus to gain insight into the missing part of the MF ap- +proximation (i.e., correlation). We note, however, that +in contrast to common density functional theory compu- +tations, the Hubbard MF approximation does not suf- +fer from an exchange problem since the ground state is +approximated by a Slater determinant that satisfies the +wave function symmetry required for the eigen-states of +the exchange operator, which commutes with the Hamil- +tonian of the electronic systems. +Our analysis highlights the ambiguous definition of the +density of states within the MF approximation depending +on the method. The MF-U method is often combined im- +plicitly with Koopman’s theorem [24, 25], which assumes +that all the states of the N-electron system are frozen +when adding/removing an electron. This assumption is +not strictly correct since even in the MF approximation, +adding or removing an electron changes the mean-field +and consequently the predicted states (both occupied and +unoccupied) as we will illustrate by comparing the DOS +obtained via MF-U and MF-MB techniques. This com- +parison highlights the fact that satellite states also ap- +pear in the MF-MB approximation, although they are +often thought as being the result of the inclusion of cor- +relation [26]. +The rest of the paper is organized as follow: we first in- +troduce the Hubbard model, the MF-U approximation, +and the ED technique. +We then explain the MF-MB +technique, based both on MF-U equations and the nu- +merical methods employed for the ED. Finally, we discuss +results of MF-MB compared with MF-U and ED and the +notion of density of states and the appearance of satellite +peaks in the MF-MB. +II. +REVIEW OF STANDARD METHODS +A. +Hubbard model +This studies focused on the single-orbital Fermi- +Hubbard model: +ˆHHubbard = −t +� +,σ +ˆc† +iσˆcjσ + U +� +i +ˆni↑ˆni↓ +(1) +where t is the hopping parameter, U is the interaction +(or Hubbard) parameter, ˆc† +iσ (resp. +ˆciσ) is a creation +(resp. +destruction) operator of an electron on atomic +site i with spin σ, and ˆniσ = ˆc† +iσˆciσ is the density op- +erator (on atomic site i and with spin σ). The atomic +site indices run from 0 to N − 1 where N is the total +number of sites. +The ⟨ . ⟩ symbol under the summa- +tion operator indicates that the sum runs over all pairs +of nearest-neighbour atomic sites. +The first term of the Hubbard Hamiltonian in eq. 1 is +the tight-binding Hamiltonian and is easily written us- +ing the one-electron basis. In contrast, the second term +(known as the Hubbard or interaction term) is the prod- +uct of two density operators (i.e., a combination of four +creation and/or destruction operators). This two-body +operator cannot be written in the one-electron basis. +B. +Mean-field approximation and single-electron +basis +In the MF approximation, the Hubbard Hamiltonian +of eq. (1) is approximated so that it only includes one- +body operators and the Hamiltonian can be written in +the one-electron basis. +In practice, the density opera- +tors are decomposed as sums of the mean value of the +operator (niσ) and the deviation (ˆniσ − niσ) from this +mean value: ˆniσ = niσ +(ˆniσ −niσ). Products of density +operators in eq. (1) are expanded and the approxima- +tion consists in dropping products of deviations from the +mean, generally assumed (without proof) to be small. +Products of mean values only induce a constant shift in +the Hamiltonian (and thus in the total energy), leading +to the Hamiltonian: +ˆHHubb,MF = − t +� +,σ +ˆc† +iσˆcjσ + U +� +i +(ni↑ˆni↓ + ni↓ˆni↑) +− U +� +i +ni↑ni↓. +(2) +In the MF-U method, the Hamiltonian of eq. (2) is +written in the form of a matrix expressed in the one- +electron basis. The basis states of the one-electron basis +are obtained by the application of the different creation +operators on the vacuum state |∅⟩ (i.e., the state contain- +ing no electron), for each site and spin value. It follows +that +|iσ⟩ = ˆc† +iσ |∅⟩ +(3) + +3 +FIG. 1. Illustration of the structure of the Hubbard Hamil- +tonian in the MF-U approximation. The Hamiltonian matrix +of size 2N ×2N is written in the single-electron basis and can +be divided into 4 blocks of size N × N: one containing pure +spin-up related terms, one containing pure spin-down terms, +and two mixing blocks between up and down spins. +are the 2N basis states of the single-electron basis. In +that basis, the matrix has dimension 2N × 2N (N being +the number of sites). The Hamiltonian matrix is com- +posed of 4 blocks of dimension N × N: the two blocks +on the diagonal are pure spin-up and spin-down blocks +and the two off-diagonal blocks mix spin up and spin +down that, according to the MF Hamiltonian, remain +equal to zero (see figure 1 for an illustration of the block +matrix). The tight-binding term implies that the Hamil- +tonian matrix has terms of amplitude −t for elements +corresponding to neighbouring sites. The second term of +the Hamiltonian is concerned with diagonal elements in +the single-electron basis. The diagonal elements in the +spin-up (-down) block involve mean densities of down +(up) spins. +One usually does not implement the last +term (which only includes mean values, not operators) +in eq. (2). We need, however, to reintroduce that term +in the computation of the total energy, as it corresponds +to a rigid shift of all energy levels. +For example, the mean-field Hubbard Hamiltonian +(eq. (2)) is expressed in the single-electron basis for the +two-site system as: +ˆHHubb,MF-U = +� +� +� +Un0↓ +−t +0 +0 +−t +Un1↓ +0 +0 +0 +0 +Un0↑ +−t +0 +0 +−t +Un1↑ +� +� +�, +(4) +if the basis states are ordered in spin blocks in the same +way as atomic sites 0 and 1: {|0 ↑⟩ , |1 ↑⟩ , |0 ↓⟩ , |1 ↓⟩}. +The Hamiltonian of eq. (2) involves mean values of +density operators and it is therefore necessary to solve it +self-consistently. Initial conditions for the mean densities +are guessed and a Hamiltonian based on these mean val- +ues is built. In the next step, the first Hamiltonian is di- +agonalized, resulting in eigen-energies and eigen-vectors. +New mean values are computed by populating the eigen- +vectors of lowest energies and a new Hamiltonian is gen- +erated, diagonalized, leading to new mean values. This +loop is repeated until the difference between the input +mean values and the output values is smaller than a given +threshold (convergence). +The density of states (DOS) in the MF-U is con- +structed as a sum of Dirac delta-peaks centered at the +converged eigen-energies of the Hamiltonian. A single- +particle states is identified at each eigen-energy. +In addition, using the single electronic eigen-energies +Ek and the coefficients ak +iσ of the eigen-states of the MF- +U Hamiltonian (k ranging from 0 to 2N), we define cre- +ation operators of each eigen-state as: +ˆd† +k = +� +i,σ +(ak +iσ)∗ˆc† +iσ. +(5) +For a system with Nel electrons, the wave-function of +the MF-U ground state is expressed as a single Slater +determinant, created by applying successively the Nel +creation operators associated with the Nel lowest eigen- +energies to the vacuum state: +|GS, MF-U⟩ = +� +k≤Nel +ˆd† +k |∅⟩ +(6) +and the mean values of density operators to be inserted +in the self-consistent Hamiltonian are given by: +niσ = +� +k≤Nel +��ak +iσ +��2. +(7) +C. +Exact diagonalization +Instead of solving for the approximated MF Hamil- +tonian described in eq. (2), the ED considers the exact +Hubbard Hamiltonian of eq. (1). The second term of the +Hamiltonian, i.e., the product of two density operators, +cannot be expressed in the single-electron basis of eq. (3) +and a many-electron basis needs to be used. As a com- +mon practice in the ED literature, the concept of sector +is introduced for a fixed number of electrons. When re- +stricted to the Nel sector, the many-electron basis states +are given by all possible combinations of Nel creation op- +erators applied to the vacuum state. For example +� +k≤Nel +ˆc† +k↑ |∅⟩ +(8) +represents one of the basis states, involving only spin-up +electrons. +As fermionic creation operators with at least one dif- +ferent index (site or spin) anti-commute, the order in +the product only affects the sign of the state (i.e., the +fermionic sign) and we find the same basis state, mod- +ulo an overall phase. Here, we conventionally choose to +define the basis states with a positive sign by ordering +all spin-up (resp. -down) creation operators to the left +(resp. right); operators with the same spin part are or- +dered from left to right in an increasing order. This re- +sults in the combination formula in combinatorics for the +number of basis states (Nb): +Nb = C2N +Nel = +(2N)! +(2N − Nel)! Nel!. +(9) + +Spin up +Spin up +X spin +N +down +2N +HMF-U += +Spin +Spin +down x +N +down +spin up +2N +N +N4 +We adopt the approach described in Refs. 6–8, and 27 +to label each state with one integer I, bijectively linked +with two other integers I↑ and I↓ by the relations: +I = 2NI↑ + I↓ +(10) +and +I↑ = I//2N +I↓ = I +mod 2N, +(11) +where // represents the integer division. +Writing I↑ (resp. +I↓) in binary notation yields the +space configuration of the state in the spin-up (resp. - +down) sector. Organizing the Hilbert space in this way +allows one to only have to deal with integers and to easily +find the effect of the creation, destruction, and density +operators on each state using simple standard binary op- +erations (e.g., bin flip, bin counting,. . . ). One can now +express the action of the full Hubbard Hamiltonian of +eq. (1) on the basis states and, in turn, calculate the +Hamiltonian matrix elements in the many-electron basis. +Formally, the Hamiltonian matrix contains elements +with value −t when they correspond to connected ba- +sis states having all the same electron creation operators +(sites and spin) but one. The different electrons have to +be of the same spin on a neighboring atomic site. The +Hamiltonian matrix also contains elements of amplitude +U on the diagonal for basis states containing two elec- +trons on the same site and of opposite spin. The U val- +ues are added if there are several doubly-occupied sites +in the basis state. +We illustrate the construction of the Hamiltonian for +a two-site system at half-filling. +The 6 basis states +are |Φ1⟩ += +ˆc† +0,↑ˆc† +1,↑ |∅⟩ , |Φ2⟩ += +ˆc† +0,↑ˆc† +1,↓ |∅⟩ , |Φ3⟩ += +ˆc† +1,↑ˆc† +0,↓ |∅⟩ , |Φ4⟩ = ˆc† +0,↑ˆc† +0,↓ |∅⟩ , |Φ5⟩ = ˆc† +1,↑ˆc† +1,↓ |∅⟩ and +|Φ6⟩ = ˆc† +0,↓ˆc† +1,↓ |∅⟩ and the Hamiltonian matrix in that +basis is given by: +HHubb,ED = +� +� +� +� +� +� +� +0 +0 +0 +0 +0 +0 +0 +0 +0 +−t −t 0 +0 +0 +0 +−t −t 0 +0 −t −t U +0 +0 +0 −t −t +0 +U +0 +0 +0 +0 +0 +0 +0 +� +� +� +� +� +� +� +, +(12) +where the element −t connects basis states |Φ2⟩ and |Φ3⟩ +to basis states |Φ4⟩ and |Φ5⟩ since they differ only by one +electron having the same spin and hopping from site 0 +to 1. U is on the diagonal for basis states |Φ4⟩ and |Φ5⟩ +both having two electrons, located on site 0 and site 1, +respectively. +The lowest eigen-energy of the Hamiltonian corre- +sponds to the ground-state and the higher ones to ex- +cited states. The corresponding eigen-vectors are by con- +struction many-electron states. In contrast to the single- +electron basis formulation of the MF-U, it is not con- +structed by populating several eigen-vectors. Likewise, +the associated eigen-energy is the total energy of the sys- +tem, without having to sum individual electron energies. +To gain access to dynamical (i.e., frequency depen- +dent) properties, we now introduce the Green’s function +with general definition [16, 28, 29]: +Giσ,jσ′(ω) = +� +ΨNel +0 +��� ˆciσ +1 +ω + (ENel +0 +− ˆHNel+1 + iη) +ˆc† +jσ′ +���ΨNel +0 +� ++ +� +ΨNel +0 +��� ˆc† +iσ +1 +ω − (ENel +0 +− ˆHNel−1 + iη) +ˆcjσ′ +���ΨNel +0 +� +, +(13) +where +���ΨNel +0 +� +and ENel +0 +are the ground state and the +ground-state energy of the system with Nel electrons and +ˆHNel±1 are the Hamiltonian operators of the system con- +taining Nel ± 1 electrons and η is a small real positive +parameter. +The first term in eq. (13) is the electron addition part +of the Green’s function: ˆc† +jσ′ +���ΨNel +0 +� +and +� +ΨNel +0 +��� ˆciσ rep- +resent both states with Nel+1 electrons and the operator +involves the Nel+1 electron Hamiltonian. This term thus +explores the possible states when an electron is added to +the Nel ground state. The second term of eq. (13) de- +scribes the situation where one electron is removed. This +is the electron removal part of the Green’s function. The +Green’s function has poles at the frequencies correspond- +ing to difference of energies between the Nel ground state +and states in the Nel ± 1 sectors. +To evaluate the Green’s function (eq. (13)), three exact +diagonalizations are completed: one in the Nel sector to +find the ground-state +���ΨNel +0 +� +and its energy ENel +0 +, and +two for the two sectors Nel ± 1 so that matrix elements +of the type +� +ΦNel±1 +k +��� ˆHNel±1 +���ΦNel±1 +k′ +� +can be computed +with +���ΦNel±1 +k′ +� +the basis vectors of the Nel±1 sector of the +Hilbert space. The Green’s function is then computed by +writing the states ˆc† +jσ′ +���ΨNel +0 +� +in the basis +���ΦNel+1 +k +� +and +the states ˆcjσ′ +���ΨNel +0 +� +in the basis +���ΦNel−1 +k +� +. +The DOS (D(ω)) is defined from the Green’s function +as: +D(ω) = − 1 +π Tr +� +Im(GR(ω)) +� +, +(14) +where Tr is the trace. +We can now identify addition and removal parts of the +density of states: the addition part is computed based on +the addition part of the Green’s function and, in an anal- +ogous way, the removal part of the DOS is constructed +from the removal part of the Green’s function. The DOS +features peaks at energies corresponding to the poles of +the Green’s function, i.e., the differences between the Nel +ground-state energy and all states (ground state and ex- +cited states) of Nel ± 1. +We can imagine this process +as probing all the possible adding or removing of energy + +5 +when adding or removing one electron, taking into ac- +count the correlation exactly in both the starting state +(ground state of Nel electron) and the final states (Nel±1 +electron states). We point out that to account for the cor- +relation, the Nel ± 1 states in the Nel ± 1 sectors cannot +be obtained simply by adding one electron independently +from the others. +III. +MEAN-FIELD APPROXIMATION IN THE +MANY-BODY BASIS +A. +Formulation in the Nel sector +As mentioned before, the MF approximation effec- +tively decouples the density operators interaction into +the interaction of one density operator with a mean-field. +This results in the possible formulation within the single- +electron basis of the MF approximation (see section II B), +reducing the basis dimension from C2N +Nel to 2N. We now +examine how the MF approximation can be expressed in +the many-body basis (MF-MB). +We consider the MF-approximated Hamiltonian of +eq. (2) and the many-electron basis described in eq. (8). +As the MF approximation does not affect the tight- +binding term of the Hamiltonian, the −t elements of +the MF-MB Hamiltonian matrix are the same as for the +ED. In contrast, the interaction term involves mean val- +ues of density operators and the diagonal in the MF-MB +Hamiltonian includes Uniσ factors for each basis states +involving a creation operator c† +iσ′, σ being one spin (up +or down) and σ′ being the opposite spin. +In general, +there will be a sum of several Uniσ terms because basis +states contain several electrons. For example, if there is +a doubly-occupied site i in a basis state, terms of the +form U(niσ′ + niσ) are present on the diagonal of the +Hamiltonian matrix. +We illustrate the construction of the Hamiltonian for +the two-site system at half-filling with the basis states ex- +plicitly written in section II C. The Hamiltonian matrix +in this case is given by: +HHubb,MF-MB = +� +� +� +� +� +� +� +U(n0↓ + n1↓) +0 +0 +0 +0 +0 +0 +U(n0↓ + n1↑) +0 +−t +−t +0 +0 +0 +U(n0↑ + n1↓) +−t +−t +0 +0 +−t +−t +U(n0↓ + n0↑) +0 +0 +0 +−t +−t +0 +U(n1↓ + n1↑) +0 +0 +0 +0 +0 +0 +U(n0↑ + n1↑) +� +� +� +� +� +� +� +, +(15) +where we removed the last term of eq. (2) as in the MF-U +case since it is only a constant shift of the Hamiltonian, +leaving the eigen-vectors unchanged. +Similar to the MF-U case, the MF-MB Hamiltonian +depends on the mean value of density operators that we +compute self-consistently from the ground state of the +preceding iteration, starting from an initial guess for the +mean densities. As the ground state is given as a linear +combination of the basis states given by eq. (8), the mean +densities of electrons at site i with spin σ are computed +by summing the square of the linear coefficients when +the basis state contains an electron at site i with spin +σ. A new Hamiltonian is also computed iteratively, and +then diagonalized to obtain to new mean densities, until +self-consistency is achieved. +Numerically, the many-electron basis is encoded ex- +actly in the same way as in ED, i.e. using the integers I, +I↑, and I↓. As in the ED case, the effect of creation, de- +struction, and density operators are implemented using +simple binary operations on the binary representation of +the three integers. +B. +Correspondence with MF-U +We are now in a position to examine the links between +MF-U and MF-MB. In this section, we explain and show +with examples how the two methods are consistent. The +MF-U method yields individual states (eigen-vectors of +the Hamiltonian) for effectively independent particles, at +given energies (eigen-energies of the Hamiltonian). The +ground state of the system containing Nel is then formed +by populating the Nel eigen-states with the lowest ener- +gies according to Pauli’s principle. The total energy of +the ground state is the sum of the individual energies. +The counterpart (i.e., the ground state) in the MF- +MB method is the lowest energy state which is a linear +combination of the many-body basis states. The total +energy of the ground state (containing Nel electrons) is +the eigen-energy of the state, i.e. the lowest eigen-energy +of the MF-MB Hamiltonian. +We illustrate this with the two-site system at half- +filling. +We first introduce the change of basis from a +localised basis to a bonding/anti-bonding basis in or- +der to better understand the relation between MF-U and +MF-MB results. In the single-electron picture, we define +bonding (b) and anti-bonding (a) states for a given spin + +6 +σ as: +|bσ⟩ = +1 +√ +2(|0σ⟩ + |1σ⟩) = +1 +√ +2(ˆc† +0σ + ˆc† +1σ) |∅⟩ and +|aσ⟩ = +1 +√ +2(|0σ⟩ − |1σ⟩) = +1 +√ +2(ˆc† +0σ − ˆc† +1σ) |∅⟩ , +(16) +such that it is intuitive to define bonding and anti- +bonding creation operators as: +ˆc† +bσ = +1 +√ +2(ˆc† +0σ + ˆc† +1σ) and +ˆc† +aσ = +1 +√ +2(ˆc† +0σ − ˆc† +1σ). +(17) +We +also +define +the +many-body +states +in +the +bonding/anti-bonding basis by the successive applica- +tions of bonding and anti-bonding operators. As for the +localised basis, it is important to choose a convention +(fermionic sign). We chose to define a positive sign state +in our convention. We consider the states presenting the +ordering of all spin-up (resp. -down) creation operators +on the left (resp. right). Within the same spin part, a +positive sign state orders the bonding and anti-bonding +operators from left to right. For example, the many-body +state containing two bonding states (of opposite spins) is +expressed as +|b ↑, b ↓⟩ = ˆc† +b↑ˆc† +b↓ |∅⟩ += 1 +2(ˆc† +0↑ + ˆc† +1↑)(ˆc† +0↓ + ˆc† +1↓) |∅⟩ += 1 +2(|Φ4⟩ + |Φ2⟩ + |Φ3⟩ + |Φ5⟩). +(18) +The eigen-states and eigen-energies for the MF ap- +proximation are listed in table I for the MF-U method +and in table II for the MF-MB method. +The MF-U +method yields four eigen-states at energies E = −0.75t +and E = 1.25t. +Each energy corresponds to doubly- +degenerated states, due to spin. +The single-particle +eigen-states are the bonding and anti-bonding states. +The MF-MB method gives six eigen-states with energies +E = −1.5t (non-degenerated), E = 0.5t (four times de- +generated), and E = 2.5t (non degenerated). The sec- +ond part of table II indicates that the states formed +by successive applications of bonding and anti-bonding +operators are the eigen-states of the Hamiltonian. The +ground state of the MF-MB (E = −1.5t) is |b ↑, b ↓⟩. +This state corresponds to the state formed by occupying +the two lowest energy single-particle states of the MF-U +method. Those states are |b ↑⟩ and |b ↓⟩, both with en- +ergies E = −0.75t. The energy of the ground state in +the MF-MB method is then the sum of the individual +energies of the two lowest energy states in MF-U. This +illustrates that, for a given number of particles, the MF- +U and MF-MB methods predict the same ground state +with the same energy, as expected. The excited many- +electron states computed in the MF-MB method (with +energies E = 0.5t and E = 2.5t) are the two-electron +Localised basis : +|0 ↑⟩ +|1 ↑⟩ +|0 ↓⟩ +|1 ↓⟩ +E = −0.75t +1/ +√ +2 +1/ +√ +2 +0 +0 +E = −0.75t +0 +0 +1/ +√ +2 +1/ +√ +2 +E = 1.25t +1/ +√ +2 −1/ +√ +2 +0 +0 +E = 1.25t +0 +0 +1/ +√ +2 −1/ +√ +2 +Bonding/anti-bonding basis : |b ↑⟩ +|b ↓⟩ +|a ↑⟩ +|a ↓⟩ +E = −0.75t +1 +0 +0 +0 +E = −0.75t +0 +1 +0 +0 +E = 1.25t +0 +0 +1 +0 +E = 1.25t +0 +0 +0 +1 +TABLE I. List of MF-U eigen-states of the two-site Hub- +bard system at half-filling with U = 0.5t. +There are four +eigen-states, two pairs of degenerate states (spin degeneracy): +one at E = −0.75t and the other at E = 1.25t. The first +part of the table shows the eigen-states’ coefficients in the +single-electron basis constructed with localised basis states +(see eq. (3)). The second part of the table gives the eigen- +states’ coefficients in the diagonal basis consisting of bonding +and anti-bonding states. +states formed by populating every other combinations of +two single-particle states of MF-U (not the two lowest +energy states). +The MF-MB thus predicts directly all the many- +electron states that can be obtained from populating +single-particle eigen-states from the MF-U method. In +the Nel sector, a correspondence is therefore established +and the two methods lead to consistent results. +IV. +DENSITY OF STATES +We now turn our attention to the notion of density +of states in the MF-U and in the MF-MB approaches. +Specifically, we highlight two possible interpretations of +the DOS. This leads to different features in the DOS +that will be discussed, such as the appearance of satellite +peaks in the DOS that are usually attributed to beyond- +MF methods. +In the MF-U method, the DOS consists of Dirac delta- +peaks at the energies of the single-particle states. The +Fermi level is positioned between the highest-occupied +and the lowest-unoccupied levels. From the MF-MB re- +sults, we can retrieve this DOS by adopting a picture +based on differences between MF-MB eigen-energies. In- +deed, starting from the ground state with E = −1.5t +and identifying the four states at E = 0.5t (see table II) +as states having exactly one electron in a bonding state +and one electron in an anti-bonding state, we obtain the +following system for single-particle state energies + +7 +Localised basis : +|Φ1⟩ = |↑, ↑⟩ |Φ2⟩ = |↑, ↓⟩ |Φ3⟩ = |↓, ↑⟩ |Φ4⟩ = |↑↓, .⟩ |Φ5⟩ = |., ↑↓⟩ |Φ6⟩ = |↓, ↓⟩ +E = −1.5t +0 +1/2 +1/2 +1/2 +1/2 +0 +E = 0.5t +0 +−1/ +√ +2 +1/ +√ +2 +0 +0 +0 +E = 0.5t +0 +0 +0 +−1/ +√ +2 +1/ +√ +2 +0 +E = 0.5t +1 +0 +0 +0 +0 +0 +E = 0.5t +0 +0 +0 +0 +0 +1 +E = 2.5t +0 +1/2 +1/2 +-1/2 +-1/2 +0 +Bonding/anti-bonding basis : +|b ↑, a ↑⟩ +|b ↑, b ↓⟩ +|b ↑, a ↓⟩ +|a ↑, b ↓⟩ +|a ↑, a ↓⟩ +|b ↓, a ↓⟩ +E = −1.5t +0 +1 +0 +0 +0 +0 +E = 0.5t +0 +0 +1 +0 +0 +0 +E = 0.5t +0 +0 +0 +1 +0 +0 +E = 0.5t +1 +0 +0 +0 +0 +0 +E = 0.5t +0 +0 +0 +0 +0 +1 +E = 2.5t +0 +0 +0 +0 +1 +0 +TABLE II. Table of the MF-MB eigen-vectors of the two-site Hubbard system for U = 0.5t. There are six eigen-states with +energies E = −1.5t, E = 0.5t (four times degenerated), and E = 2.5t. The first part of the table shows the linear coefficients of +the states in the many-body basis presented in sec. II C) (localised basis). The second part of the table gives the eigen-states’ +coefficients in the bonding/anti-bonding basis, the basis constructed from the tensor product of the diagonal basis for the +single-electron picture (MF-U). +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Eb↑ + Eb↓ = −1.5t +(I) +Eb↑ + Ea↓ = 0.5t +(II) +Ea↑ + Eb↓ = 0.5t +(III) +Eb↑ + Ea↑ = 0.5t +(IV) +Eb↓ + Ea↓ = 0.5t +(V) +Ea↑ + Ea↓ = 2.5t +(VI). +Using these six equations, we can find the single- +particle state energies: +Eb↑ = Eb↓ = −0.75t +Ea↑ = Ea↓ = 1.25t. +(19) +This reasoning is somewhat inconvenient because it +cannot be easily generalized to large systems. However, +it presents the advantage of explaining how the MF-U +DOS (see table I) can be recovered from the MF-MB +method. However, we emphasize that this interpretation +of the single-particle DOS does not match with the defi- +nition of the DOS in a many-electron basis, constructed +from the Green’s function (see eqs. 13 and 14). In the +MB basis, the DOS explores, from the ground state in +the Nel sector, all possible states in the Nel ± 1 sectors, +corresponding to all possible ways of adding or removing +one electron. +To adapt this definition of the DOS to MF-MB +method, we need (as in the case of ED) to compute MF- +MB results in the Nel, Nel + 1, and Nel − 1 sectors in- +dependently and then compute the Green’s function and +the DOS from eqs. 13 and 14. The fundamental difference +between this approach of the DOS and the approach of +MF-U (with the correspondence with MF-MB explained +before) is that: +• in MF-U, all single-particle states are computed as +interacting with the mean-field calculated from the +ground state of the Nel sector; +• in MF-MB, peaks in the DOS corresponding to the +addition (resp. removal) of one electron take into +account the fact that when one electron is added +(resp. removed), the states of the Nel + 1 (resp. +Nel − 1) sector interact with the mean-field con- +structed from the ground state of the Nel +1 (resp. +Nel − 1) sector. +In MF-U, adding one electron to the ground state (con- +taining Nel) in the first unoccupied states will not result +in the ground state of the system containing Nel +1 elec- +trons. Indeed, we would have to perform another MF- +U calculation using a self-consistent procedure with the +mean-field created from Nel + 1 electrons. This would +even change the Nel lowest energy states from the previ- +ous MF-U calculation with Nel electrons. +In the MF-U method, the occupied states in the DOS +have a clear physical meaning: they are the individual +states composing the ground state. However, the inter- +pretation of the unoccupied states is more difficult since +they can’t be attributed to physical states that will be +occupied if we add an electron. +In contrast, the definition of the DOS in MF-MB, in +analogy with the ED technique, provides a clear physical +interpretation of all the peaks: They represent all pos- +sible ways of adding or removing one electron from the +Nel-electron ground state. +We conclude this discussion with an important re- +mark about the computation of the DOS in the MF-MB +method. For some systems, starting from different ini- +tial conditions, it is possible to converge towards several +different ground states, degenerated in energy. It seems +unphysical to take into account only one of the ground- +state solutions in the calculation of the DOS (eqs. (13) +and (14)). At the same time, this reasoning might lead +to the exclusion of states in some sectors. This idea is +better explained by examining the example of the two- + +8 +Bonding/anti-bonding basis +spin-up ground state : +|b ↑⟩ |b ↓⟩ |a ↑⟩ |a ↓⟩ +E = −t +1 +0 +0 +0 +E = −0.75t +0 +1 +0 +0 +E = t +0 +0 +1 +0 +E = 1.25t +0 +0 +0 +1 +Bonding/anti-bonding basis +spin-down ground state : +|b ↑⟩ |b ↓⟩ |a ↑⟩ |a ↓⟩ +E = −t +0 +1 +0 +0 +E = −0.75t +1 +0 +0 +0 +E = t +0 +0 +0 +1 +E = 1.25t +0 +0 +1 +0 +TABLE III. List of eigen-states of the two-site Hubbard +system at quarter-filling from the MF-U or MF-MB meth- +ods for U = 0.5t. +There are four eigen-states at energies +E = −t, −0.75t, t, and 1.25t. The first (resp. second) part +of the table shows the eigen-states’ coefficients in the single- +electron bonding/anti-bonding basis where the ground state +was chosen to be spin up (resp. down). +site system at half-filling. One needs to perform a MF- +MB calculation in the one-electron and three-electron +sectors to construct the MF-MB DOS. For illustration +purposes, we only focus on the one-electron sector and +on the removal part of the DOS. The one-electron sector +eigen-states of the MF-MB method are given directly in +the bonding/anti-bonding basis in table III. The single- +particle eigen-states for the one-electron sector are the +same as for the two-electron sector, but we observe that +depending on the choice of the ground-state polarization +(spin up or spin down), all other states are affected and +spin-flipped. +When computing the Green’s function (eq. (13)), we +have to consider all accessible states (in the Nel = 1 sec- +tor) from the Nel = 2 ground state |b ↑, b ↓⟩ (see table II). +For example, removing a spin-down electron from the +state |b ↑, b ↓⟩ results in the state |b ↑⟩ that is present at +different energies in the one-electron sector, depending on +the ground-state polarization (E = −t and E = −0.75t), +see table III. It appears that only the state at E = −t has +to be taken into account since the other one is an unoccu- +pied state when the ground state is spin-down polarised. +Since we precisely chose to remove a spin-down electron +from the Nel = 2 ground state, the one-electron ground +state to be taken into account should be only the spin-up +polarized one. The same reasoning holds for the removal +of the spin-up electron. For the two-site system, there +is a symmetry in the one-electron and three-electron sec- +tors and thus in the addition and removal parts of the +DOS. +The density of states computed in ED, MF-U, and +MF-MB (with and without the exclusion of states in the +Nat = 1 and Nat = 3 sectors) are given in fig. 2. This +illustrates the process of state exclusion when comparing +the two middle curves: the first one with no exclusion +(second curve from the top at fig. 2) exhibits two peaks +below the Fermi level, corresponding to a transition from +FIG. 2. Density of states of the two-site Hubbard system at +half-filling with U = 0.5t for MF-U (blue), MF-MB without +state exclusion in the Nat = 1 and Nat = 3 sectors (orange), +MF-MB with state exclusion (green), and ED (red). +the Nel = 2 ground state |b ↑, b ↓⟩ to the states |b ↑⟩ and +|b ↓⟩ for the two possible choices of ground state in the +one-electron sector. +In contrast, with the exclusion of +states, the third curve from the top of fig. 2 shows only +one peak below the Fermi level, corresponding to the +transition from the Nel = 2 ground state to the states +|b ↑⟩ and |b ↓⟩ of E = −t in the one-electron sector (the +E = −0.75t have been excluded). We note that for this +specific case, we recover the MF-U result from the MF- +MB with states exclusion but this is not the case in gen- +eral. We also stress that ED predicts satellites (smaller +peaks) for the two-site Hubbard model at half-filling, lo- +cated at E ≃ −2.77t and E ≃ 3.27t (see Ref. 11, 12, and +30). One satellite is visible in the insert of fig. 2 but no +satellite is present in either of the MF densities of states +presented. We explore the presence of satellites in further +detail in the next section. +V. +COMPARISON BETWEEN MF METHODS +AND ED +We now illustrate the formalism described in this paper +for the case of small atomic linear chains containing up to +3 atomic sites, showcasing the main differences between +the MF-U, MF-MB, and ED methods. +A. +MF-MB vs. ED states and DOS: the two-site +Hubbard system +We have shown above the results of the two-site prob- +lem for the MF-MB technique in the one-electron and +two-electron sectors as well as the DOS in the half-filling +case. Here, we start by recalling ED results that have +been extensively studied before [9–12, 14]. We then de- +scribe the differences between MF-U, MF-MB, and ED + +MF-U +MF-MB exclusion +MF-MB no exclusion +ED +x50 +x50 +人 +x50 +x50 +-3 +-2 +-1 +0 +1 +2 +3 +4 +E [t units]9 +Bonding/anti-bonding basis : |b ↑⟩ |b ↓⟩ |a ↑⟩ |a ↓⟩ +E = −t +1 +0 +0 +0 +E = −t +0 +1 +0 +0 +E = t +0 +0 +1 +0 +E = t +0 +0 +0 +1 +TABLE IV. List of ED eigen-states of the two-site Hub- +bard system at quarter-filling for U = 0.5t. There are four +eigen-states at energies E = −t and t (each doubly spin- +degenerated). +states based on this simple model as well their conse- +quences on the DOS. +The eigen-vectors of the one-electron sector of the two- +site Hubbard model with U = 0.5t are listed in table IV in +the bonding/anti-bonding basis. They have to be com- +pared to MF-MB eigen-states shown in table III. +We +observe that the main difference between the ED and +MF-MB results is the lifting of spin-degeneracy in the +MF-MB for both bonding and anti-bonding states. This +can be understood as follows: in MF-MB, a specific po- +larized ground state is chosen, which leads to a mean-field +interaction term that is only present for the opposite spin +states. In ED, there is no interaction term since there is +only one electron and no double occupation in the basis +states. +The eigen-vectors of the two-site Hubbard model with +U = 0.5t at half-filling are listed in table V in the lo- +calised and in the bonding/anti-bonding bases. +These +eigen-states have to be compared with MF-MB results +of table II. Note that if we want to compare the eigen- +energies, we have to remember to take into account the +constant term of eq. (2) that consists in a constant shift +in the energies. In the case of table II, one has to apply +a constant shift of −0.25t for all eigen-energies. We ob- +serve three main effects of the MF approximation when +comparing results from tables II and V. The first MF +effect is a well-known property: the MF approximation +overestimates the ground-state energy. +The difference +between MF and ED ground-state energies is usually +called the correlation energy. +The correlation energy +Ecorr = EMF − EED depend on the parameter U and, +in our example, is given by Ecorr ≃ 0.01556t. The second +effect is that the three degenerate states at E = 0 and +the singly-degenerated state at E = 0.5t in ED are de- +generate in MF at E = 0.5t. This can be understood by +noting that the basis states |↑↓, .⟩ and |., ↑↓⟩ are treated +in the same manner as basis states |↑, ↓⟩ and |↓, ↑⟩ in MF, +whereas it is not the case in ED: the E = 0.5t singlet is +formed by the basis states |↑↓, .⟩ and |., ↑↓⟩ that are the +only basis states introducing a Hubbard term since they +induce a doubly-occupied site. This difference of treat- +ment in MF and in ED is also responsible for the third +observed difference: the MF ground state (at E = −1.5t) +has equal weight for |↑, ↓⟩ and |↓, ↑⟩ than for |↑↓, .⟩ and +|., ↑↓⟩ whereas there is an asymmetry in the ED ground +state due to the Hubbard term induced by the doubly- +occupied site. +The equal weight in MF results in the +FIG. 3. +Density of states of the two-site Hubbard system +at quarter-filling with U = 0.5t for MF-U (blue), MF-MB +(green), and ED (red). +ground state being a product state (i.e., a Slater de- +terminant) of single-electron eigen-states (bonding and +anti-bonding states) and the ED ground states is a lin- +ear combination of Slater determinant of single-electron +eigen-states. +Because the ED ground state is a superposition of +|b ↑, b ↓⟩ and |a ↑, a ↓⟩, the Green’s function (eq. (13)) +features two poles for the part related to the Nel − 1- +electron sector since the ground state couples with all +the possible states of table IV. The same holds for the +Nel+1 electron sector part of the Green’s function due to +the symmetry of the two-site Hubbard model for Nel = 1 +and Nel = 3. The coupling between the Nel = 2 ground +state and the excited states of the Nel = 1 sector (E = t +states shown in table IV) results in the so-called satellite +peaks in the DOS of the half-filled two-site Hubbard sys- +tem. A key feature of this peak is that it disappears at +the U = 0 limit: this can be explained since at U = 0, +the asymmetry between the two states |↑, ↓⟩ and |↓, ↑⟩ +and the two states |↑↓, .⟩ and |., ↑↓⟩ is canceled and the +ground state is found to be a single Slater determinant +of single-electron eigen-states |b ↑, b ↓⟩. +At the U = 0 +limit as well as in MF, the ground state is a pure prod- +uct state of single-particle eigen-states such that it only +couples to |b ↑⟩ and |b ↓⟩ listed in tables III and IV. Only +one peak on each side of the Fermi level is observed as in +the MF-MB exclusion curve of fig. 2. +We now turn to the quarter-filling DOS of the two- +site Hubbard system. The ground state is either |b ↑⟩ or +|b ↓⟩. +Specifically, we chose the Nat = 1 ground state +to be |b ↑⟩. The Green’s function of eq. (13) has poles +corresponding to energy transitions between the single- +particle ground state and all two-particle states listed in +tables II or V for MF and ED respectively as well as +at energy transitions between the single-particle ground +state and all zero-particle states. There exists only one +zero-particle state: it is the vacuum state and has zero +energy. +Figure 3 shows the DOS of the quarter-filling two-site +Hubbard system for U = 0.5t using the MF-U, MF-MB, +and ED methods. +Since there is only one electron in + +x50 +MF-U +MF-MB +x50 +ED +x50 +-4 +-2 +-1 +0 +1 +2 +3 +4 +E [t units]10 +Localised basis : +|↑, ↑⟩ +|↑, ↓⟩ +|↓, ↑⟩ +|↑↓, .⟩ +|., ↑↓⟩ +|↓, ↓⟩ +E = −1.76556t +0 +0.5301 +0.5301 +0.467970 0.46797 +0 +E = 0 +0 +−1/ +√ +2 +1/ +√ +2 +0 +0 +0 +E = 0 +1 +0 +0 +0 +0 +0 +E = 0 +0 +0 +0 +0 +0 +1 +E = 0.5t +0 +0 +0 +−1/ +√ +2 +1/ +√ +2 +0 +E = 2.26556t +0 +-0.46797 -0.46797 +0.5301 +0.5301 +0 +Bonding/anti-bonding basis : |b ↑, a ↑⟩ |b ↑, b ↓⟩ |b ↑, a ↓⟩ |a ↑, b ↓⟩ |a ↑, a ↓⟩ |b ↓, a ↓⟩ +E = −1.76556t +0 +0.99807 +0 +0 +-0.06214 +0 +E = 0 +0 +0 +1/ +√ +2 +−1/ +√ +2 +0 +0 +E = 0 +1 +0 +0 +0 +0 +0 +E = 0 +0 +0 +0 +0 +0 +1 +E = 0.5t +0 +0 +1/ +√ +2 +1/ +√ +2 +0 +0 +E = 2.26556t +0 +0.06214 +0 +0 +0.99807 +0 +TABLE V. List of the ED eigen-vectors of the two-site Hubbard system calculated for U = 0.5t. There are six eigen-states +with energies E ≃ −1.76556t, E = 0 (three times degenerated), E = 0.5t, and E ≃ 2.26556t. The first part of the table shows +the linear coefficients of the states in the many-body basis of sec. II C) (localised basis). The second part of the table gives the +eigen-states’ coefficients in the bonding/anti-bonding basis. +the system, the MF-U DOS exhibits peaks at the single- +electron eigen-energies that are also found in MF-MB and +given in table III. The lowest energy peak in MF-MB and +ED methods are both removal peaks at the ground-state +energy of the Nel = 1 sector of tables III and IV. +The second lowest energy peak in MF-MB and ED cor- +responds to the coupling in the Green’s function between +the Nel = 1 and the Nel = 2 ground states. Since in +the Nel = 2 sector the ED ground state is ≃ 0.02t lower +than the MF-MB one (correlation energy), the DOS peak +is also found at a lower energy. In each case, it corre- +sponds to the addition of a |b ↓⟩ electron to the Nel = 1 +ground state |b ↑⟩. In MF-MB, the ground state is the +only state in the Nel = 2 sector that contains that ba- +sis state |b ↑, b ↓⟩. In ED, due to the asymmetry between +states with and without doubly occupied states, the high- +est excited state also contains a part of the basis state +|b ↑, b ↓⟩. This is responsible for the presence of a satel- +lite in the DOS, clearly visible in the insert of fig. 3. +Alternatively, this satellite might also be interpreted as +a contribution from the addition of a |b ↓⟩ electron. The +other consequence of this asymmetry is the fact that the +second lowest-energy peak in ED does not integrate to 1, +since it does not simply represent a single-electron state. +Finally, the highest energy peak of the MF-MB DOS +(E = 1.25t) originates from the coupling between the +Nel = 1 ground state |b ↑⟩ and two of the four states +at E = 0.5t listed in table II. +The states |a ↑, b ↓⟩ +and |b ↓, a ↓⟩ indeed cannot be reached from |b ↑⟩ in the +Green’s function expression of eq. 13. +The observed +peak in the DOS thus comes from the addition of |a ↑⟩ +and |a ↓⟩ electrons. In MF-U, the addition of |a ↑⟩ and +|a ↓⟩ electrons is non-degenerate because these unoccu- +pied states are built by interacting with the mean-field +produced by only one electron (the |b ↑⟩ electron of the +ground state) such that there is an interaction term for +the addition of |a ↓⟩ but not for the addition of |a ↑⟩. +In contrast, in MF-MB, the many-body states of the +Nel = 2 sector are all constructed based on the mean- +field produced by the Nel = 2 ground state (|b ↑, b ↓⟩) +that induces the same interaction term for the consid- +ered excited states. +In ED, the peak is also split into +two peaks in the DOS because of the lifting of degener- +acy between E = 0 and E = 0.5t energy states listed +in table V. The E = t peak in ED reflects the addition +of a |a ↑⟩ electron and the half of the addition of a |a ↓⟩ +electron. The other half is contained in the next peak +in the DOS at E = 1.5t. +This splitting is due to the +fact that adding a |a ↓⟩ electron to a system containing +a |b ↑⟩ state involves contributions both from states with +and without doubly-occupied sites translated into differ- +ent interaction terms. +The peak corresponding to the +addition of a |a ↑⟩ electron is not split since there is no +interaction between two spin-up electrons. +B. +The emergence of satellites in MF-MB +The MF approximation treats particles as independent +particles interacting with a mean-field, neglecting all the +correlation. It is often thought that satellites and quasi- +particles are features that can only be observed when +including correlation effects via approximations or exact +treatment [9, 10, 26, 31, 32]. Satellites have zero weight +in the DOS for U = 0 and an increasing weight when the +interaction increases whereas quasi-particle peaks can be +linked to particle peaks in the U = 0 limit. Their weight +might decrease as the interaction increases: +a weight +transfer from quasi-particle to satellite peaks then oc- +curs [32]. +We now show that the MF-MB description presented +here can also induce the emergence of satellites in linear +chains containing Nat = 3, showing that the satellite +peaks are not correlation peaks. +The DOS for the Nat = 3 linear chain with one elec- +tron are shown in fig. 4 for the MF-U, MF-MB, and ED + +11 +FIG. 4. Density of states of the three-site Hubbard system +with one electron for U = 0.5t for MF-U (blue), MF-MB +(green), and ED (red). +methods. The DOS can be understood from the Nel = 1 +and Nel = 2 eigen-states for MF-MB and ED given in the +SI (tables 1 to 4). The general differences between the +two methods have already been described for the two-site +system in section V A and can be applied to the three-site +system. +Similar to the two-site system at quarter-filling (see +fig. 3), the two lowest peaks (i.e., the removal and the +lowest-energy addition peaks) have approximately the +same energies. The higher-energy pairs of peaks in MF-U +look degenerated in MF-MB. However, the term degen- +erated is not well-suited in the MF-MB case since, as for +the ED DOS, each peak does not necessarily integrate +to an integer value. This is related to the weight trans- +fer from quasi-particle peaks to emergent satellites (at +E ≃ 2.96t and E ≃ 4.43t) that can be visualized on the +zoomed-in MF-MB curve shown in fig. 4. +In ED, one observes that the two higher energy pairs +of single-particle states in MF-U correspond to a prin- +cipal peak that is more intense than the single-particle +peaks, plus several smaller peaks located near the princi- +pal peaks and satellites that can be seen on the zoomed-in +ED curve (see fig. 4). The small peaks near the principal +ones are not reproduced in MF-MB for the same rea- +son why the E ≃ 1.5t peak in the two-site system for +ED is not present in the MF-MB (see fig. 3). +Specif- +ically, in MF-MB (Nel = 2 sector), there are fourfold- +degenerated states that show a lifting of degeneracy +and result in threefold-degenerated plus non-degenerated +states (as discussed in section V A and can be seen in ta- +bles II and V). The main observation for the three-site +system with one electron is thus the emergence of satel- +lites in MF-MB, that are also present in ED but absent in +MF-U. The emergence of satellites in MF-MB is also ob- +served for the three-site Hubbard system with filling up +to half-filling (see SI). We expect satellites to be present +also for larger systems since they contain an increasing +number of accessible excited states. +VI. +CONCLUSION +In this work, we develop a method to compute the MF +approximation in a many-body basis instead of a single- +body basis. The MF approximation is a well-known and +broadly used approximation in the single-body basis such +that the interest of doing MF computations in a many- +body basis, at much more expensive computational cost, +might not be so straightforward. However, since ED tech- +niques require a many-body basis treatment, the devel- +oped method allows for a better comparison between MF +and ED. +We focus on the definition and the signification of the +DOS and show how the usual MF approximation DOS +can be found from MF-MB. We also highlight the pos- +sible ambiguity in a definition inspired by the ED DOS +definition, coming from the use or not of Koopman’s the- +orem (i.e., frozen single-particle states) in the prediction +of the Nel ± 1 ground states. Attention is paid to the +exclusion process, physically motivated by the lack of co- +herence in accounting for some of the different possible +ground states, in the MF-MB DOS. The exclusion pro- +cess is implemented manually in this work and an auto- +mated treatment of the process would be necessary for +large-scale investigations. +We also observe that our MF-MB method induces +satellites structure in the DOS. Satellites are usually seen +as a sign of correlation since they appear in ED or ap- +proximations beyond MF (GW, T-matrix, DMFT,. . . ). +We demonstrate in this work that satellites could appear +in purely MF computed DOS, when adopting the ED- +like definition. This result assumes the computation of +states in Nel + 1 (resp. Nel − 1) sector using the mean- +field based on the of the Nel + 1 (resp. Nel − 1) sector +instead of using only one mean-field from the Nel sector. +The approach presented here contributes to a better +understanding of the fundamental differences between +MF and ED Hamiltonians and the associated energies +and wave-functions. +Our formulation includes Green’s +function expression that is the bridge between MF and +many-body corrections. +We therefore expect that this +work could pave the way for developing other levels of ap- +proximation that introduce many-body correlations, such +as GW or T-matrix, in the many-body basis in order to +gain a deeper understanding of their mathematical and +physical underpinnings. As further perspective, we sug- +gest that revisiting and implementing known approxima- +tions from a refreshed point of view will deepen their un- +derstanding and lead to new approximations that would +be deduced either numerically or from physical intuition. +Specifically, we intend to explore the process of symme- +try restoration of Refs. 19 and 20 rewritten in MB basis. +Finally, we posit that providing an in-depth and pedagog- +ical analysis of a simple case of electronic correlation will +help readers appreciate the difficulties associated with +many-body electronic systems, especially since those are +a cornerstone of quantum information and quantum com- +puting developments. + +x200 +MF-U +MF-MB +x200 +ED +x200 +-6 +-4 +-2 +0 +2 +4 +6 +E [t units]12 +ACKNOWLEDGEMENTS +A.H. is a Research Fellow of the Fonds de la Recherche +Scientifique - FNRS. 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Ro- +maniello, “Reduced density-matrix functional theory: +Correlation and spectroscopy,” The Journal of Chemical +Physics, vol. 143, no. 2, p. 024108, 2015. + +Mean-field approximation of the Hubbard model expressed in a many-body basis: +Supplementary information. +Antoine Honet and Luc Henrard +Department of Physics and Namur Institute of Structured Materials, +University of Namur, Rue de Bruxelles 51, 5000 Namur, Belgium +Vincent Meunier +Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA +(Dated: January 10, 2023) +I. +MF-MB AND ED EIGEN-STATES FOR THE +THREE-SITE HUBBARD SYSTEM +We consider here the three-site Hubbard system such that +the single-particle localized basis states are given by |↓, ., .⟩, +|., ↓, .⟩, |., ., ↓⟩, |↑, ., .⟩, |., ↑, .⟩ and |., ., ↑⟩. We also define the +eigen-states (named d, e and f) for the one-electron sector at +U = 0 case (i.e. tight-binding): +|dσ⟩ = 1 +2 |σ, ., .⟩ + 1 +√ +2 |., σ, .⟩ + 1 +2 |., ., σ⟩ +|eσ⟩ = −1 +√ +2 |σ, ., .⟩ + 1 +√ +2 |., ., σ⟩ +|fσ⟩ = −1 +2 |σ, ., .⟩ + 1 +√ +2 |., σ, .⟩ + −1 +2 |., ., σ⟩ +(1) +where σ stand for the spin and is eiter ↑ or ↓. +The MF-MB and ED eigen-states for the three-site Hubbard +with one electron are given at tables I and II, respectively. The +MF-MB eigen-states are equivalant to MF-U eigen-states since +there is only one electron in the system. The main difference +between MF-MB and ED results is the lifting of spin degen- +eracy and the influence of the spin-polarized ground state on +state with opposite spin in MF-MB. All eigen-states remain +unchanged in ED when turning only the interaction (see ta- +ble II for U = 0.5t), since there is no interaction for this case +(no double occupied state). On the contrary, the eigen-states +of spin opposite to the ground state spin in MF-MB are af- +fected when increasing U due to there interaction with the +mean field based on the ground state. The eigen-state |eσ⟩ +remains however eigen-state for all U values because the mean +field is constant on all the localized sites where the state |eσ⟩ +is defined. +The eigen-states in the Nel = 2 sector in MF-MB and ED are +given at tables III and IV, respectively. The main difference +is the lifting of degeneracy of four times degenerated states in +MF-MB to three times degenerated states and non degenerated +state in ED. We also observe the mixing of a pure product state +of U = 0 eigen-states (|e ↑, e ↓⟩ at E ≃ 0.26t in MF-MB) with +other states in ED. +II. +DOS OF FOR TWO AND THREE ELECTRONS +ON THREE SITES +The DOS for the three-site Hubbard system filled with two +electrons with U = 0.5t is given at fig. 1. Since the Nel = 2±1 +sectors present polarized ground state, the exclusion mecha- +nism explained in the main text is used such that MF-MB +DOS is computed both with and without exclusion. We note +the inclusion of satellites in the addition part of the spectrum. +There are no satellites in the removal part since it correspond +FIG. 1. Density of states of the three-site Hubbard system with +two electrons with U = 0.5t for MF-U (blue), MF-MB (green) and +ED (red). +FIG. 2. Density of states of the three-site Hubbard system with +three electrons with U = 0.5t for MF-U (blue), MF-MB (green) +and ED (red). +to exploring the Nel = 1 sector in which the eigen-states are +single-particle eigen-states. +The DOS for the three-site Hubbard system filled with three +electrons with U = 0.5t is given at fig. 2. As in the case of +the three-site system with one electron exposed in the main +text, we observe in MF-MB the pairing of states that are non- +degenerated in MF-U. These states are split in ED and a part +of the weight is transfered to satellites both in MF-MB and +ED. In fig. 2, only the addition satellites are shown since there +is a removal/addition symmetry. +arXiv:2301.03223v1 [cond-mat.str-el] 9 Jan 2023 + +MF-U +MF-MB exclusion +MF-MB no exclusion +ED +x200 +x200 +x200 +x200 +x200 +x200 +x200 +x200 +6 +-4 +-2 +0 +2 +4 +6 +E[t units]x200 +MF-U +MF-MB +x200 +ED +x200 +-6 +-2 +-4 +0 +2 +4 +6 +E[t units]2 +Localised basis : +|↓, ., .⟩ |., ↓, .⟩ |., ., ↓⟩ +|↑, ., .⟩ +|., ↑, .⟩ +|., ., ↑⟩ +E = -1.41421 t +1/2 +1/ +√ +2 +1/2 +0 +0 +0 +E = -1.22809t +0 +0 +0 +0.51092 0.69132 0.51092 +E = 0 +−1/ +√ +2 +0 +1/ +√ +2 +0 +0 +0 +E = 0.125t +0 +0 +0 +−1/ +√ +2 +0 +1/ +√ +2 +E = 1.41421 t +−1/2 +1/ +√ +2 +-1/2 +0 +0 +0 +E = 1.60309 t +0 +0 +0 +-0.48884 0.72255 -0.48884 +U = 0 diagonal basis : +|d ↓⟩ +|e ↓⟩ +|f ↓⟩ +|d ↑⟩ +|e ↑⟩ +|f ↑⟩ +E = -1.41421 t +1 +0 +0 +0 +0 +0 +E = -1.22809t +0 +0 +0 +0.99976 +0 +0.02208 +E = 0 +0 +1 +0 +0 +0 +0 +E = 0.125t +0 +0 +0 +0 +1 +0 +E = 1.41421 t +0 +0 +1 +0 +0 +0 +E = 1.60309 t +0 +0 +0 +0.02208 +0 +-0.99976 +TABLE I. Table representing the eigen-states of the three-site Hubbard system with one electron in either the MF-U or the MF-MB +methods and with U = 0.5t. +Localised basis : +|↓, ., .⟩ |., ↓, .⟩ |., ., ↓⟩ |↑, ., .⟩ |., ↑, .⟩ |., ., ↑⟩ +E = -1.41421 t +1/2 +1/ +√ +2 +1/2 +0 +0 +0 +E = -1.41421 t +0 +0 +0 +1/2 +1/ +√ +2 +1/2 +E = 0 +−1/ +√ +2 +0 +1/ +√ +2 +0 +0 +0 +E = 0 +0 +0 +0 +−1/ +√ +2 +0 +1/ +√ +2 +E = 1.41421 t +−1/2 +1/ +√ +2 +-1/2 +0 +0 +0 +E = 1.41421 t +0 +0 +0 +−1/2 +1/ +√ +2 +-1/2 +U = 0 diagonal basis : +|d ↓⟩ +|e ↓⟩ +|f ↓⟩ +|d ↑⟩ +|e ↑⟩ +|f ↑⟩ +E = -1.41421 t +1 +0 +0 +0 +0 +0 +E = -1.41421 t +0 +0 +0 +1 +0 +0 +E = 0 +0 +1 +0 +0 +0 +0 +E = 0 +0 +0 +0 +0 +1 +0 +E = 1.41421 t +0 +0 +1 +0 +0 +0 +E = 1.41421 t +0 +0 +0 +0 +0 +1 +TABLE II. Table representing the eigen-states of the three-site Hubbard system with one electron in ED and U = 0.5t. + +3 +Loc. basis : +|., ↓, ↓⟩ +|↓, ., ↓⟩ +|↓, ↓, .⟩ +|., ., ↑↓⟩ +|., ↓, ↑⟩ +|↓, ., ↑⟩ +|., ↑, ↓⟩ +|., ↑↓, .⟩ +|↓, ↑, .⟩ +|↑, ., ↓⟩ +|↑, ↓, .⟩ +|↑↓, ., .⟩ +|., ↑, ↑⟩ +|↑, ., ↑⟩ +|↑, ↑, .⟩ +E = -2.46 t +0.25975 0.35328 0.25975 +0.35328 +0.4805 +0.35328 0.25975 0.35328 0.25975 +E = -1.10 t +0.50935 +0.3341 +-0.01804 0.35864 +-0.35864 0.01804 -0.33410 -0.50934 +E = -1.10 t 0.49015 0.72076 0.49015 +E = -1.10 t +0.49015 0.72076 0.49015 +E = -1.10 t +0.01804 0.35864 0.50934 +-0.3341 +0.3341 -0.50934 -0.35864 -0.01804 +E = 0.26 t +−1/2 +1/2 +1/2 +-1/2 +E = 0.37 t +1/ +√ +2 +−1/ +√ +2 +E = 0.37 t +1/ +√ +2 +−1/ +√ +2 +E = 0.37 t +0.35324 -0.02724 0.35324 +0.01175 -0.70648 -0.01175 0.35324 -0.02724 0.35324 +E = 0.37 t +-0.00547 -0.49964 -0.00547 0.50024 0.10943 0.50024 -0.00547 -0.49964 -0.00547 +E = 1.73 t -0.50966 0.69318 -0.50966 +E = 1.73 t +-0.50966 0.69318 -0.50966 +E = 1.73 t +0.48992 -0.37131 +0.0151 +-0.3491 +0.3491 +-0.0151 +0.3713 -0.48992 +E = 1.73 t +-0.0151 +-0.3491 +0.48992 +0.37131 +-0.37131 -0.48992 0.3491 +0.0151 +E = 3.20 t +-0.24025 0.35328 -0.24025 0.35328 +-0.5195 0.35328 -0.24025 0.35328 -0.24025 +U = 0 +diag. basis : |d ↓, e ↓⟩ |d ↓, f ↓⟩ |e ↓, f ↓⟩ |d ↑, d ↓⟩ |d ↑, f ↓⟩ |f ↑, d ↓⟩ |f ↑, f ↓⟩ |e ↑, e ↓⟩ |d ↑, e ↓⟩ |f ↑, e ↓⟩ |e ↑, d ↓⟩ |e ↑, f ↓⟩ |d ↑, e ↑⟩ |d ↑, f ↑⟩ |e ↑, f ↑⟩ +E = -2.46 t +0.99962 +0.0195 +0.0195 +0.0004 +E = -1.10 t +-0.01125 0.70604 0.03881 0.70702 +E = -1.10 t -0.99981 +0.0195 +E = -1.10 t +-0.99981 +0.0195 +E = -1.10 t +0.70702 0.03881 -0.70604 0.01125 +E = 0.26 t +1 +E = 0.37 t +1 +E = 0.37 t +1 +E = 0.37 t +-0.02757 0.71744 0.69553 +0.02757 +E = 0.37 t +0.00043 0.69608 -0.71796 0.00043 +E = 1.73 t +0.0195 +0.99981 +E = 1.73 t +0.0195 +0.99981 +E = 1.73 t +0.70621 +0.008 +0.70706 -0.03557 +E = 1.73 t +-0.03557 0.70706 +-0.008 +-0.70621 +E = 3.20 t +-0.00038 +0.0195 +0.0195 +-0.99962 +TABLE III. Table representing the eigen-states of the three-site Hubbard system with two electrons in MF-MB and U = 0.5t. +Loc. basis : +|., ↓, ↓⟩ +|↓, ., ↓⟩ +|↓, ↓, .⟩ +|., ., ↑↓⟩ +|., ↓, ↑⟩ +|↓, ., ↑⟩ +|., ↑, ↓⟩ +|., ↑↓, .⟩ +|↓, ↑, .⟩ +|↑, ., ↓⟩ +|↑, ↓, .⟩ +|↑↓, ., .⟩ +|., ↑, ↑⟩ +|↑, ., ↑⟩ +|↑, ↑, .⟩ +E = -2.66 t +0.23077 0.36404 0.27422 +0.36404 0.46153 0.36404 0.27422 0.36404 0.23077 +E = -1.41 t +0.35355 +0.5 +-0.35355 +0.35355 +-0.5 +-0.35355 +E = -1.41 t +1/2 +1/ +√ +2 +1/2 +E = -1.41 t +1/2 +1/ +√ +2 +1/2 +E = -1.19 t +0.45440 0.38309 +0.38309 +-0.38309 +-0.38309 -0.45440 +E = 0 +1/ +√ +2 +−1/ +√ +2 +E = 0 +1/ +√ +2 +−1/ +√ +2 +E = 0 +1/2 +-1/2 +-1/2 +1/2 +E = 0.12 t +-0.20235 -0.03801 0.61178 -0.03801 -0.40470 -0.03801 0.61178 -0.03801 -0.20235 +E = 0.5 t +0.57735 +-0.57735 +0.57735 +E = 1.41 t +-1/2 +1/ +√ +2 +-1/2 +E = 1.41 t +-1/2 +1/ +√ +2 +-1/2 +E = 1.41 t +0.35355 +-0.50 +-0.35355 +0.35355 +0.50 +-0.35355 +E = 1.69 t +0.54177 -0.32131 +-0.32131 +0.32131 +0.32131 -0.54177 +E = 3.03 t +-0.26920 0.34064 -0.22479 -0.34064 -0.53840 0.34064 -0.22479 0.34064 -0.26920 +U = 0 +diag. basis : |d ↓, e ↓⟩ |d ↓, f ↓⟩ |e ↓, f ↓⟩ |d ↑, d ↓⟩ |d ↑, f ↓⟩ |f ↑, d ↓⟩ |f ↑, f ↓⟩ |e ↑, e ↓⟩ |d ↑, e ↓⟩ |f ↑, e ↓⟩ |e ↑, d ↓⟩ |e ↑, f ↓⟩ |d ↑, e ↑⟩ |d ↑, f ↑⟩ |e ↑, f ↑⟩ +E = -2.66 t +0.99808 0.02173 0.02173 -0.03156 0.02173 +E = -1.41 t +1/ +√ +2 +−1/ +√ +2 +E = -1.41 t +-1 +E = -1.41 t +-1 +E = -1.19 t +-0.70440 -0.06178 -0.70440 -0.06178 +E = 0 +1 +E = 0 +1 +E = 0 +−1/ +√ +2 +1/ +√ +2 +E = 0.12 t +-0.05140 0.35330 0.35330 +0.05612 -0.81413 +E = 0.50 t +0.57735 0.57735 +0.57735 +E = 1.41 t +1 +E = 1.41 t +1 +E = 1.41 t +−1/ +√ +2 +1/ +√ +2 +E = 1.69 t +-0.06178 -0.70440 -0.06178 -0.70440 +E = 3.03 t +-0.03446 0.02221 0.02221 -0.99792 -0.04441 +TABLE IV. Table representing the eigen-states of the three-site Hubbard system with two electrons in ED and U = 0.5t. + +4 +Loc. basis : +|↓, ↓, ↓⟩ +|., ↓, ↑↓⟩ +|↓, ., ↑↓⟩ +|., ., ↑↓⟩ +|↓, ↓, ↑⟩ +|., ↑↓, ↓⟩ +|↓, ↑, ↓⟩ +|↓, ↑↓, .⟩ +|↑, ↓, ↓⟩ +|↑↓, ., ↓⟩ +|↑, ↓, .⟩ +|↑↓, ↓, .⟩ +|., ↑, ↑↓⟩ +|., ↑↓, ↑⟩ +|↓, ↑, ↑⟩ +|↑, ., ↑↓⟩ +|↑, ↓, ↑⟩ +|↑↓, ., ↑⟩ +|↑, ↑, ↓⟩ +|↑, ↑↓, .⟩ +|↑↓, ↑, .⟩ |↑, ↑, ↑⟩ +E = -2.21 t +0.23728 +0.3531 +0.23728 +0.3531 +0.52544 +0.3531 +0.23728 +0.3531 +0.23728 +E = -1.95 t +0.26272 +0.3531 +0.26272 +0.3531 +0.47456 +0.3531 +0.26272 +0.3531 +0.26272 +E = -0.73 t +-0.02544 +0.34301 +0.48645 +-0.38087 +0.38087 +-0.48645 +-0.34301 +0.02544 +E = -0.73 t +-0.48645 +-0.38087 +-0.02544 +-0.34301 +0.34301 +0.02544 +0.38087 +0.48645 +E = -0.61 t +0.51242 +0.33637 +-0.01188 +0.35234 +- 0.35234 +0.01188 +-0.33637 +-0.51242 +E = -0.61 t +0.01188 +0.35234 +0.51242 +-0.33637 +0.33637 +-0.51242 +-0.35234 +-0.01188 +E = 0.5 t +1 +E = 0.62 t +-0.00666 +0.49943 +-0.00666 +0.50039 +0.01332 +-0.50039 +-0.00666 +0.49943 +-0.00666 +E = 0.62 t +-0.35303 +-0.03486 +-0.35303 +-0.016 +0.70607 +-0.016 +-0.03486 +-0.35303 +E = 0.74 t +0.5 +-0.5 +-0.5 +0.5 +E = 0.76 t +0.5 +-0.5 +-0.5 +0.5 +E = 0.88 t +0.35303 +-0.03537 +0.35303 +-0.0155 +-0.70605 +-0.0155 +0.35303 +-0.03537 +0.35303 +E = 0.88 t +-0.00702 +-0.4994 +-0.00702 +0.50041 +0.01403 +0.50041 +-0.00702 +-0.4994 +-0.00702 +E = t +1 +E = 2.11 t +0.02053 +-0.35796 +0.51215 +0.33037 +-0.33037 +-0.51215 +0.35796 +-0.02053 +E = 2.11 t +0.51215 +-0.33037 +-0.02053 +-0.35796 +0.35796 +0.02053 +0.33037 +-0.51215 +E = 2.23 t +0.4869 +-0.37302 +0.01444 +-0.35153 +0.35153 +-0.01444 +0.37302 +-0.4869 +E = 2.23 t +-0.01444 +-0.35153 +0.4869 +0.37302 +-0.37302 +-0.4869 +0.35153 +0.01444 +E = 3.45 t +-0.26272 +0.35310 +-0.26272 +0.35310 +-0.47456 +0.35310 +-0.26272 +0.35310 +-0.26272 +E = 3.71 t +-0.23728 +0.35310 +-0.23728 +0.35310 +-0.52544 +0.35310 +-0.23728 +0.35310 +-0.23728 +U = 0 +diag. basis : |d ↓, e ↓, f ↓⟩ |d ↑, d ↓, f ↓⟩ |d ↑, d ↓, e ↓⟩ |d ↑, e ↓, f ↓⟩ |f ↑, d ↓, f ↓⟩ |f ↑, d ↓, e ↓⟩ |f ↑, e ↓, f ↓⟩ |e ↑, d ↓, f ↓⟩ |e ↑, d ↓, e ↓⟩ |e ↑, e ↓, f ↓⟩ |d ↑, f ↑, d ↓⟩ |d ↑, f ↑, f ↓⟩ |d ↑, f ↑, e ↓⟩ |d ↑, e ↑, d ↓⟩ |d ↑, e ↑, f ↓⟩ |d ↑, e ↑, e ↓⟩ |e ↑, f ↑, d ↓⟩ |e ↑, f ↑, f ↓⟩ |e ↑, f ↑, e ↓⟩ |d ↑, f ↑, d ↑⟩ +E = -2.21 t +0.99935 +-0.02544 +-0.02544 +0.00065 +E = -1.95 t +0.99935 +0.02544 +0.02544 +0.00065 +E = -0.73 t +-0.74283 +0.01891 +0.66899 +-0.01703 +E = -0.73 t +-0.66899 +0.01703 +-0.74283 +0.01891 +E = -0.61 t +0.6903 +0.01757 +0.72308 +0.01841 +E = -0.61 t +0.72308 +0.01841 +-0.6903 +-0.01757 +E = 0.5 t +1 +E = 0.62 t +0.00068 +0.7203 +-0.69366 +-0.00068 +E = 0.62 t +0.03597 +0.69273 +0.7194 +-0.03597 +E = 0.74 t +-1 +E = 0.76 t +-1 +E = 0.88 t +0.03597 +-0.7201 +-0.692 +-0.03597 +E = 0.88 t +-0.00071 +-0.69293 +0.721 +0.00071 +E = t +1 +E = 2.11 t +-0.01726 +-0.678 +0.0187 +0.73462 +E = 2.11 t +0.0187 +0.73462 +0.01726 +0.678 +E = 2.23 t +-0.01852 +0.72753 +-0.01745 +0.68561 +E = 2.23 t +-0.01745 +0.68561 +0.01852 +-0.72753 +E = 3.45 t +0.00065 +0.02544 +0.02544 +0.99935 +E = 3.71 t +0.00065 +-0.02544 +-0.02544 +0.99935 +TABLE V. Table representing the eigen-states of the three-site Hubbard system with three electrons in MF-MB and U = 0.5t. + diff --git a/i9E1T4oBgHgl3EQffwT8/content/tmp_files/load_file.txt b/i9E1T4oBgHgl3EQffwT8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2b94789ef541394c2ea316453913ffa006e8003 --- /dev/null +++ b/i9E1T4oBgHgl3EQffwT8/content/tmp_files/load_file.txt @@ -0,0 +1,1518 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf,len=1517 +page_content='Mean-field approximation of the Hubbard model expressed in a many-body basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Antoine Honet and Luc Henrard Department of Physics and Namur Institute of Structured Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' University of Namur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Rue de Bruxelles 51,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 5000 Namur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Belgium Vincent Meunier Department of Engineering Science and Mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' USA (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2023) The effective independent-particle (mean-field) approximation of the Hubbard Hamiltonian is described in a many-body basis to develop a formal comparison with the exact diagonalization of the full Hubbard model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' using small atomic chain as test systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This allows for the development of an intuitive understanding of the shortcomings of the mean-field approximation and of how critical correlation effects are missed in this popular approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The description in the many-body basis highlights a potential ambiguity related to the definition of the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Specifically, satellite peaks are shown to emerge in the mean-field approximation, in departure from the common belief that they characterize correlation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The scheme emphasizes the importance of correlation and how different many-body corrections can improve the mean-field description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The pedagogical treatment is expected to make it possible for researchers to acquire an improved understanding of many-body effects as found in various areas related to electronic properties of molecules and solids, which is highly relevant to current efforts in quantum information and quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Keywords: Hubbard model, mean-field approximation, exact diagonalization, many-electron basis I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' INTRODUCTION The Hubbard model [1] is a popular and simple model to describe electron correlation in solids, molecules, and nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The exact description of correlation is a tremendous task in the field of electronics, and it is of paramount importance for the accurate description of magnetism, optical properties, electron transport, and plasmonics [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In spite of its apparent simplicity, finding exact so- lutions of the Hubbard model is a formidable effort in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Starting from the Hubbard Hamiltonian, the simplest conceptual way to solve it is the exact diag- onalization (ED) method [6–8], that can however only be performed analytically or numerically for very small systems [6–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This method hinges on the diagonaliza- tion of the Hubbard Hamiltonian expressed in a many- electron basis and yields the eigen-energies and eigen- vectors of the Hamiltonian, expressed as linear combina- tions of the (many-body) basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The major issue with this method is that the number of basis states grows exponentially with the number of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For example, at half-filling, the many-body basis for the single-orbital Hubbard model of a two-site system has dimension 6, di- mension 20 for three-site system, and already dimension 924 for a six-site system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The dimension of the Hilbert space for a ten-site system reaches 184, 756, leading to a Hamiltonian matrix with more that 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='4 × 1010 ele- ments [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This illustrates how the numerical resolution of the method is rapidly limited, even for modest size systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For this reason, researchers have realized the need for approximation methods to render the compu- tational treatment of the Hubbard model at a tractable computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The mean-field approximation (MF) is one of the sim- plest approximations for the Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' It consists in replacing the two-body interaction term of the Hub- bard model as an interaction between one electron and a mean-field due to the other electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' As a result, a given electron no longer interacts directly with other electrons but rather with a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This makes it possible to write the Hamiltonian in a single-electron basis and to consider electrons as particles that are effectively free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The main advantage of MF is that the corresponding dimension of the Hilbert space is reduced to 2Nel, where Nel is the number of electrons and the factor 2 accounts for the spin degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The MF approximation is often used to describe electronic systems [2, 3, 15] and we will refer to the formulation of the MF approximation in the single-electron basis as MF-U for mean-field usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' One major drawback of the MF-U approximation is that it misses all the correlation between electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Sev- eral techniques have been developed to move beyond MF-U to include (at least a part of the) correlation, such as the Green’s function many-body approxima- tion (GFMBA), which consists of a sum of Feynman diagrams [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This family of approximations in- cludes the second-order Born approximation, the GW approximation, and the T-matrix approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' There exists several other approaches such as the dynamical mean-field theory and the quantum Monte Carlo ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' [15, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Furthermore, other methods are based on the MF-U computation augmented by a symmetry restoration procedure [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' What’s more, machine learning based self-energy construction have also been investigated more recently [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Up to now, the discrepancies between MF-U and ED wave functions have not been fully described or under- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03223v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='str-el] 9 Jan 2023 2 stood in a general framework despite their importance for the development of intuitive and accurate corrections to the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We believe that this lack of a deeper understanding is partly due to the fact that ED methods must be expressed in the many-electron basis whereas the MF-U is, by design, usually implemented in the single-electron basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' It is therefore challenging to compare the two methods since this change of basis is not a simple unitary transformation but a complete change of paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The objective of this paper is to propose an in-depth comparison between MF and ED by formulating the MF approximation in the many-body basis framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This approximation will be referred to as the MF-MB in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The goal of this formulation is not to reduce the computational cost of the method but to build a MF approximation in the same basis as the ED, and thus to gain insight into the missing part of the MF ap- proximation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', correlation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We note, however, that in contrast to common density functional theory compu- tations, the Hubbard MF approximation does not suf- fer from an exchange problem since the ground state is approximated by a Slater determinant that satisfies the wave function symmetry required for the eigen-states of the exchange operator, which commutes with the Hamil- tonian of the electronic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Our analysis highlights the ambiguous definition of the density of states within the MF approximation depending on the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The MF-U method is often combined im- plicitly with Koopman’s theorem [24, 25], which assumes that all the states of the N-electron system are frozen when adding/removing an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This assumption is not strictly correct since even in the MF approximation, adding or removing an electron changes the mean-field and consequently the predicted states (both occupied and unoccupied) as we will illustrate by comparing the DOS obtained via MF-U and MF-MB techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This com- parison highlights the fact that satellite states also ap- pear in the MF-MB approximation, although they are often thought as being the result of the inclusion of cor- relation [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The rest of the paper is organized as follow: we first in- troduce the Hubbard model, the MF-U approximation, and the ED technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We then explain the MF-MB technique, based both on MF-U equations and the nu- merical methods employed for the ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Finally, we discuss results of MF-MB compared with MF-U and ED and the notion of density of states and the appearance of satellite peaks in the MF-MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' REVIEW OF STANDARD METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Hubbard model This studies focused on the single-orbital Fermi- Hubbard model: ˆHHubbard = −t � ,σ ˆc† iσˆcjσ + U � i ˆni↑ˆni↓ (1) where t is the hopping parameter, U is the interaction (or Hubbard) parameter, ˆc† iσ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' ˆciσ) is a creation (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' destruction) operator of an electron on atomic site i with spin σ, and ˆniσ = ˆc† iσˆciσ is the density op- erator (on atomic site i and with spin σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The atomic site indices run from 0 to N − 1 where N is the total number of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The ⟨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' ⟩ symbol under the summa- tion operator indicates that the sum runs over all pairs of nearest-neighbour atomic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The first term of the Hubbard Hamiltonian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 1 is the tight-binding Hamiltonian and is easily written us- ing the one-electron basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In contrast, the second term (known as the Hubbard or interaction term) is the prod- uct of two density operators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', a combination of four creation and/or destruction operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This two-body operator cannot be written in the one-electron basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Mean-field approximation and single-electron basis In the MF approximation, the Hubbard Hamiltonian of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (1) is approximated so that it only includes one- body operators and the Hamiltonian can be written in the one-electron basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In practice, the density opera- tors are decomposed as sums of the mean value of the operator (niσ) and the deviation (ˆniσ − niσ) from this mean value: ˆniσ = niσ +(ˆniσ −niσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Products of density operators in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (1) are expanded and the approxima- tion consists in dropping products of deviations from the mean, generally assumed (without proof) to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Products of mean values only induce a constant shift in the Hamiltonian (and thus in the total energy), leading to the Hamiltonian: ˆHHubb,MF = − t � ,σ ˆc† iσˆcjσ + U � i (ni↑ˆni↓ + ni↓ˆni↑) − U � i ni↑ni↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2) In the MF-U method, the Hamiltonian of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2) is written in the form of a matrix expressed in the one- electron basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The basis states of the one-electron basis are obtained by the application of the different creation operators on the vacuum state |∅⟩ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', the state contain- ing no electron), for each site and spin value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' It follows that |iσ⟩ = ˆc† iσ |∅⟩ (3) 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Illustration of the structure of the Hubbard Hamil- tonian in the MF-U approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Hamiltonian matrix of size 2N ×2N is written in the single-electron basis and can be divided into 4 blocks of size N × N: one containing pure spin-up related terms, one containing pure spin-down terms, and two mixing blocks between up and down spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' are the 2N basis states of the single-electron basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In that basis, the matrix has dimension 2N × 2N (N being the number of sites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Hamiltonian matrix is com- posed of 4 blocks of dimension N × N: the two blocks on the diagonal are pure spin-up and spin-down blocks and the two off-diagonal blocks mix spin up and spin down that, according to the MF Hamiltonian, remain equal to zero (see figure 1 for an illustration of the block matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The tight-binding term implies that the Hamil- tonian matrix has terms of amplitude −t for elements corresponding to neighbouring sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The second term of the Hamiltonian is concerned with diagonal elements in the single-electron basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The diagonal elements in the spin-up (-down) block involve mean densities of down (up) spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' One usually does not implement the last term (which only includes mean values, not operators) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We need, however, to reintroduce that term in the computation of the total energy, as it corresponds to a rigid shift of all energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For example, the mean-field Hubbard Hamiltonian (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2)) is expressed in the single-electron basis for the two-site system as: ˆHHubb,MF-U = � � � Un0↓ −t 0 0 −t Un1↓ 0 0 0 0 Un0↑ −t 0 0 −t Un1↑ � � �, (4) if the basis states are ordered in spin blocks in the same way as atomic sites 0 and 1: {|0 ↑⟩ , |1 ↑⟩ , |0 ↓⟩ , |1 ↓⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Hamiltonian of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2) involves mean values of density operators and it is therefore necessary to solve it self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Initial conditions for the mean densities are guessed and a Hamiltonian based on these mean val- ues is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In the next step, the first Hamiltonian is di- agonalized, resulting in eigen-energies and eigen-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' New mean values are computed by populating the eigen- vectors of lowest energies and a new Hamiltonian is gen- erated, diagonalized, leading to new mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This loop is repeated until the difference between the input mean values and the output values is smaller than a given threshold (convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The density of states (DOS) in the MF-U is con- structed as a sum of Dirac delta-peaks centered at the converged eigen-energies of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' A single- particle states is identified at each eigen-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In addition, using the single electronic eigen-energies Ek and the coefficients ak iσ of the eigen-states of the MF- U Hamiltonian (k ranging from 0 to 2N), we define cre- ation operators of each eigen-state as: ˆd† k = � i,σ (ak iσ)∗ˆc† iσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (5) For a system with Nel electrons, the wave-function of the MF-U ground state is expressed as a single Slater determinant, created by applying successively the Nel creation operators associated with the Nel lowest eigen- energies to the vacuum state: |GS, MF-U⟩ = � k≤Nel ˆd† k |∅⟩ (6) and the mean values of density operators to be inserted in the self-consistent Hamiltonian are given by: niσ = � k≤Nel ��ak iσ ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (7) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Exact diagonalization Instead of solving for the approximated MF Hamil- tonian described in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2), the ED considers the exact Hubbard Hamiltonian of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The second term of the Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', the product of two density operators, cannot be expressed in the single-electron basis of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (3) and a many-electron basis needs to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' As a com- mon practice in the ED literature, the concept of sector is introduced for a fixed number of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' When re- stricted to the Nel sector, the many-electron basis states are given by all possible combinations of Nel creation op- erators applied to the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For example � k≤Nel ˆc† k↑ |∅⟩ (8) represents one of the basis states, involving only spin-up electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' As fermionic creation operators with at least one dif- ferent index (site or spin) anti-commute, the order in the product only affects the sign of the state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', the fermionic sign) and we find the same basis state, mod- ulo an overall phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Here, we conventionally choose to define the basis states with a positive sign by ordering all spin-up (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' -down) creation operators to the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' operators with the same spin part are or- dered from left to right in an increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This re- sults in the combination formula in combinatorics for the number of basis states (Nb): Nb = C2N Nel = (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2N − Nel)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Nel!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='. (9) Spin up Spin up X spin N down 2N HMF-U = Spin Spin down x N down spin up 2N N N4 We adopt the approach described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 6–8, and 27 to label each state with one integer I, bijectively linked with two other integers I↑ and I↓ by the relations: I = 2NI↑ + I↓ (10) and I↑ = I//2N I↓ = I mod 2N, (11) where // represents the integer division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Writing I↑ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' I↓) in binary notation yields the space configuration of the state in the spin-up (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' - down) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Organizing the Hilbert space in this way allows one to only have to deal with integers and to easily find the effect of the creation, destruction, and density operators on each state using simple standard binary op- erations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', bin flip, bin counting,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' One can now express the action of the full Hubbard Hamiltonian of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (1) on the basis states and, in turn, calculate the Hamiltonian matrix elements in the many-electron basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Formally, the Hamiltonian matrix contains elements with value −t when they correspond to connected ba- sis states having all the same electron creation operators (sites and spin) but one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The different electrons have to be of the same spin on a neighboring atomic site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Hamiltonian matrix also contains elements of amplitude U on the diagonal for basis states containing two elec- trons on the same site and of opposite spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The U val- ues are added if there are several doubly-occupied sites in the basis state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We illustrate the construction of the Hamiltonian for a two-site system at half-filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The 6 basis states are |Φ1⟩ = ˆc† 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↑ˆc† 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↑ |∅⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' |Φ2⟩ = ˆc† 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↑ˆc† 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↓ |∅⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' |Φ3⟩ = ˆc† 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↑ˆc† 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↓ |∅⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' |Φ4⟩ = ˆc† 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↑ˆc† 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↓ |∅⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' |Φ5⟩ = ˆc† 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↑ˆc† 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↓ |∅⟩ and |Φ6⟩ = ˆc† 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↓ˆc† 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='↓ |∅⟩ and the Hamiltonian matrix in that basis is given by: HHubb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='ED = � � � � � � � 0 0 0 0 0 0 0 0 0 −t −t 0 0 0 0 −t −t 0 0 −t −t U 0 0 0 −t −t 0 U 0 0 0 0 0 0 0 � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (12) where the element −t connects basis states |Φ2⟩ and |Φ3⟩ to basis states |Φ4⟩ and |Φ5⟩ since they differ only by one electron having the same spin and hopping from site 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' U is on the diagonal for basis states |Φ4⟩ and |Φ5⟩ both having two electrons, located on site 0 and site 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The lowest eigen-energy of the Hamiltonian corre- sponds to the ground-state and the higher ones to ex- cited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The corresponding eigen-vectors are by con- struction many-electron states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In contrast to the single- electron basis formulation of the MF-U, it is not con- structed by populating several eigen-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Likewise, the associated eigen-energy is the total energy of the sys- tem, without having to sum individual electron energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' To gain access to dynamical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' frequency depen- dent) properties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' we now introduce the Green’s function with general definition [16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 29]: Giσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='jσ′(ω) = � ΨNel 0 ��� ˆciσ 1 ω + (ENel 0 − ˆHNel+1 + iη) ˆc† jσ′ ���ΨNel 0 � + � ΨNel 0 ��� ˆc† iσ 1 ω − (ENel 0 − ˆHNel−1 + iη) ˆcjσ′ ���ΨNel 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (13) where ���ΨNel 0 � and ENel 0 are the ground state and the ground-state energy of the system with Nel electrons and ˆHNel±1 are the Hamiltonian operators of the system con- taining Nel ± 1 electrons and η is a small real positive parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The first term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (13) is the electron addition part of the Green’s function: ˆc† jσ′ ���ΨNel 0 � and � ΨNel 0 ��� ˆciσ rep- resent both states with Nel+1 electrons and the operator involves the Nel+1 electron Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This term thus explores the possible states when an electron is added to the Nel ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The second term of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (13) de- scribes the situation where one electron is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This is the electron removal part of the Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Green’s function has poles at the frequencies correspond- ing to difference of energies between the Nel ground state and states in the Nel ± 1 sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' To evaluate the Green’s function (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (13)), three exact diagonalizations are completed: one in the Nel sector to find the ground-state ���ΨNel 0 � and its energy ENel 0 , and two for the two sectors Nel ± 1 so that matrix elements of the type � ΦNel±1 k ��� ˆHNel±1 ���ΦNel±1 k′ � can be computed with ���ΦNel±1 k′ � the basis vectors of the Nel±1 sector of the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Green’s function is then computed by writing the states ˆc† jσ′ ���ΨNel 0 � in the basis ���ΦNel+1 k � and the states ˆcjσ′ ���ΨNel 0 � in the basis ���ΦNel−1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The DOS (D(ω)) is defined from the Green’s function as: D(ω) = − 1 π Tr � Im(GR(ω)) � , (14) where Tr is the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We can now identify addition and removal parts of the density of states: the addition part is computed based on the addition part of the Green’s function and, in an anal- ogous way, the removal part of the DOS is constructed from the removal part of the Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The DOS features peaks at energies corresponding to the poles of the Green’s function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', the differences between the Nel ground-state energy and all states (ground state and ex- cited states) of Nel ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We can imagine this process as probing all the possible adding or removing of energy 5 when adding or removing one electron, taking into ac- count the correlation exactly in both the starting state (ground state of Nel electron) and the final states (Nel±1 electron states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We point out that to account for the cor- relation, the Nel ± 1 states in the Nel ± 1 sectors cannot be obtained simply by adding one electron independently from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' MEAN-FIELD APPROXIMATION IN THE MANY-BODY BASIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Formulation in the Nel sector As mentioned before, the MF approximation effec- tively decouples the density operators interaction into the interaction of one density operator with a mean-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This results in the possible formulation within the single- electron basis of the MF approximation (see section II B), reducing the basis dimension from C2N Nel to 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We now examine how the MF approximation can be expressed in the many-body basis (MF-MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We consider the MF-approximated Hamiltonian of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2) and the many-electron basis described in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' As the MF approximation does not affect the tight- binding term of the Hamiltonian, the −t elements of the MF-MB Hamiltonian matrix are the same as for the ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In contrast, the interaction term involves mean val- ues of density operators and the diagonal in the MF-MB Hamiltonian includes Uniσ factors for each basis states involving a creation operator c† iσ′, σ being one spin (up or down) and σ′ being the opposite spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In general, there will be a sum of several Uniσ terms because basis states contain several electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For example, if there is a doubly-occupied site i in a basis state, terms of the form U(niσ′ + niσ) are present on the diagonal of the Hamiltonian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We illustrate the construction of the Hamiltonian for the two-site system at half-filling with the basis states ex- plicitly written in section II C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Hamiltonian matrix in this case is given by: HHubb,MF-MB = � � � � � � � U(n0↓ + n1↓) 0 0 0 0 0 0 U(n0↓ + n1↑) 0 −t −t 0 0 0 U(n0↑ + n1↓) −t −t 0 0 −t −t U(n0↓ + n0↑) 0 0 0 −t −t 0 U(n1↓ + n1↑) 0 0 0 0 0 0 U(n0↑ + n1↑) � � � � � � � , (15) where we removed the last term of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2) as in the MF-U case since it is only a constant shift of the Hamiltonian, leaving the eigen-vectors unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Similar to the MF-U case, the MF-MB Hamiltonian depends on the mean value of density operators that we compute self-consistently from the ground state of the preceding iteration, starting from an initial guess for the mean densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' As the ground state is given as a linear combination of the basis states given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (8), the mean densities of electrons at site i with spin σ are computed by summing the square of the linear coefficients when the basis state contains an electron at site i with spin σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' A new Hamiltonian is also computed iteratively, and then diagonalized to obtain to new mean densities, until self-consistency is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Numerically, the many-electron basis is encoded ex- actly in the same way as in ED, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' using the integers I, I↑, and I↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' As in the ED case, the effect of creation, de- struction, and density operators are implemented using simple binary operations on the binary representation of the three integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Correspondence with MF-U We are now in a position to examine the links between MF-U and MF-MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In this section, we explain and show with examples how the two methods are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The MF-U method yields individual states (eigen-vectors of the Hamiltonian) for effectively independent particles, at given energies (eigen-energies of the Hamiltonian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The ground state of the system containing Nel is then formed by populating the Nel eigen-states with the lowest ener- gies according to Pauli’s principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The total energy of the ground state is the sum of the individual energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The counterpart (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', the ground state) in the MF- MB method is the lowest energy state which is a linear combination of the many-body basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The total energy of the ground state (containing Nel electrons) is the eigen-energy of the state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' the lowest eigen-energy of the MF-MB Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We illustrate this with the two-site system at half- filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We first introduce the change of basis from a localised basis to a bonding/anti-bonding basis in or- der to better understand the relation between MF-U and MF-MB results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In the single-electron picture, we define bonding (b) and anti-bonding (a) states for a given spin 6 σ as: |bσ⟩ = 1 √ 2(|0σ⟩ + |1σ⟩) = 1 √ 2(ˆc† 0σ + ˆc† 1σ) |∅⟩ and |aσ⟩ = 1 √ 2(|0σ⟩ − |1σ⟩) = 1 √ 2(ˆc† 0σ − ˆc† 1σ) |∅⟩ , (16) such that it is intuitive to define bonding and anti- bonding creation operators as: ˆc† bσ = 1 √ 2(ˆc† 0σ + ˆc† 1σ) and ˆc† aσ = 1 √ 2(ˆc† 0σ − ˆc† 1σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (17) We also define the many-body states in the bonding/anti-bonding basis by the successive applica- tions of bonding and anti-bonding operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' As for the localised basis, it is important to choose a convention (fermionic sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We chose to define a positive sign state in our convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We consider the states presenting the ordering of all spin-up (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' -down) creation operators on the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Within the same spin part, a positive sign state orders the bonding and anti-bonding operators from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For example, the many-body state containing two bonding states (of opposite spins) is expressed as |b ↑, b ↓⟩ = ˆc† b↑ˆc† b↓ |∅⟩ = 1 2(ˆc† 0↑ + ˆc† 1↑)(ˆc† 0↓ + ˆc† 1↓) |∅⟩ = 1 2(|Φ4⟩ + |Φ2⟩ + |Φ3⟩ + |Φ5⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (18) The eigen-states and eigen-energies for the MF ap- proximation are listed in table I for the MF-U method and in table II for the MF-MB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The MF-U method yields four eigen-states at energies E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t and E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Each energy corresponds to doubly- degenerated states, due to spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The single-particle eigen-states are the bonding and anti-bonding states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The MF-MB method gives six eigen-states with energies E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (non-degenerated), E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (four times de- generated), and E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (non degenerated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The sec- ond part of table II indicates that the states formed by successive applications of bonding and anti-bonding operators are the eigen-states of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The ground state of the MF-MB (E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t) is |b ↑, b ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This state corresponds to the state formed by occupying the two lowest energy single-particle states of the MF-U method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Those states are |b ↑⟩ and |b ↓⟩, both with en- ergies E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The energy of the ground state in the MF-MB method is then the sum of the individual energies of the two lowest energy states in MF-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This illustrates that, for a given number of particles, the MF- U and MF-MB methods predict the same ground state with the same energy, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The excited many- electron states computed in the MF-MB method (with energies E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t and E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t) are the two-electron Localised basis : |0 ↑⟩ |1 ↑⟩ |0 ↓⟩ |1 ↓⟩ E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t 1/ √ 2 1/ √ 2 0 0 E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t 0 0 1/ √ 2 1/ √ 2 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t 1/ √ 2 −1/ √ 2 0 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t 0 0 1/ √ 2 −1/ √ 2 Bonding/anti-bonding basis : |b ↑⟩ |b ↓⟩ |a ↑⟩ |a ↓⟩ E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t 1 0 0 0 E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t 0 1 0 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t 0 0 1 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t 0 0 0 1 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' List of MF-U eigen-states of the two-site Hub- bard system at half-filling with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' There are four eigen-states, two pairs of degenerate states (spin degeneracy): one at E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t and the other at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The first part of the table shows the eigen-states’ coefficients in the single-electron basis constructed with localised basis states (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The second part of the table gives the eigen- states’ coefficients in the diagonal basis consisting of bonding and anti-bonding states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' states formed by populating every other combinations of two single-particle states of MF-U (not the two lowest energy states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The MF-MB thus predicts directly all the many- electron states that can be obtained from populating single-particle eigen-states from the MF-U method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In the Nel sector, a correspondence is therefore established and the two methods lead to consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' DENSITY OF STATES We now turn our attention to the notion of density of states in the MF-U and in the MF-MB approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Specifically, we highlight two possible interpretations of the DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This leads to different features in the DOS that will be discussed, such as the appearance of satellite peaks in the DOS that are usually attributed to beyond- MF methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In the MF-U method, the DOS consists of Dirac delta- peaks at the energies of the single-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Fermi level is positioned between the highest-occupied and the lowest-unoccupied levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' From the MF-MB re- sults, we can retrieve this DOS by adopting a picture based on differences between MF-MB eigen-energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In- deed, starting from the ground state with E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t and identifying the four states at E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (see table II) as states having exactly one electron in a bonding state and one electron in an anti-bonding state, we obtain the following system for single-particle state energies 7 Localised basis : |Φ1⟩ = |↑, ↑⟩ |Φ2⟩ = |↑, ↓⟩ |Φ3⟩ = |↓, ↑⟩ |Φ4⟩ = |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |Φ5⟩ = |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ |Φ6⟩ = |↓, ↓⟩ E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 1/2 1/2 1/2 1/2 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 −1/ √ 2 1/ √ 2 0 0 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 0 0 −1/ √ 2 1/ √ 2 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 1 0 0 0 0 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 0 0 0 0 1 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 1/2 1/2 1/2 1/2 0 Bonding/anti-bonding basis : |b ↑, a ↑⟩ |b ↑, b ↓⟩ |b ↑, a ↓⟩ |a ↑, b ↓⟩ |a ↑, a ↓⟩ |b ↓, a ↓⟩ E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 1 0 0 0 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 0 1 0 0 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 0 0 1 0 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 1 0 0 0 0 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 0 0 0 0 1 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 0 0 0 1 0 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Table of the MF-MB eigen-vectors of the two-site Hubbard system for U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' There are six eigen-states with energies E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t, E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (four times degenerated), and E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The first part of the table shows the linear coefficients of the states in the many-body basis presented in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' II C) (localised basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The second part of the table gives the eigen-states’ coefficients in the bonding/anti-bonding basis, the basis constructed from the tensor product of the diagonal basis for the single-electron picture (MF-U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' � � � � � � � � � � � � � � � � � Eb↑ + Eb↓ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (I) Eb↑ + Ea↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (II) Ea↑ + Eb↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (III) Eb↑ + Ea↑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (IV) Eb↓ + Ea↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (V) Ea↑ + Ea↓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t (VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Using these six equations, we can find the single- particle state energies: Eb↑ = Eb↓ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t Ea↑ = Ea↓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (19) This reasoning is somewhat inconvenient because it cannot be easily generalized to large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' However, it presents the advantage of explaining how the MF-U DOS (see table I) can be recovered from the MF-MB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' However, we emphasize that this interpretation of the single-particle DOS does not match with the defi- nition of the DOS in a many-electron basis, constructed from the Green’s function (see eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 13 and 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In the MB basis, the DOS explores, from the ground state in the Nel sector, all possible states in the Nel ± 1 sectors, corresponding to all possible ways of adding or removing one electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' To adapt this definition of the DOS to MF-MB method, we need (as in the case of ED) to compute MF- MB results in the Nel, Nel + 1, and Nel − 1 sectors in- dependently and then compute the Green’s function and the DOS from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The fundamental difference between this approach of the DOS and the approach of MF-U (with the correspondence with MF-MB explained before) is that: in MF-U, all single-particle states are computed as interacting with the mean-field calculated from the ground state of the Nel sector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' in MF-MB, peaks in the DOS corresponding to the addition (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' removal) of one electron take into account the fact that when one electron is added (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' removed), the states of the Nel + 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Nel − 1) sector interact with the mean-field con- structed from the ground state of the Nel +1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Nel − 1) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In MF-U, adding one electron to the ground state (con- taining Nel) in the first unoccupied states will not result in the ground state of the system containing Nel +1 elec- trons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Indeed, we would have to perform another MF- U calculation using a self-consistent procedure with the mean-field created from Nel + 1 electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This would even change the Nel lowest energy states from the previ- ous MF-U calculation with Nel electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In the MF-U method, the occupied states in the DOS have a clear physical meaning: they are the individual states composing the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' However, the inter- pretation of the unoccupied states is more difficult since they can’t be attributed to physical states that will be occupied if we add an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In contrast, the definition of the DOS in MF-MB, in analogy with the ED technique, provides a clear physical interpretation of all the peaks: They represent all pos- sible ways of adding or removing one electron from the Nel-electron ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We conclude this discussion with an important re- mark about the computation of the DOS in the MF-MB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For some systems, starting from different ini- tial conditions, it is possible to converge towards several different ground states, degenerated in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' It seems unphysical to take into account only one of the ground- state solutions in the calculation of the DOS (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (13) and (14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' At the same time, this reasoning might lead to the exclusion of states in some sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This idea is better explained by examining the example of the two- 8 Bonding/anti-bonding basis spin-up ground state : |b ↑⟩ |b ↓⟩ |a ↑⟩ |a ↓⟩ E = −t 1 0 0 0 E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t 0 1 0 0 E = t 0 0 1 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t 0 0 0 1 Bonding/anti-bonding basis spin-down ground state : |b ↑⟩ |b ↓⟩ |a ↑⟩ |a ↓⟩ E = −t 0 1 0 0 E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t 1 0 0 0 E = t 0 0 0 1 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t 0 0 1 0 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' List of eigen-states of the two-site Hubbard system at quarter-filling from the MF-U or MF-MB meth- ods for U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' There are four eigen-states at energies E = −t, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t, t, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' second) part of the table shows the eigen-states’ coefficients in the single- electron bonding/anti-bonding basis where the ground state was chosen to be spin up (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' site system at half-filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' One needs to perform a MF- MB calculation in the one-electron and three-electron sectors to construct the MF-MB DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For illustration purposes, we only focus on the one-electron sector and on the removal part of the DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The one-electron sector eigen-states of the MF-MB method are given directly in the bonding/anti-bonding basis in table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The single- particle eigen-states for the one-electron sector are the same as for the two-electron sector, but we observe that depending on the choice of the ground-state polarization (spin up or spin down), all other states are affected and spin-flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' When computing the Green’s function (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (13)), we have to consider all accessible states (in the Nel = 1 sec- tor) from the Nel = 2 ground state |b ↑, b ↓⟩ (see table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For example, removing a spin-down electron from the state |b ↑, b ↓⟩ results in the state |b ↑⟩ that is present at different energies in the one-electron sector, depending on the ground-state polarization (E = −t and E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t), see table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' It appears that only the state at E = −t has to be taken into account since the other one is an unoccu- pied state when the ground state is spin-down polarised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Since we precisely chose to remove a spin-down electron from the Nel = 2 ground state, the one-electron ground state to be taken into account should be only the spin-up polarized one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The same reasoning holds for the removal of the spin-up electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' For the two-site system, there is a symmetry in the one-electron and three-electron sec- tors and thus in the addition and removal parts of the DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The density of states computed in ED, MF-U, and MF-MB (with and without the exclusion of states in the Nat = 1 and Nat = 3 sectors) are given in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This illustrates the process of state exclusion when comparing the two middle curves: the first one with no exclusion (second curve from the top at fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2) exhibits two peaks below the Fermi level, corresponding to a transition from FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Density of states of the two-site Hubbard system at half-filling with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t for MF-U (blue), MF-MB without state exclusion in the Nat = 1 and Nat = 3 sectors (orange), MF-MB with state exclusion (green), and ED (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' the Nel = 2 ground state |b ↑, b ↓⟩ to the states |b ↑⟩ and |b ↓⟩ for the two possible choices of ground state in the one-electron sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In contrast, with the exclusion of states, the third curve from the top of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2 shows only one peak below the Fermi level, corresponding to the transition from the Nel = 2 ground state to the states |b ↑⟩ and |b ↓⟩ of E = −t in the one-electron sector (the E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='75t have been excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We note that for this specific case, we recover the MF-U result from the MF- MB with states exclusion but this is not the case in gen- eral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We also stress that ED predicts satellites (smaller peaks) for the two-site Hubbard model at half-filling, lo- cated at E ≃ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='77t and E ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='27t (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 11, 12, and 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' One satellite is visible in the insert of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2 but no satellite is present in either of the MF densities of states presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We explore the presence of satellites in further detail in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' COMPARISON BETWEEN MF METHODS AND ED We now illustrate the formalism described in this paper for the case of small atomic linear chains containing up to 3 atomic sites, showcasing the main differences between the MF-U, MF-MB, and ED methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' MF-MB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' ED states and DOS: the two-site Hubbard system We have shown above the results of the two-site prob- lem for the MF-MB technique in the one-electron and two-electron sectors as well as the DOS in the half-filling case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Here, we start by recalling ED results that have been extensively studied before [9–12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We then de- scribe the differences between MF-U, MF-MB, and ED MF-U MF-MB exclusion MF-MB no exclusion ED x50 x50 人 x50 x50 3 2 1 0 1 2 3 4 E [t units]9 Bonding/anti-bonding basis : |b ↑⟩ |b ↓⟩ |a ↑⟩ |a ↓⟩ E = −t 1 0 0 0 E = −t 0 1 0 0 E = t 0 0 1 0 E = t 0 0 0 1 TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' List of ED eigen-states of the two-site Hub- bard system at quarter-filling for U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' There are four eigen-states at energies E = −t and t (each doubly spin- degenerated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' states based on this simple model as well their conse- quences on the DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The eigen-vectors of the one-electron sector of the two- site Hubbard model with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t are listed in table IV in the bonding/anti-bonding basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' They have to be com- pared to MF-MB eigen-states shown in table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We observe that the main difference between the ED and MF-MB results is the lifting of spin-degeneracy in the MF-MB for both bonding and anti-bonding states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This can be understood as follows: in MF-MB, a specific po- larized ground state is chosen, which leads to a mean-field interaction term that is only present for the opposite spin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In ED, there is no interaction term since there is only one electron and no double occupation in the basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The eigen-vectors of the two-site Hubbard model with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t at half-filling are listed in table V in the lo- calised and in the bonding/anti-bonding bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' These eigen-states have to be compared with MF-MB results of table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Note that if we want to compare the eigen- energies, we have to remember to take into account the constant term of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (2) that consists in a constant shift in the energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In the case of table II, one has to apply a constant shift of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t for all eigen-energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We ob- serve three main effects of the MF approximation when comparing results from tables II and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The first MF effect is a well-known property: the MF approximation overestimates the ground-state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The difference between MF and ED ground-state energies is usually called the correlation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The correlation energy Ecorr = EMF − EED depend on the parameter U and, in our example, is given by Ecorr ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01556t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The second effect is that the three degenerate states at E = 0 and the singly-degenerated state at E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t in ED are de- generate in MF at E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This can be understood by noting that the basis states |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ and |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ are treated in the same manner as basis states |↑, ↓⟩ and |↓, ↑⟩ in MF, whereas it is not the case in ED: the E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t singlet is formed by the basis states |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ and |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ that are the only basis states introducing a Hubbard term since they induce a doubly-occupied site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This difference of treat- ment in MF and in ED is also responsible for the third observed difference: the MF ground state (at E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t) has equal weight for |↑, ↓⟩ and |↓, ↑⟩ than for |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ and |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ whereas there is an asymmetry in the ED ground state due to the Hubbard term induced by the doubly- occupied site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The equal weight in MF results in the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Density of states of the two-site Hubbard system at quarter-filling with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t for MF-U (blue), MF-MB (green), and ED (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' ground state being a product state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', a Slater de- terminant) of single-electron eigen-states (bonding and anti-bonding states) and the ED ground states is a lin- ear combination of Slater determinant of single-electron eigen-states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Because the ED ground state is a superposition of |b ↑, b ↓⟩ and |a ↑, a ↓⟩, the Green’s function (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (13)) features two poles for the part related to the Nel − 1- electron sector since the ground state couples with all the possible states of table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The same holds for the Nel+1 electron sector part of the Green’s function due to the symmetry of the two-site Hubbard model for Nel = 1 and Nel = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The coupling between the Nel = 2 ground state and the excited states of the Nel = 1 sector (E = t states shown in table IV) results in the so-called satellite peaks in the DOS of the half-filled two-site Hubbard sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' A key feature of this peak is that it disappears at the U = 0 limit: this can be explained since at U = 0, the asymmetry between the two states |↑, ↓⟩ and |↓, ↑⟩ and the two states |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ and |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ is canceled and the ground state is found to be a single Slater determinant of single-electron eigen-states |b ↑, b ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' At the U = 0 limit as well as in MF, the ground state is a pure prod- uct state of single-particle eigen-states such that it only couples to |b ↑⟩ and |b ↓⟩ listed in tables III and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Only one peak on each side of the Fermi level is observed as in the MF-MB exclusion curve of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We now turn to the quarter-filling DOS of the two- site Hubbard system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The ground state is either |b ↑⟩ or |b ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Specifically, we chose the Nat = 1 ground state to be |b ↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The Green’s function of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' (13) has poles corresponding to energy transitions between the single- particle ground state and all two-particle states listed in tables II or V for MF and ED respectively as well as at energy transitions between the single-particle ground state and all zero-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' There exists only one zero-particle state: it is the vacuum state and has zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Figure 3 shows the DOS of the quarter-filling two-site Hubbard system for U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t using the MF-U, MF-MB, and ED methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Since there is only one electron in x50 MF-U MF-MB x50 ED x50 4 2 1 0 1 2 3 4 E [t units]10 Localised basis : |↑, ↑⟩ |↑, ↓⟩ |↓, ↑⟩ |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ |↓, ↓⟩ E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='76556t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='467970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='46797 0 E = 0 0 −1/ √ 2 1/ √ 2 0 0 0 E = 0 1 0 0 0 0 0 E = 0 0 0 0 0 0 1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 0 0 −1/ √ 2 1/ √ 2 0 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26556t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='46797 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='46797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5301 0 Bonding/anti-bonding basis : |b ↑, a ↑⟩ |b ↑, b ↓⟩ |b ↑, a ↓⟩ |a ↑, b ↓⟩ |a ↑, a ↓⟩ |b ↓, a ↓⟩ E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='76556t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99807 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='06214 0 E = 0 0 0 1/ √ 2 −1/ √ 2 0 0 E = 0 1 0 0 0 0 0 E = 0 0 0 0 0 0 1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t 0 0 1/ √ 2 1/ √ 2 0 0 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26556t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='06214 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99807 0 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' List of the ED eigen-vectors of the two-site Hubbard system calculated for U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' There are six eigen-states with energies E ≃ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='76556t, E = 0 (three times degenerated), E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t, and E ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26556t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The first part of the table shows the linear coefficients of the states in the many-body basis of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' II C) (localised basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The second part of the table gives the eigen-states’ coefficients in the bonding/anti-bonding basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' the system, the MF-U DOS exhibits peaks at the single- electron eigen-energies that are also found in MF-MB and given in table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The lowest energy peak in MF-MB and ED methods are both removal peaks at the ground-state energy of the Nel = 1 sector of tables III and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The second lowest energy peak in MF-MB and ED cor- responds to the coupling in the Green’s function between the Nel = 1 and the Nel = 2 ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Since in the Nel = 2 sector the ED ground state is ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02t lower than the MF-MB one (correlation energy), the DOS peak is also found at a lower energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In each case, it corre- sponds to the addition of a |b ↓⟩ electron to the Nel = 1 ground state |b ↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In MF-MB, the ground state is the only state in the Nel = 2 sector that contains that ba- sis state |b ↑, b ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In ED, due to the asymmetry between states with and without doubly occupied states, the high- est excited state also contains a part of the basis state |b ↑, b ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This is responsible for the presence of a satel- lite in the DOS, clearly visible in the insert of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Alternatively, this satellite might also be interpreted as a contribution from the addition of a |b ↓⟩ electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The other consequence of this asymmetry is the fact that the second lowest-energy peak in ED does not integrate to 1, since it does not simply represent a single-electron state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Finally, the highest energy peak of the MF-MB DOS (E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25t) originates from the coupling between the Nel = 1 ground state |b ↑⟩ and two of the four states at E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t listed in table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The states |a ↑, b ↓⟩ and |b ↓, a ↓⟩ indeed cannot be reached from |b ↑⟩ in the Green’s function expression of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The observed peak in the DOS thus comes from the addition of |a ↑⟩ and |a ↓⟩ electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In MF-U, the addition of |a ↑⟩ and |a ↓⟩ electrons is non-degenerate because these unoccu- pied states are built by interacting with the mean-field produced by only one electron (the |b ↑⟩ electron of the ground state) such that there is an interaction term for the addition of |a ↓⟩ but not for the addition of |a ↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In contrast, in MF-MB, the many-body states of the Nel = 2 sector are all constructed based on the mean- field produced by the Nel = 2 ground state (|b ↑, b ↓⟩) that induces the same interaction term for the consid- ered excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In ED, the peak is also split into two peaks in the DOS because of the lifting of degener- acy between E = 0 and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t energy states listed in table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The E = t peak in ED reflects the addition of a |a ↑⟩ electron and the half of the addition of a |a ↓⟩ electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The other half is contained in the next peak in the DOS at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This splitting is due to the fact that adding a |a ↓⟩ electron to a system containing a |b ↑⟩ state involves contributions both from states with and without doubly-occupied sites translated into differ- ent interaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The peak corresponding to the addition of a |a ↑⟩ electron is not split since there is no interaction between two spin-up electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The emergence of satellites in MF-MB The MF approximation treats particles as independent particles interacting with a mean-field, neglecting all the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' It is often thought that satellites and quasi- particles are features that can only be observed when including correlation effects via approximations or exact treatment [9, 10, 26, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Satellites have zero weight in the DOS for U = 0 and an increasing weight when the interaction increases whereas quasi-particle peaks can be linked to particle peaks in the U = 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Their weight might decrease as the interaction increases: a weight transfer from quasi-particle to satellite peaks then oc- curs [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We now show that the MF-MB description presented here can also induce the emergence of satellites in linear chains containing Nat = 3, showing that the satellite peaks are not correlation peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The DOS for the Nat = 3 linear chain with one elec- tron are shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 4 for the MF-U, MF-MB, and ED 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Density of states of the three-site Hubbard system with one electron for U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t for MF-U (blue), MF-MB (green), and ED (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The DOS can be understood from the Nel = 1 and Nel = 2 eigen-states for MF-MB and ED given in the SI (tables 1 to 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The general differences between the two methods have already been described for the two-site system in section V A and can be applied to the three-site system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Similar to the two-site system at quarter-filling (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 3), the two lowest peaks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', the removal and the lowest-energy addition peaks) have approximately the same energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The higher-energy pairs of peaks in MF-U look degenerated in MF-MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' However, the term degen- erated is not well-suited in the MF-MB case since, as for the ED DOS, each peak does not necessarily integrate to an integer value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This is related to the weight trans- fer from quasi-particle peaks to emergent satellites (at E ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='96t and E ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='43t) that can be visualized on the zoomed-in MF-MB curve shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In ED, one observes that the two higher energy pairs of single-particle states in MF-U correspond to a prin- cipal peak that is more intense than the single-particle peaks, plus several smaller peaks located near the princi- pal peaks and satellites that can be seen on the zoomed-in ED curve (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The small peaks near the principal ones are not reproduced in MF-MB for the same rea- son why the E ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t peak in the two-site system for ED is not present in the MF-MB (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Specif- ically, in MF-MB (Nel = 2 sector), there are fourfold- degenerated states that show a lifting of degeneracy and result in threefold-degenerated plus non-degenerated states (as discussed in section V A and can be seen in ta- bles II and V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The main observation for the three-site system with one electron is thus the emergence of satel- lites in MF-MB, that are also present in ED but absent in MF-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The emergence of satellites in MF-MB is also ob- served for the three-site Hubbard system with filling up to half-filling (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We expect satellites to be present also for larger systems since they contain an increasing number of accessible excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' CONCLUSION In this work, we develop a method to compute the MF approximation in a many-body basis instead of a single- body basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The MF approximation is a well-known and broadly used approximation in the single-body basis such that the interest of doing MF computations in a many- body basis, at much more expensive computational cost, might not be so straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' However, since ED tech- niques require a many-body basis treatment, the devel- oped method allows for a better comparison between MF and ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We focus on the definition and the signification of the DOS and show how the usual MF approximation DOS can be found from MF-MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We also highlight the pos- sible ambiguity in a definition inspired by the ED DOS definition, coming from the use or not of Koopman’s the- orem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', frozen single-particle states) in the prediction of the Nel ± 1 ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Attention is paid to the exclusion process, physically motivated by the lack of co- herence in accounting for some of the different possible ground states, in the MF-MB DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The exclusion pro- cess is implemented manually in this work and an auto- mated treatment of the process would be necessary for large-scale investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We also observe that our MF-MB method induces satellites structure in the DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Satellites are usually seen as a sign of correlation since they appear in ED or ap- proximations beyond MF (GW, T-matrix, DMFT,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We demonstrate in this work that satellites could appear in purely MF computed DOS, when adopting the ED- like definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This result assumes the computation of states in Nel + 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Nel − 1) sector using the mean- field based on the of the Nel + 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Nel − 1) sector instead of using only one mean-field from the Nel sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The approach presented here contributes to a better understanding of the fundamental differences between MF and ED Hamiltonians and the associated energies and wave-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Our formulation includes Green’s function expression that is the bridge between MF and many-body corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We therefore expect that this work could pave the way for developing other levels of ap- proximation that introduce many-body correlations, such as GW or T-matrix, in the many-body basis in order to gain a deeper understanding of their mathematical and physical underpinnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' As further perspective, we sug- gest that revisiting and implementing known approxima- tions from a refreshed point of view will deepen their un- derstanding and lead to new approximations that would be deduced either numerically or from physical intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Specifically, we intend to explore the process of symme- try restoration of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 19 and 20 rewritten in MB basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Finally, we posit that providing an in-depth and pedagog- ical analysis of a simple case of electronic correlation will help readers appreciate the difficulties associated with many-body electronic systems, especially since those are a cornerstone of quantum information and quantum com- puting developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' x200 MF-U MF-MB x200 ED x200 6 4 2 0 2 4 6 E [t units]12 ACKNOWLEDGEMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' is a Research Fellow of the Fonds de la Recherche Scientifique - FNRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' This research used resources of the ”Plateforme Technologique de Calcul Intensif (PTCI)” (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='ptci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='unamur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='be) located at the Univer- sity of Namur, Belgium, and of the Universit´e catholique de Louvain (CISM/UCL) which are supported by the F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='-FNRS under the convention No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The PTCI and CISM are member of the ”Consortium des ´Equipements de Calcul Intensif (C´ECI)” (http://www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Reining, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Ro- maniello, “Reduced density-matrix functional theory: Correlation and spectroscopy,” The Journal of Chemical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 143, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 024108, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Mean-field approximation of the Hubbard model expressed in a many-body basis: Supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Antoine Honet and Luc Henrard Department of Physics and Namur Institute of Structured Materials, University of Namur, Rue de Bruxelles 51, 5000 Namur, Belgium Vincent Meunier Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA (Dated: January 10, 2023) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' MF-MB AND ED EIGEN-STATES FOR THE THREE-SITE HUBBARD SYSTEM We consider here the three-site Hubbard system such that the single-particle localized basis states are given by |↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩, |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩, |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓⟩, |↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩, |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ and |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We also define the eigen-states (named d, e and f) for the one-electron sector at U = 0 case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' tight-binding): |dσ⟩ = 1 2 |σ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ + 1 √ 2 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', σ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ + 1 2 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', σ⟩ |eσ⟩ = −1 √ 2 |σ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ + 1 √ 2 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', σ⟩ |fσ⟩ = −1 2 |σ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ + 1 √ 2 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', σ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ + −1 2 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', σ⟩ (1) where σ stand for the spin and is eiter ↑ or ↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The MF-MB and ED eigen-states for the three-site Hubbard with one electron are given at tables I and II, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The MF-MB eigen-states are equivalant to MF-U eigen-states since there is only one electron in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The main difference between MF-MB and ED results is the lifting of spin degen- eracy and the influence of the spin-polarized ground state on state with opposite spin in MF-MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' All eigen-states remain unchanged in ED when turning only the interaction (see ta- ble II for U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t), since there is no interaction for this case (no double occupied state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' On the contrary, the eigen-states of spin opposite to the ground state spin in MF-MB are af- fected when increasing U due to there interaction with the mean field based on the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The eigen-state |eσ⟩ remains however eigen-state for all U values because the mean field is constant on all the localized sites where the state |eσ⟩ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The eigen-states in the Nel = 2 sector in MF-MB and ED are given at tables III and IV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The main difference is the lifting of degeneracy of four times degenerated states in MF-MB to three times degenerated states and non degenerated state in ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We also observe the mixing of a pure product state of U = 0 eigen-states (|e ↑, e ↓⟩ at E ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26t in MF-MB) with other states in ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' DOS OF FOR TWO AND THREE ELECTRONS ON THREE SITES The DOS for the three-site Hubbard system filled with two electrons with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t is given at fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Since the Nel = 2±1 sectors present polarized ground state, the exclusion mecha- nism explained in the main text is used such that MF-MB DOS is computed both with and without exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' We note the inclusion of satellites in the addition part of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' There are no satellites in the removal part since it correspond FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Density of states of the three-site Hubbard system with two electrons with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t for MF-U (blue), MF-MB (green) and ED (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Density of states of the three-site Hubbard system with three electrons with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t for MF-U (blue), MF-MB (green) and ED (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' to exploring the Nel = 1 sector in which the eigen-states are single-particle eigen-states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' The DOS for the three-site Hubbard system filled with three electrons with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t is given at fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' As in the case of the three-site system with one electron exposed in the main text, we observe in MF-MB the pairing of states that are non- degenerated in MF-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' These states are split in ED and a part of the weight is transfered to satellites both in MF-MB and ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 2, only the addition satellites are shown since there is a removal/addition symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03223v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='str-el] 9 Jan 2023 MF-U MF-MB exclusion MF-MB no exclusion ED x200 x200 x200 x200 x200 x200 x200 x200 6 4 2 0 2 4 6 E[t units]x200 MF-U MF-MB x200 ED x200 6 2 4 0 2 4 6 E[t units]2 Localised basis : |↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓⟩ |↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑⟩ E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 1/2 1/ √ 2 1/2 0 0 0 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='22809t 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='51092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='51092 E = 0 −1/ √ 2 0 1/ √ 2 0 0 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='125t 0 0 0 −1/ √ 2 0 1/ √ 2 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t −1/2 1/ √ 2 1/2 0 0 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='60309 t 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='48884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='72255 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='48884 U = 0 diagonal basis : |d ↓⟩ |e ↓⟩ |f ↓⟩ |d ↑⟩ |e ↑⟩ |f ↑⟩ E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 1 0 0 0 0 0 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='22809t 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99976 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02208 E = 0 0 1 0 0 0 0 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='125t 0 0 0 0 1 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 0 0 1 0 0 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='60309 t 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02208 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99976 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Table representing the eigen-states of the three-site Hubbard system with one electron in either the MF-U or the MF-MB methods and with U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Localised basis : |↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓⟩ |↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑⟩ E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 1/2 1/ √ 2 1/2 0 0 0 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 0 0 0 1/2 1/ √ 2 1/2 E = 0 −1/ √ 2 0 1/ √ 2 0 0 0 E = 0 0 0 0 −1/ √ 2 0 1/ √ 2 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t −1/2 1/ √ 2 1/2 0 0 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 0 0 0 −1/2 1/ √ 2 1/2 U = 0 diagonal basis : |d ↓⟩ |e ↓⟩ |f ↓⟩ |d ↑⟩ |e ↑⟩ |f ↑⟩ E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 1 0 0 0 0 0 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 0 0 0 1 0 0 E = 0 0 1 0 0 0 0 E = 0 0 0 0 0 1 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 0 0 1 0 0 0 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41421 t 0 0 0 0 0 1 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Table representing the eigen-states of the three-site Hubbard system with one electron in ED and U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 3 Loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' basis : |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓, ↓⟩ |↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓⟩ |↓, ↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓, ↑⟩ |↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑, ↓⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↓, ↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓⟩ |↑, ↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑, ↑⟩ |↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑⟩ |↑, ↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ E = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='46 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='4805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='25975 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01804 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='33410 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50934 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='49015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='72076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='49015 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='49015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='72076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='49015 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3341 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50934 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35864 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01804 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26 t −1/2 1/2 1/2 1/2 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37 t 1/ √ 2 −1/ √ 2 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37 t 1/ √ 2 −1/ √ 2 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35324 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01175 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70648 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35324 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35324 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00547 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='49964 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='10943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50024 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00547 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='49964 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00547 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69318 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50966 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69318 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50966 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='48992 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3713 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='48992 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='48992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37131 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='48992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0151 E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='20 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='24025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35328 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='24025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35328 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='24025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35328 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='24025 U = 0 diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' basis : |d ↓, e ↓⟩ |d ↓, f ↓⟩ |e ↓, f ↓⟩ |d ↑, d ↓⟩ |d ↑, f ↓⟩ |f ↑, d ↓⟩ |f ↑, f ↓⟩ |e ↑, e ↓⟩ |d ↑, e ↓⟩ |f ↑, e ↓⟩ |e ↑, d ↓⟩ |e ↑, f ↓⟩ |d ↑, e ↑⟩ |d ↑, f ↑⟩ |e ↑, f ↑⟩ E = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='46 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0004 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70702 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='10 t -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0195 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0195 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03881 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01125 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26 t 1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37 t 1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37 t 1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='71744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02757 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69608 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='71796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00043 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99981 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99981 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70621 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70706 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03557 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70621 E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='20 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99962 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Table representing the eigen-states of the three-site Hubbard system with two electrons in MF-MB and U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' basis : |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓, ↓⟩ |↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓⟩ |↓, ↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓, ↑⟩ |↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑, ↓⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↓, ↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓⟩ |↑, ↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑, ↑⟩ |↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑⟩ |↑, ↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ E = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='66 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='36404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='27422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='36404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='46153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='36404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='27422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='36404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23077 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35355 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 1/2 1/ √ 2 1/2 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 1/2 1/ √ 2 1/2 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='19 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='45440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='38309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='38309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='38309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='38309 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='45440 E = 0 1/ √ 2 −1/ √ 2 E = 0 1/ √ 2 −1/ √ 2 E = 0 1/2 1/2 1/2 1/2 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='12 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='20235 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='61178 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03801 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='40470 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='61178 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03801 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='20235 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='57735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='57735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='57735 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 1/2 1/ √ 2 1/2 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 1/2 1/ √ 2 1/2 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35355 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='54177 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='32131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='32131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='32131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='32131 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='54177 E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='34064 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='22479 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='34064 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='53840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='34064 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='22479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='34064 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26920 U = 0 diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' basis : |d ↓, e ↓⟩ |d ↓, f ↓⟩ |e ↓, f ↓⟩ |d ↑, d ↓⟩ |d ↑, f ↓⟩ |f ↑, d ↓⟩ |f ↑, f ↓⟩ |e ↑, e ↓⟩ |d ↑, e ↓⟩ |f ↑, e ↓⟩ |e ↑, d ↓⟩ |e ↑, f ↓⟩ |d ↑, e ↑⟩ |d ↑, f ↑⟩ |e ↑, f ↑⟩ E = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='66 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02173 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02173 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 1/ √ 2 −1/ √ 2 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 1 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 1 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='19 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70440 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='06178 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70440 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='06178 E = 0 1 E = 0 1 E = 0 −1/ √ 2 1/ √ 2 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='12 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='05140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='05612 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='81413 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='57735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='57735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='57735 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 1 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t 1 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='41 t −1/ √ 2 1/ √ 2 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='06178 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70440 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='06178 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='70440 E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02221 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99792 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='04441 TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Table representing the eigen-states of the three-site Hubbard system with two electrons in ED and U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' 4 Loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' basis : |↓, ↓, ↓⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓, ↑↓⟩ |↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ |↓, ↓, ↑⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓, ↓⟩ |↓, ↑, ↓⟩ |↓, ↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↑, ↓, ↓⟩ |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↓⟩ |↑, ↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↑↓, ↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑, ↑↓⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓, ↑⟩ |↓, ↑, ↑⟩ |↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑↓⟩ |↑, ↓, ↑⟩ |↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=', ↑⟩ |↑, ↑, ↓⟩ |↑, ↑↓, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↑↓, ↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='⟩ |↑, ↑, ↑⟩ E = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='21 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='52544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23728 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='95 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='47456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='3531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26272 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35303 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='74 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='76 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='88 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35303 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='51215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='33037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='33037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='51215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02053 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='11 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='51215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='33037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='33037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='51215 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='4869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='4869 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='4869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='37302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='4869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01444 E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='45 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='47456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='26272 E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='71 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='52544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='35310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23728 U = 0 diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' basis : |d ↓, e ↓, f ↓⟩ |d ↑, d ↓, f ↓⟩ |d ↑, d ↓, e ↓⟩ |d ↑, e ↓, f ↓⟩ |f ↑, d ↓, f ↓⟩ |f ↑, d ↓, e ↓⟩ |f ↑, e ↓, f ↓⟩ |e ↑, d ↓, f ↓⟩ |e ↑, d ↓, e ↓⟩ |e ↑, e ↓, f ↓⟩ |d ↑, f ↑, d ↓⟩ |d ↑, f ↑, f ↓⟩ |d ↑, f ↑, e ↓⟩ |d ↑, e ↑, d ↓⟩ |d ↑, e ↑, f ↓⟩ |d ↑, e ↑, e ↓⟩ |e ↑, f ↑, d ↓⟩ |e ↑, f ↑, f ↓⟩ |e ↑, f ↑, e ↓⟩ |d ↑, f ↑, d ↑⟩ E = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='21 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00065 E = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='95 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00065 E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='74283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='66899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01703 E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='66899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='74283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01891 E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='61 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='6903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='72308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01841 E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='61 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='72308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='6903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01757 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5 t 1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='62 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='7203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00068 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='62 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='7194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03597 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='74 t 1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='76 t 1 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='88 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='7201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='692 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='03597 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='88 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='69293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00071 E = t 1 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='11 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73462 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='11 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='0187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='73462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='678 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='72753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='68561 E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='23 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='68561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='01852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='72753 E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='45 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99935 E = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='71 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='00065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='02544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='99935 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content=' Table representing the eigen-states of the three-site Hubbard system with three electrons in MF-MB and U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} +page_content='5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E1T4oBgHgl3EQffwT8/content/2301.03223v1.pdf'} diff --git a/idFLT4oBgHgl3EQfay-h/vector_store/index.faiss b/idFLT4oBgHgl3EQfay-h/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..453d6518a42cc573ce1b736de898994f8ee7730f --- /dev/null +++ b/idFLT4oBgHgl3EQfay-h/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d745c56c2471521e47e776405929036397c1abf6aaa033c65bfd53836c1c4abb +size 3276845 diff --git a/jdAyT4oBgHgl3EQf_Po5/content/tmp_files/2301.00904v1.pdf.txt b/jdAyT4oBgHgl3EQf_Po5/content/tmp_files/2301.00904v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..05cf200346a556e0c4acfc45bde99fb10e53a5b2 --- /dev/null +++ b/jdAyT4oBgHgl3EQf_Po5/content/tmp_files/2301.00904v1.pdf.txt @@ -0,0 +1,852 @@ +Safe Reinforcement Learning for an +Energy-Efficient Driver Assistance System +Habtamu Hailemichael ∗ Beshah Ayalew ∗ Lindsey Kerbel ∗ +Andrej Ivanco ∗∗ Keith Loiselle ∗∗ +∗ Automotive Engineering, Clemson University, Greenville, SC 29607, +USA (hhailem, beshah, lsutto2)@clemson.edu. +∗∗ Allison Transmission Inc., One Allison Way, Indianapolis, IN, +46222, USA (andrej.ivanco, keith.loiselle)@allisontransmission.com +Abstract: Reinforcement learning (RL)-based driver assistance systems seek to improve fuel +consumption via continual improvement of powertrain control actions considering experiential +data from the field. However, the need to explore diverse experiences in order to learn optimal +policies often limits the application of RL techniques in safety-critical systems like vehicle +control. In this paper, an exponential control barrier function (ECBF) is derived and utilized +to filter unsafe actions proposed by an RL-based driver assistance system. The RL agent freely +explores and optimizes the performance objectives while unsafe actions are projected to the +closest actions in the safe domain. The reward is structured so that driver’s acceleration requests +are met in a manner that boosts fuel economy and doesn’t compromise comfort. The optimal +gear and traction torque control actions that maximize the cumulative reward are computed +via the Maximum a Posteriori Policy Optimization (MPO) algorithm configured for a hybrid +action space. The proposed safe-RL scheme is trained and evaluated in car following scenarios +where it is shown that it effectively avoids collision both during training and evaluation while +delivering on the expected fuel economy improvements for the driver assistance system. +Keywords: RL driver-assist, Safe reinforcement learning, Safety filtering, Control barrier +functions +1. INTRODUCTION +Reliable, safe, and efficient commercial vehicles are es- +sential for the transportation industry to have a positive +impact on the environment, the economy, and road safety. +Given the estimated increase in freight demand of 16% +by 2030 (Bureau of Transportation Statistics, 2017), there +is clearly a need and an opportunity to reduce emission +and fuel usage as more of these vehicles get on the roads +to meet this demand. Furthermore, ensuring safety via +accident prevention is critical. Advanced driver assistance +systems (ADAS) such as emergency braking, adaptive +cruise control (ACC), and lane keeping assist have been +developed to primarily address the safety concerns. More +advanced systems additionally aid the driver with ecologi- +cal (fuel saving) driving behaviors such as reduced braking +and accelerations (Barkenbus, 2010), optimizing velocity +profiles for ACC (Maamria et al., 2016; Nie and Farzaneh, +2020) and optimizing gear shifting (Ngo et al., 2013). +As typical commercial routes include frequent stopping +and starting along with various required speeds, the ACC +approach may be cumbersome for a driver to use. In Yoon +et al. (2020), a driver assistance implementation is pro- +posed that uses radar information and motion models +to directly modulate the torque request to the power- +train/braking system. To this end, an MPC scheme is em- +ployed to optimize traction/braking torque/power while +tracking the driver’s desired acceleration and maintaining +a safe distance to a leading vehicle. In Kerbel et al. (2022), +a similar driver assistance objective is pursued in a model- +free reinforcement learning approach to learn both optimal +gear selection and torque request using fuel usage and +other reward signals in the vehicle’s experience. Although +this study demonstrated a fuel consumption improvement +of up to 12%, it did not include provisions to guarantee +collision avoidance. Unlike typical optimal control schemes +such as MPC, where hard constraints are set based on +a dynamic model, it is generally difficult to enforce such +constraints in RL controllers where learning an optimal +policy requires exploration of different actions and states. +However, unlimited exploration is unacceptable for safety- +critical systems such as vehicle control. In this paper, +we construct a driver-assist RL agent that targets fuel +efficiency and driver accommodation and incorporates el- +ements that ensure safety. +Different approaches are proposed to properly constrain +the exploration of the RL agent within a safe set. In +Li and G¨orges (2020), a supervisor is used to simply +enforce (override) the gear and engine speed constraints +to control the transmission, yet the RL agent never really +learns these limits. Often, reward shaping approaches are +utilized by assigning a penalty to safety violations that +discourage policies leading to constraint violation. Since +the RL agent with reward shaping learns the penalties +only after experiencing them, this approach does not guar- +antee safety, especially during initial training. Another +approach to enforcing safety is to pose the problem as +a constrained Markov decision process (CMDP) where a +arXiv:2301.00904v1 [cs.RO] 3 Jan 2023 + +constraint cost is assigned for each state-action pair and +the RL agent learns to keep the discounted constraint +cost over the horizon below a certain threshold (Altman, +1999). Many implementations of the CMDP then involve +joint optimizations of the main performance task and +the constraint reward, and this entails trade-offs between +safety and performance. In this work, we seek to somewhat +decouple the two goals by adopting what is known as a +safety filtering approach. This approach configures the RL +agent to focus on maximizing performance (reward), while +a safety layer/filter is designed to project the outputs of +the RL agent onto a safe set. Although the filter does not +typically interfere with the inner workings of the RL agent, +it will influence performance as it often determines the +extent of the safe set and subsequent interactions of the +RL agent with the system under control. Evaluations of +the proposed actions in the safety layer could be based on +learning constraints (Dalal et al., 2018) and safety indexes +(Thananjeyan et al., 2021; Srinivasan et al., 2020) from +offline data or using a dynamic model of the system. +Of the dynamic model-based approaches to safety filtering, +control barrier functions (CBF) provide scalable and com- +putationally light safety filters (Li, 2021). A CBF applies +hard safety constraints by forcing the system to operate +in the invariant safe-set defined by a super-level set of a +continuously differentiable function h(x) : Rn → R. The +actions selected by the RL agent are projected into the +safe set in such a manner that the proposed actions are +minimally modified +(Ames et al., 2019), and no unsafe +actions are passed to the controlled system. One could +come up with handcrafted CBFs considering the dynamics +of the system; a case in point is the relationship between +the maximum deceleration available to the vehicle and the +distance gap in the collision avoidance problems (Ames +et al., 2014; Cheng et al., 2019). For high relative degree +nonlinear systems, as in the present application, tailored +CBFs known as exponential barrier functions (ECBF) +have been proposed (Nguyen and Sreenath, 2016). +In this paper, we derive a specific ECBF structure that +works in conjunction with the RL driver-assist agent in +order to take explicit consideration of inertia effects which +are relevant for the safety of commercial vehicles in traffic. +The main performance goal of the driver-assist RL agent +is given by a multi-objective reward function that is struc- +tured to balance driver accommodation, fuel economy, +and smooth vehicle operation. In addition, driveability +is encouraged by introducing an additional reward for +reserve power. In this regard, Ngo (2012) characterizes +acceleration potential at a given speed by merely analyzing +different standard drive cycles. In this paper, we propose +to learn the power reserve reward to customize the vehi- +cle’s response to the driving conditions and the driver’s +tendencies. +To summarise, the contributions of this paper are: 1) +formulation of a driver assist RL agent configured for +reward optimal gear selection and torque control of a +commercial vehicle, 2) derivation of an ECBF safety filter +to work with this RL agent and 3) demonstration of the +potential of learning power reserve attributes to further +customize the system to the driver and driving conditions. +The rest of the paper is organized as follows: Section 2 +discusses the vehicle model and the driver-assist RL agent. +Fig. 1. Proposed safe RL-based Eco-Assist system set up +Section 3 discusses the design of the safety filter and the +subsequent projection of the output of the RL agent onto +the safe set. Section 4 presents simulation and training +settings and results are discussed in Section 5. Finally, +Section 6 concludes the paper. +2. VEHICLE ENVIRONMENT AND RL +CONTROLLER +A schematic of the proposed RL-based driver assistance +system, including the safety filter, is shown in Fig.1. In this +section, we detail the different computational components +of the Driver-Assist RL agent; the next section deals with +the ECBF safety filter. +The vehicle-driver-environment is modeled as Markov de- +cision process (MDP) with state s, actions a, rewards +r and a discount factor γ. The states are included in +s = {vl, vrel, ades, a, z, ng, mv, θ, f} which, respectively, are +the ego vehicle velocity, the relative velocity between the +preceding and ego vehicle, the driver demanded acceler- +ation, the actual vehicle acceleration, the separation dis- +tance with the preceding vehicle, transmission gear, mass +of the vehicle, road grade and a flag to alert if a preceding +vehicle is the sensing range of the ego-vehicle’s radar. The +RL controller is designed to maximize the vehicle’s perfor- +mance objectives through wheel traction torque Tt control +and gear change selection ∆ng, i.e., the action vector is: +a = {Tt, ∆ng}. The velocity and the wheel traction torque +are propagated back to calculate the engine torque and +speed using the transmission ratio of the current gear and +the final drive ratio. A fuel rate map is then utilized to +solve for the fuel consumption at the given engine torque +and speed. +The reward function, given by (1) below, is structured to +capture the performance objectives of the driver-assist RL- +agent. The major objective of the RL agent is to fulfill the +driver’s acceleration request, and consequently, an accel- +eration error term is given a higher weight, wa. Through +the fuel rate reward term, weighted by wf, the RL agent is +encouraged to operate the engine at fuel-efficient operating +points while fulfilling the driver-demanded acceleration. +Smooth torque changes are weighted with wt, and gear +hunting and the associated rough vehicle operation are +mitigated by including a shifting frequency penalty term +weighted by wg. Note relevant reward signals are normal- +ized by their corresponding maximum values as noted by +the max subscripts. ˙mf and ∆Tt are the fuel rate and +torque change respectively. +r = wa0.1 +|a−ades| +ades,max + wf0.1 +˙ +mf +mf,max + wT 0.1 +|∆Tt| +Tt,max + +wg0.1 +|∆ng| +ng,max + rpr, +(1) + +Driver +ng +ades +Eco-Assist +Ta +T +RL Agent +ECBF Filter +statewhere rpr models the power reserve reward term that +accounts for enhanced driveability. We define it as: rpr = +wpr0.1 +Pres,req−Pres +Pres,req +if Pres < Pres,req, else rpr = wpr, +where wpr is the corresponding weight. Pres is the actual +available power which is given in terms of engine speed and +engine torque as Pres = (Te,max(ω) − Te)ω; and Pres,req is +the required power reserve which we discuss next. +To adapt the Pres,req with the different acceleration de- +mands in different driving conditions, Ngo (2012) models +acceleration potential as varying with the vehicle veloc- +ity. To this end, the speed of the vehicle is discretized +and acceleration requests for each speed level in multiple +standard cycles are collected to be fitted in a cumulative +probability distribution. The maximum acceleration at a +given design confidence level (usually 90%) is taken as the +required acceleration potential at that velocity, areq(v). +Given areq(v), the required power reserve is then modeled +with Pres,req = mvvareq(v). In our work, rather than +using acceleration data from standard drive cycles, we +propose using the data generated by the driver in the +prevailing driving conditions. The demanded acceleration +of the driver is continuously fitted to get areq(v) that +adapts to the driver’s demand. Observing that polynomial +fits of areq suggested in Ngo (2012) lead to overfitting +issues when used with driver generated training data, we +instead use a logistic function that is easier to parametrize +and learn: +areq(v) = +k1 +1 + k2k−v +3 +. +(2) +Next, we briefly describe the framework we adopted for +training the driver assist RL agent. The states, control +actions, next states, and associated rewards are continu- +ously stored in the memory buffer R. We use actor-critic +architecture proposed by Kerbel et al. (2022) that utilizes +the off-policy algorithm known as maximum posteriori +optimization (MPO) (Abdolmaleki et al., 2018; Neunert +et al., 2020) for training. Even if it is possible to use other +state of the art algorithms, we use MPO for its sample +efficiency and robustness to hyper-parameters as well as +ease of use with the hybrid action space for the present +problem. The algorithm starts with a policy evaluation +step where a critic network approximates the state-action- +value (Q-value) for the policy. A squared loss function is +minimized between the current Q-value and an estimated +target Q-value, Qtarget(s, a). For this study, we adopted +the Retrace algorithm, known for efficiency and stability, +as described in Munos et al. (2016) for our target Q-value. +For policy improvement (actor network), the MPO algo- +rithm uses an expectation-maximization scheme. By tak- +ing samples from the memory buffer, we construct a non- +parametric policy q that maximizes Eq[Qθk(s, a)]. +max +q +Eq(a|s)[Qθk(s, a)] +s.t. Eµ(s) [KL (q(a|s)||πθk(a|s))] < ϵ, +(3) +where µ(s) is the visitation distribution given in the replay +buffer. Then a new parametric policy πθ is fitted to q with +a Kullback–Leibler (KL) divergence constraint to limit ex- +cessive deviations from the current policy. The parameters +of the actor network are updated via a gradient-based +optimization in Adam solver (Kingma and Ba, 2015). More +detailed explanation of the MPO algorithm can be found +in Neunert et al. (2020) and +Abdolmaleki et al. (2018). +Further implementation details for the present application +can also be found in our straight RL implementation +in Kerbel et al. (2022). As noted above, other state of +the art RL training algorithms can also be applied for the +driver-assist RL agent and this is independent of the safety +filter discussed next. +3. EXPONENTIAL CBF SAFETY FILTER +In this section, we give the derivation of the ECBF filter +for our application. We start with a brief review of the +definition of CBF and ECBF. We refer readers to Nguyen +and Sreenath (2016) for more detailed discussions on these +topics. Consider a nonlinear control affine system: +˙x = f (x) + g (x) u, +(4) +where f and g are locally Lipschitz, x ∈ Rn is the state +of the system, u ∈ Rm is the control input. Assume a safe +set defined by C = {x ∈ Rn|h (x) ≥ 0}, where h : Rn → R +is a continuously differentiable function. Then h is a CBF +if there exists an extended class κ∞ function α such that +for all x ∈ Int (C) = {x ∈ Rn : h (x) > 0} : +sup +u∈U +[Lfh (x) + Lgh (x) u] ≥ −α (h (x)). +(5) +The fact that h is a CBF ensures the safe set C is forward +invariant and we are able to guarantee safety. ECBFs +use input-output (IO) linearization of nonlinear systems +with relative degree r in order to generate CBFs. As +detailed in Nguyen and Sreenath (2016), the new virtual +linear system (after IO linearization) has state variables +ηb := [h(x), ˙h(x), · · ·, hr(x)]T , input µ and output h (x) : +˙ηb = Fηb (x) + Gµ, +h(x) = Cηb, +(6) +where F and G are matrices representing an integrator +chain, C = [1, 0, · · · , 0]. +The control action for the virtual linear system µ is +the rth derivative of the control output; µ = Lr +fh(x) + +LgLr−1 +f +h(x)u. When µ is set by state feedback control +with gain Kα as µ = −Kαηb, the control output evolves +with time as h (x (t)) = Ce(F −GKα)tηb (x0). For initial +condition h (x0) > 0, by imposing µ > −Kαηb (x), it +possible to guarantee h(x(t)) ≥ Ce(F −GKα )tηb (x0 ). This +relationship leads to the definition of ECBF. Considering +the dynamic system (4) and the set C = {x ∈ Rn| h (x) ≥ +0}, h (x) is an ECBF if there exists Kα ∈ Rr×1 +sup +� +Lr +fh (x) + LgLr−1 +f +h (x) u +� +≥ −Kαηb (x) , ∀ x ∈ C. +(7) +If Kα makes the closed-loop system matrix F − GKα +stronger than Hurwitz and for favorable initial conditions +choosing, µ ≥ −Kαηb (x) guarantees h (x) is an ECBF. +Pole placement strategies of linear feedback control can +then be used to design the ECBF. +We construct the collision avoidance model with the sep- +aration distance z, the velocity of the ego vehicle vh and +velocity of the leading vehicle vl as state variables. The +model follows: +˙z = vl − vh, +(8a) +˙vl = al, +(8b) + +˙vh = +Tt +rwmv +− Fr (vh, mv, θ) +mv +, +(8c) +Fr = ρAcdv2 +h +2 ++ mvgf cos θ + mvg sin θ, +(9) +where Fr is the total resistance force that includes grav- +itational, rolling friction and aerodynamics resistances. +Tt, cd, f, θ, mv, ρ, Av, rw, al are the traction torque at the +wheels, aerodynamic coefficient, rolling resistance coeffi- +cient, road grade, the mass of the vehicle, the density of +air, the frontal area of the vehicle, the radius of the wheels, +and acceleration of the leading vehicle, respectively. +Having affine dynamics, the above state space represen- +tation could also be separated into unactuated dynamics +f (x) and actuated dynamic g (x) components when writ- +ten as (4). With the choice of a minimum inter-vehicle +distance objective z0, a natural choice is h (x) = z − z0. +Input-output linearization is then employed to transform +the nonlinear dynamics into a virtual linear system as +in (6) and following the accompanying discussions above, +with the feedback gain Kα = [kα1, kα2], we have: +˙h(x) = vl − vh, +(10) +µ = ¨h (x) = Fr (vh, mv, θ) +mv ++ al − +Tt +mvrw +, +(11) +−Kαηb (x) = −kα1h(x) − kα2 ˙h(x) += −kα1 (z − z0) − kα2 (vl−vh) . +(12) +Using these with (7), we arrive at the ECBF filter. The +traction torque actions proposed by the RL agent are then +passed through this safety filter before being sent to the +vehicle environment. As shown in Fig. 1, the ECBF filter +enforces safety by projecting the action proposed by the +RL agent Ta(s) to the safe control traction torque Tt in a +way that introduces minimal change, as given by the QP +problem below. +T ∗ +t = arg min +Tt +1 +2 ∥Tt − Ta(s)∥2 +s.t. +al + Fr (vh, mv, θ) +mv +− +Tt +mvrw +≥ −kα1 (z − z0) +− kα2 (vl − vh) +(13) +4. SIMULATION AND TRAINING SETTINGS +Simulations of a medium-duty truck with a 10-speed au- +tomated manual transmission (AMT) are used to demon- +strate the workings and performance of RL driver-assist +agent with the ECBF safety filter. The actor and critic +are represented with deep neural networks with three +hidden layers, and each layer consists of 256 nodes. The +RL controller is trained in a scenario in which the truck +driven by an imperfect driver follows a preceding vehi- +cle under Federal Test Procedure (FTP-75) drive cycle +(Barlow et al., 2009). The agent is trained for a weight +range of 5 to 10 tons, as commercial vehicles have to +operate in significant load fluctuations. In order to help +capture different driving experiences, the training data is +randomized by adding noise to the velocity profile of the +preceding vehicle, using random initial separation distance +and road grade, and manipulating the parameters of the +driver model, which in this work is taken as the intelligent +Fig. 2. RL training setup +driver model (IDM)(Treiber et al., 2000) described by the +equations below. The desired minimum gap for IDM is +calculated as in (15), in which the approach term zapp (16) +dominates as the host vehicle approaches the preceding +vehicle. By only adding zapp for distance gaps closer than +a certain threshold and ignoring this term beyond that, we +emulate a distracted driver requesting unsafe actions. +a = amax +� +1 − +� v +v0 +�4 +− +�z∗ (v, vrel) +z +�2 � +, +(14) +z∗ (v, vrel) = z0 + τv − zapp, +(15) +zapp = +vrelv +2√amaxb, +(16) +where amax is the maximum acceleration, b is a comfort- +able deceleration level, τ is the time headway. +In each simulation step, as shown in Fig. 2, the IDM driver +requests acceleration based on the distance gap and veloc- +ity of the preceding vehicle. To fulfill the driver’s demand, +the actor network outputs {Ta, ∆ng} and the proposed +actions are filtered by the ECBF safety layer. The safe +actions are then implemented in the vehicle environment +and rewards are observed. As mentioned previously, the +reward aims to fulfill the driver’s acceleration demand +in a manner that promotes fuel economy, driveability, +and smooth vehicle operation. Accordingly, the reward +objectives are weighted as [wa = 0.65, wf = 0.2, wT = +0.05, wg = 0.05, wpr = 0.05]. Concurrently, the acceler- +ation demand of the driver and the vehicle velocity are +used to characterize the power reserve. The maximum +acceleration request of the driver (with 90% confidence +level) is fitted to a logistic function to continually adapt +the power reserve term as described in Section 2. +For the vehicle described above and with the parameters +given in Table 1, we designed an ECBF safety layer with +the gain vector Kα = [0.8, 2]. We found that, for the given +vehicle, the filter is effective in projecting unsafe actions +with no collisions to report throughout the training. We +observed that as the training progresses, the RL agent +learns to control the vehicle’s acceleration to align with the +driver’s request. The MPG is also improved with training, +which shows the RL agent managed to learn to achieve the + +Power reserve +fitting +[ades, V} +Memory buffer +[state,action,reward? +Load actor +{Tt, Ang} +parameters +RL +Training +ECBF filter +π(s) = {Ta,Ang} +Load critic +Q(s,a) +parameters +Actor +network +π(s) +State +ades +[z, Vrel }] +Critic +network +Driveracceleration-tracking objective in a fuel-efficient manner. +We omit the details of this training progression for space +reasons, and instead present comparative evaluations in +the next section. +The parameters we used for the simulation and training of +the driver assist RL agent are given in Table 1. +Table 1. Vehicle environment and RL hyperpa- +rameter setting +Vehicle Parameters +MPO Hyperparameters +Mass +5 - 10 tons +Actor, critic learning rate +10−5, 10−5 +Au +7.71m2 +Dual constraint +0.1 +Cd +0.08 +Retrace steps +15 +rw +0.498 +KL constraints ϵµ, ϵσ, ϵd +0.1, 0.001, 0.1 +f +0.015 +αd, αc +10 +zsr +350 +γ +0.99 +As a baseline, we also consider and simulate the same +driving scenarios without the safe RL-assist agent in +the loop (IDM only). The baseline powertrain control +generates traction torque that compensates for resistances +and fulfills the IDM driver’s requested acceleration. For +gear decisions, an optimal gear with the lowest fuel rate +is selected according to a scheme described in Yoon et al. +(2020), which is model-based and has full knowledge of the +engine fuel consumption map. Note that our RL agent has +no such knowledge of the engine’s fuel consumption map +or any of the modeled dynamics. +5. EVALUATION RESULTS AND DISCUSSIONS +The safety performance of the RL-ECBF assist is eval- +uated during and after training following the preceding +vehicle under ARTEMIS Urban drive cycle (Barlow et al., +2009), which is different from the FTP cycle used for train- +ing. Fig. 3 illustrates how the RL-ECBF assist handles the +worst case of training in which both the RL exploration +and distracted driver are the sources of unsafe actions. +The distracted driver is modeled by IDM with a τ of +2seconds that only considers the approach term (zapp) +for distance gaps less than 50m. Such a driver closes the +initial 350m separation and collides with the preceding +vehicle (red star on Fig. 3.A). However, when RL-ECBF +assist is introduced, no unsafe action is sent to the vehicle +environment due to significant ECBF projections (Fig. +3.D) and the fact that the RL becomes aware of the safety +boundaries. +We also looked at the performance of the RL-ECBF +assist system with respect to meeting driver demand and +improving fuel efficiency when paired with a conscientious +driver. To evaluate this aspect, we model a relatively +conscientious (good) driver via an IDM driver with the +approach term (zapp) activated for distance gaps less than +100m. Fig.4 and Table 2 show the performance comparison +of the good IDM driver only case (with model-based gear +and torque control) and when the same driver is assisted +by a well-trained RL with ECBF safety filter. For the IDM +driver only case, the root mean square error between the +driver demand and the actual vehicle acceleration is 0.38, +and this value is improved to 0.17 when the assist system is +introduced. In addition to enhancing driveability, the RL +agent tends to operate at higher gears, resulting in MPG +Fig. 3. Evaluation of RL assist with ECBF filter after a +few training episodes for a shortsighted IDM driver. +The red star in (A) shows collision for a case without +the RL agent (IDM only). +Fig. 4. Evaluation of a well trained RL assist with ECBF +filter for a conscientious driver +improvement of 6.34% over the baseline (IDM only with +model-based powertrain control). The RL agent eventually +learns to confine operations predominantly within the +safe set defined by the ECBF filter, as the projection +of unsafe actions usually produces suboptimal behavior +(poor reward). This fact is illustrated in Fig. 4.D in which +the ECBF projections are relatively infrequent, small and +limited to fast approaches in close proximities. +Table 2. Performance comparison between con- +scious driver with and without RL-ECBF as- +sist +IDM without +RL-ECBF assist +IDM +with +RL-ECBF +assist +MPG +6.875(−) +7.31(6.34%) +arms(m) +0.38 +0.17 +Zmean(m) +15.5 +16.7 +Zmin(m) +2 +2 + +IDM only +RL with ECBF +Preceding vehicle +Ta +collision +Driver's desire +A +10 +200 +300 +305 +310 +315 +0 +B +20 +((s/ +10 +0 +c +2.5 +yw +(zs/u)e +0.0 +2.5 +2 +D +1e4 +Torque (Nm) +1 +0 +E +8 +Gear +6 +4 +2 +0 +100 +200 +300 +400 +500 +Time (s)RL with ECBF +Driver's desire +IDM only +Preceding vehicle +Ta +A +10 +0 +300 +310 +0 +B +v(m/s)) +20 +10 +0 +c +2 +(zs/w)e +0 +1e4 +1e3 +Torque (Nm) +1 +0 +5 +580 +600 +620 +8 +Gear +6 +JH +2 +Hi +0 +200 +400 +600 +800 +1000 +Time (s)6. CONCLUSION +In this paper, a Driver-assist RL agent is formulated and +demonstrated that can assist drivers in achieving better +fuel economy and driveability. Safety is instilled into the +RL agent by filtering unsafe actions using exponential +control barrier functions (ECBF) both during training and +actual operation. The RL-ECBF assist system is trained to +maximize a multi-objective reward structure that balances +the fulfillment of the driver’s acceleration demands, fuel +economy, smooth operation and power reserve objectives. +The acceleration request profile for a given driver is +continuously adapted during the training of the RL agent +to ensure enough acceleration potential is available for the +particular driver. Evaluations on a different drive cycle +than the agent is trained on demonstrated that the RL- +ECBF assist system successfully boosted fuel economy and +driveability while ensuring safety, even when considering +distracted drivers that would cause collisions without the +driver assist system in the loop. +In future works, we intend to use randomized traffic data +and simulation for training and evaluation of the proposed +RL-ECBF agent. Furthermore, it is necessary to consider +uncertainties that are inherent in the model-based ECBF +projection approach outlined here. +REFERENCES +Abdolmaleki, A., Springenberg, J.T., Tassa, Y., Munos, +R., Heess, N., and Riedmiller, M.A. (2018). Maximum +a posteriori policy optimisation. CoRR, abs/1806.06920. +URL http://arxiv.org/abs/1806.06920. +Altman, E. (1999). +Constrained Markov Decision Pro- +cesses . +Ames, A.D., Coogan, S., Egerstedt, M., Notomista, G., +Sreenath, K., and Tabuada, P. (2019). Control barrier +functions: Theory and applications. 2019 18th European +Control Conference, ECC 2019, 3420–3431. +Ames, A.D., Grizzle, J.W., and Tabuada, P. (2014). Con- +trol barrier function based quadratic programs with +application to adaptive cruise control. +Proceedings of +the IEEE Conference on Decision and Control, 2015- +Febru(February), 6271–6278. +Barkenbus, J. (2010). Eco-driving: An overlooked climate +change initiative. Energy Policy, 38, 762–769. +Barlow, T.J., Latham, S., Mccrae, I.S., and Boulter, P.G. +(2009). A reference book of driving cycles for use in the +measurement of road vehicle emissions. +Bureau of Transportation Statistics (2017). Freight anal- +ysis framework, version 5. +Cheng, R., Orosz, G., Murray, R.M., and Burdick, J.W. +(2019). 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Gear Shift Strategies for Automotive +Transmissions. 2012. doi:10.6100/IR735458. +Ngo, V.D., Navarrete, J.A.C., Hofman, T., Steinbuch, M., +and Serrarens, A. (2013). Optimal gear shift strategies +for fuel economy and driveability. +Proceedings of the +Institution of Mechanical Engineers, Part D: Journal +of Automobile Engineering, 227(10), 1398–1413. +doi: +10.1177/0954407013491240. +Nguyen, Q. and Sreenath, K. (2016). Exponential Con- +trol Barrier Functions for enforcing high relative-degree +safety-critical constraints. Proceedings of the American +Control Conference, 2016-July(3), 322–328. +Nie, Z. and Farzaneh, H. (2020). +Adaptive cruise con- +trol for eco-driving based on model predictive con- +trol algorithm. +Applied Sciences, 10, 5271. +doi: +10.3390/app10155271. +Srinivasan, K., Eysenbach, B., Ha, S., Tan, J., and Finn, +C. (2020). Learning to be Safe: Deep RL with a Safety +Critic. 1–16. +Thananjeyan, B., Balakrishna, A., Nair, S., Luo, M., +Srinivasan, K., Hwang, M., Gonzalez, J.E., Ibarz, J., +Finn, C., and Goldberg, K. (2021). Recovery RL: Safe +Reinforcement Learning with Learned Recovery Zones. +IEEE Robotics and Automation Letters, 6(3). +Treiber, M., Hennecke, A., and Helbing, D. (2000). Con- +gested traffic states in empirical observations and micro- +scopic simulations. Physical Review E, 62, 1805–1824. +doi:10.1103/PhysRevE.62.1805. +Yoon, D.D., Ayalew, B., Ivanco, A., and Loiselle, K. +(2020). +Predictive kinetic energy management for an +add-on driver assistance eco-driving of heavy vehicles. +IET Intelligent Transport Systems, 14(13), 1824–1834. + diff --git a/jdAyT4oBgHgl3EQf_Po5/content/tmp_files/load_file.txt b/jdAyT4oBgHgl3EQf_Po5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..691a92400bb44dc9687c1eb4dd89f2ab75aec2be --- /dev/null +++ b/jdAyT4oBgHgl3EQf_Po5/content/tmp_files/load_file.txt @@ -0,0 +1,467 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf,len=466 +page_content='Safe Reinforcement Learning for an Energy-Efficient Driver Assistance System Habtamu Hailemichael ∗ Beshah Ayalew ∗ Lindsey Kerbel ∗ Andrej Ivanco ∗∗ Keith Loiselle ∗∗ ∗ Automotive Engineering, Clemson University, Greenville, SC 29607, USA (hhailem, beshah, lsutto2)@clemson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' ∗∗ Allison Transmission Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', One Allison Way, Indianapolis, IN, 46222, USA (andrej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='ivanco, keith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='loiselle)@allisontransmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='com Abstract: Reinforcement learning (RL)-based driver assistance systems seek to improve fuel consumption via continual improvement of powertrain control actions considering experiential data from the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' However, the need to explore diverse experiences in order to learn optimal policies often limits the application of RL techniques in safety-critical systems like vehicle control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In this paper, an exponential control barrier function (ECBF) is derived and utilized to filter unsafe actions proposed by an RL-based driver assistance system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The RL agent freely explores and optimizes the performance objectives while unsafe actions are projected to the closest actions in the safe domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The reward is structured so that driver’s acceleration requests are met in a manner that boosts fuel economy and doesn’t compromise comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The optimal gear and traction torque control actions that maximize the cumulative reward are computed via the Maximum a Posteriori Policy Optimization (MPO) algorithm configured for a hybrid action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The proposed safe-RL scheme is trained and evaluated in car following scenarios where it is shown that it effectively avoids collision both during training and evaluation while delivering on the expected fuel economy improvements for the driver assistance system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Keywords: RL driver-assist, Safe reinforcement learning, Safety filtering, Control barrier functions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' INTRODUCTION Reliable, safe, and efficient commercial vehicles are es- sential for the transportation industry to have a positive impact on the environment, the economy, and road safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Given the estimated increase in freight demand of 16% by 2030 (Bureau of Transportation Statistics, 2017), there is clearly a need and an opportunity to reduce emission and fuel usage as more of these vehicles get on the roads to meet this demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Furthermore, ensuring safety via accident prevention is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Advanced driver assistance systems (ADAS) such as emergency braking, adaptive cruise control (ACC), and lane keeping assist have been developed to primarily address the safety concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' More advanced systems additionally aid the driver with ecologi- cal (fuel saving) driving behaviors such as reduced braking and accelerations (Barkenbus, 2010), optimizing velocity profiles for ACC (Maamria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Nie and Farzaneh, 2020) and optimizing gear shifting (Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' As typical commercial routes include frequent stopping and starting along with various required speeds, the ACC approach may be cumbersome for a driver to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2020), a driver assistance implementation is pro- posed that uses radar information and motion models to directly modulate the torque request to the power- train/braking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' To this end, an MPC scheme is em- ployed to optimize traction/braking torque/power while tracking the driver’s desired acceleration and maintaining a safe distance to a leading vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In Kerbel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2022), a similar driver assistance objective is pursued in a model- free reinforcement learning approach to learn both optimal gear selection and torque request using fuel usage and other reward signals in the vehicle’s experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Although this study demonstrated a fuel consumption improvement of up to 12%, it did not include provisions to guarantee collision avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Unlike typical optimal control schemes such as MPC, where hard constraints are set based on a dynamic model, it is generally difficult to enforce such constraints in RL controllers where learning an optimal policy requires exploration of different actions and states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' However, unlimited exploration is unacceptable for safety- critical systems such as vehicle control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In this paper, we construct a driver-assist RL agent that targets fuel efficiency and driver accommodation and incorporates el- ements that ensure safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Different approaches are proposed to properly constrain the exploration of the RL agent within a safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In Li and G¨orges (2020), a supervisor is used to simply enforce (override) the gear and engine speed constraints to control the transmission, yet the RL agent never really learns these limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Often, reward shaping approaches are utilized by assigning a penalty to safety violations that discourage policies leading to constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Since the RL agent with reward shaping learns the penalties only after experiencing them, this approach does not guar- antee safety, especially during initial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Another approach to enforcing safety is to pose the problem as a constrained Markov decision process (CMDP) where a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='00904v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='RO] 3 Jan 2023 constraint cost is assigned for each state-action pair and the RL agent learns to keep the discounted constraint cost over the horizon below a certain threshold (Altman, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Many implementations of the CMDP then involve joint optimizations of the main performance task and the constraint reward, and this entails trade-offs between safety and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In this work, we seek to somewhat decouple the two goals by adopting what is known as a safety filtering approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' This approach configures the RL agent to focus on maximizing performance (reward), while a safety layer/filter is designed to project the outputs of the RL agent onto a safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Although the filter does not typically interfere with the inner workings of the RL agent, it will influence performance as it often determines the extent of the safe set and subsequent interactions of the RL agent with the system under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Evaluations of the proposed actions in the safety layer could be based on learning constraints (Dalal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2018) and safety indexes (Thananjeyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Srinivasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2020) from offline data or using a dynamic model of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Of the dynamic model-based approaches to safety filtering, control barrier functions (CBF) provide scalable and com- putationally light safety filters (Li, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' A CBF applies hard safety constraints by forcing the system to operate in the invariant safe-set defined by a super-level set of a continuously differentiable function h(x) : Rn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The actions selected by the RL agent are projected into the safe set in such a manner that the proposed actions are minimally modified (Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2019), and no unsafe actions are passed to the controlled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' One could come up with handcrafted CBFs considering the dynamics of the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' a case in point is the relationship between the maximum deceleration available to the vehicle and the distance gap in the collision avoidance problems (Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' For high relative degree nonlinear systems, as in the present application, tailored CBFs known as exponential barrier functions (ECBF) have been proposed (Nguyen and Sreenath, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In this paper, we derive a specific ECBF structure that works in conjunction with the RL driver-assist agent in order to take explicit consideration of inertia effects which are relevant for the safety of commercial vehicles in traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The main performance goal of the driver-assist RL agent is given by a multi-objective reward function that is struc- tured to balance driver accommodation, fuel economy, and smooth vehicle operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In addition, driveability is encouraged by introducing an additional reward for reserve power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In this regard, Ngo (2012) characterizes acceleration potential at a given speed by merely analyzing different standard drive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In this paper, we propose to learn the power reserve reward to customize the vehi- cle’s response to the driving conditions and the driver’s tendencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' To summarise, the contributions of this paper are: 1) formulation of a driver assist RL agent configured for reward optimal gear selection and torque control of a commercial vehicle, 2) derivation of an ECBF safety filter to work with this RL agent and 3) demonstration of the potential of learning power reserve attributes to further customize the system to the driver and driving conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The rest of the paper is organized as follows: Section 2 discusses the vehicle model and the driver-assist RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Proposed safe RL-based Eco-Assist system set up Section 3 discusses the design of the safety filter and the subsequent projection of the output of the RL agent onto the safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Section 4 presents simulation and training settings and results are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Finally, Section 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' VEHICLE ENVIRONMENT AND RL CONTROLLER A schematic of the proposed RL-based driver assistance system, including the safety filter, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In this section, we detail the different computational components of the Driver-Assist RL agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' the next section deals with the ECBF safety filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The vehicle-driver-environment is modeled as Markov de- cision process (MDP) with state s, actions a, rewards r and a discount factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The states are included in s = {vl, vrel, ades, a, z, ng, mv, θ, f} which, respectively, are the ego vehicle velocity, the relative velocity between the preceding and ego vehicle, the driver demanded acceler- ation, the actual vehicle acceleration, the separation dis- tance with the preceding vehicle, transmission gear, mass of the vehicle, road grade and a flag to alert if a preceding vehicle is the sensing range of the ego-vehicle’s radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The RL controller is designed to maximize the vehicle’s perfor- mance objectives through wheel traction torque Tt control and gear change selection ∆ng, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', the action vector is: a = {Tt, ∆ng}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The velocity and the wheel traction torque are propagated back to calculate the engine torque and speed using the transmission ratio of the current gear and the final drive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' A fuel rate map is then utilized to solve for the fuel consumption at the given engine torque and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The reward function, given by (1) below, is structured to capture the performance objectives of the driver-assist RL- agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The major objective of the RL agent is to fulfill the driver’s acceleration request, and consequently, an accel- eration error term is given a higher weight, wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Through the fuel rate reward term, weighted by wf, the RL agent is encouraged to operate the engine at fuel-efficient operating points while fulfilling the driver-demanded acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Smooth torque changes are weighted with wt, and gear hunting and the associated rough vehicle operation are mitigated by including a shifting frequency penalty term weighted by wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Note relevant reward signals are normal- ized by their corresponding maximum values as noted by the max subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' ˙mf and ∆Tt are the fuel rate and torque change respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' r = wa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='1 |a−ades| ades,max + wf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='1 ˙ mf mf,max + wT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='1 |∆Tt| Tt,max + wg0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='1 |∆ng| ng,max + rpr, (1) Driver ng ades Eco-Assist Ta T RL Agent ECBF Filter statewhere rpr models the power reserve reward term that accounts for enhanced driveability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' We define it as: rpr = wpr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='1 Pres,req−Pres Pres,req if Pres < Pres,req, else rpr = wpr, where wpr is the corresponding weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Pres is the actual available power which is given in terms of engine speed and engine torque as Pres = (Te,max(ω) − Te)ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' and Pres,req is the required power reserve which we discuss next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' To adapt the Pres,req with the different acceleration de- mands in different driving conditions, Ngo (2012) models acceleration potential as varying with the vehicle veloc- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' To this end, the speed of the vehicle is discretized and acceleration requests for each speed level in multiple standard cycles are collected to be fitted in a cumulative probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The maximum acceleration at a given design confidence level (usually 90%) is taken as the required acceleration potential at that velocity, areq(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Given areq(v), the required power reserve is then modeled with Pres,req = mvvareq(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In our work, rather than using acceleration data from standard drive cycles, we propose using the data generated by the driver in the prevailing driving conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The demanded acceleration of the driver is continuously fitted to get areq(v) that adapts to the driver’s demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Observing that polynomial fits of areq suggested in Ngo (2012) lead to overfitting issues when used with driver generated training data, we instead use a logistic function that is easier to parametrize and learn: areq(v) = k1 1 + k2k−v 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2) Next, we briefly describe the framework we adopted for training the driver assist RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The states, control actions, next states, and associated rewards are continu- ously stored in the memory buffer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' We use actor-critic architecture proposed by Kerbel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2022) that utilizes the off-policy algorithm known as maximum posteriori optimization (MPO) (Abdolmaleki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Neunert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2020) for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Even if it is possible to use other state of the art algorithms, we use MPO for its sample efficiency and robustness to hyper-parameters as well as ease of use with the hybrid action space for the present problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The algorithm starts with a policy evaluation step where a critic network approximates the state-action- value (Q-value) for the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' A squared loss function is minimized between the current Q-value and an estimated target Q-value, Qtarget(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' For this study, we adopted the Retrace algorithm, known for efficiency and stability, as described in Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2016) for our target Q-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' For policy improvement (actor network), the MPO algo- rithm uses an expectation-maximization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' By tak- ing samples from the memory buffer, we construct a non- parametric policy q that maximizes Eq[Qθk(s, a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' max q Eq(a|s)[Qθk(s, a)] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Eµ(s) [KL (q(a|s)||πθk(a|s))] < ϵ, (3) where µ(s) is the visitation distribution given in the replay buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Then a new parametric policy πθ is fitted to q with a Kullback–Leibler (KL) divergence constraint to limit ex- cessive deviations from the current policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The parameters of the actor network are updated via a gradient-based optimization in Adam solver (Kingma and Ba, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' More detailed explanation of the MPO algorithm can be found in Neunert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2020) and Abdolmaleki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Further implementation details for the present application can also be found in our straight RL implementation in Kerbel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' As noted above, other state of the art RL training algorithms can also be applied for the driver-assist RL agent and this is independent of the safety filter discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' EXPONENTIAL CBF SAFETY FILTER In this section, we give the derivation of the ECBF filter for our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' We start with a brief review of the definition of CBF and ECBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' We refer readers to Nguyen and Sreenath (2016) for more detailed discussions on these topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Consider a nonlinear control affine system: ˙x = f (x) + g (x) u, (4) where f and g are locally Lipschitz, x ∈ Rn is the state of the system, u ∈ Rm is the control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Assume a safe set defined by C = {x ∈ Rn|h (x) ≥ 0}, where h : Rn → R is a continuously differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Then h is a CBF if there exists an extended class κ∞ function α such that for all x ∈ Int (C) = {x ∈ Rn : h (x) > 0} : sup u∈U [Lfh (x) + Lgh (x) u] ≥ −α (h (x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (5) The fact that h is a CBF ensures the safe set C is forward invariant and we are able to guarantee safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' ECBFs use input-output (IO) linearization of nonlinear systems with relative degree r in order to generate CBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' As detailed in Nguyen and Sreenath (2016), the new virtual linear system (after IO linearization) has state variables ηb := [h(x), ˙h(x), · · ·, hr(x)]T , input µ and output h (x) : ˙ηb = Fηb (x) + Gµ, h(x) = Cηb, (6) where F and G are matrices representing an integrator chain, C = [1, 0, · · · , 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The control action for the virtual linear system µ is the rth derivative of the control output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' µ = Lr fh(x) + LgLr−1 f h(x)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' When µ is set by state feedback control with gain Kα as µ = −Kαηb, the control output evolves with time as h (x (t)) = Ce(F −GKα)tηb (x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' For initial condition h (x0) > 0, by imposing µ > −Kαηb (x), it possible to guarantee h(x(t)) ≥ Ce(F −GKα )tηb (x0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' This relationship leads to the definition of ECBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Considering the dynamic system (4) and the set C = {x ∈ Rn| h (x) ≥ 0}, h (x) is an ECBF if there exists Kα ∈ Rr×1 sup � Lr fh (x) + LgLr−1 f h (x) u � ≥ −Kαηb (x) , ∀ x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (7) If Kα makes the closed-loop system matrix F − GKα stronger than Hurwitz and for favorable initial conditions choosing, µ ≥ −Kαηb (x) guarantees h (x) is an ECBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Pole placement strategies of linear feedback control can then be used to design the ECBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' We construct the collision avoidance model with the sep- aration distance z, the velocity of the ego vehicle vh and velocity of the leading vehicle vl as state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The model follows: ˙z = vl − vh, (8a) ˙vl = al, (8b) ˙vh = Tt rwmv − Fr (vh, mv, θ) mv , (8c) Fr = ρAcdv2 h 2 + mvgf cos θ + mvg sin θ, (9) where Fr is the total resistance force that includes grav- itational, rolling friction and aerodynamics resistances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Tt, cd, f, θ, mv, ρ, Av, rw, al are the traction torque at the wheels, aerodynamic coefficient, rolling resistance coeffi- cient, road grade, the mass of the vehicle, the density of air, the frontal area of the vehicle, the radius of the wheels, and acceleration of the leading vehicle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Having affine dynamics, the above state space represen- tation could also be separated into unactuated dynamics f (x) and actuated dynamic g (x) components when writ- ten as (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' With the choice of a minimum inter-vehicle distance objective z0, a natural choice is h (x) = z − z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Input-output linearization is then employed to transform the nonlinear dynamics into a virtual linear system as in (6) and following the accompanying discussions above, with the feedback gain Kα = [kα1, kα2], we have: ˙h(x) = vl − vh, (10) µ = ¨h (x) = Fr (vh, mv, θ) mv + al − Tt mvrw , (11) −Kαηb (x) = −kα1h(x) − kα2 ˙h(x) = −kα1 (z − z0) − kα2 (vl−vh) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (12) Using these with (7), we arrive at the ECBF filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The traction torque actions proposed by the RL agent are then passed through this safety filter before being sent to the vehicle environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 1, the ECBF filter enforces safety by projecting the action proposed by the RL agent Ta(s) to the safe control traction torque Tt in a way that introduces minimal change, as given by the QP problem below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' T ∗ t = arg min Tt 1 2 ∥Tt − Ta(s)∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' al + Fr (vh, mv, θ) mv − Tt mvrw ≥ −kα1 (z − z0) − kα2 (vl − vh) (13) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' SIMULATION AND TRAINING SETTINGS Simulations of a medium-duty truck with a 10-speed au- tomated manual transmission (AMT) are used to demon- strate the workings and performance of RL driver-assist agent with the ECBF safety filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The actor and critic are represented with deep neural networks with three hidden layers, and each layer consists of 256 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The RL controller is trained in a scenario in which the truck driven by an imperfect driver follows a preceding vehi- cle under Federal Test Procedure (FTP-75) drive cycle (Barlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The agent is trained for a weight range of 5 to 10 tons, as commercial vehicles have to operate in significant load fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In order to help capture different driving experiences, the training data is randomized by adding noise to the velocity profile of the preceding vehicle, using random initial separation distance and road grade, and manipulating the parameters of the driver model, which in this work is taken as the intelligent Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' RL training setup driver model (IDM)(Treiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2000) described by the equations below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The desired minimum gap for IDM is calculated as in (15), in which the approach term zapp (16) dominates as the host vehicle approaches the preceding vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' By only adding zapp for distance gaps closer than a certain threshold and ignoring this term beyond that, we emulate a distracted driver requesting unsafe actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' a = amax � 1 − � v v0 �4 − �z∗ (v, vrel) z �2 � , (14) z∗ (v, vrel) = z0 + τv − zapp, (15) zapp = vrelv 2√amaxb, (16) where amax is the maximum acceleration, b is a comfort- able deceleration level, τ is the time headway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In each simulation step, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 2, the IDM driver requests acceleration based on the distance gap and veloc- ity of the preceding vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' To fulfill the driver’s demand, the actor network outputs {Ta, ∆ng} and the proposed actions are filtered by the ECBF safety layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The safe actions are then implemented in the vehicle environment and rewards are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' As mentioned previously, the reward aims to fulfill the driver’s acceleration demand in a manner that promotes fuel economy, driveability, and smooth vehicle operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Accordingly, the reward objectives are weighted as [wa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='65, wf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='2, wT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='05, wg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='05, wpr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Concurrently, the acceler- ation demand of the driver and the vehicle velocity are used to characterize the power reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The maximum acceleration request of the driver (with 90% confidence level) is fitted to a logistic function to continually adapt the power reserve term as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' For the vehicle described above and with the parameters given in Table 1, we designed an ECBF safety layer with the gain vector Kα = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='8, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' We found that, for the given vehicle, the filter is effective in projecting unsafe actions with no collisions to report throughout the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' We observed that as the training progresses, the RL agent learns to control the vehicle’s acceleration to align with the driver’s request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The MPG is also improved with training, which shows the RL agent managed to learn to achieve the Power reserve fitting [ades, V} Memory buffer [state,action,reward?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Load actor {Tt, Ang} parameters RL Training ECBF filter π(s) = {Ta,Ang} Load critic Q(s,a) parameters Actor network π(s) State ades [z, Vrel }] Critic network Driveracceleration-tracking objective in a fuel-efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' We omit the details of this training progression for space reasons, and instead present comparative evaluations in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The parameters we used for the simulation and training of the driver assist RL agent are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Vehicle environment and RL hyperpa- rameter setting Vehicle Parameters MPO Hyperparameters Mass 5 - 10 tons Actor, critic learning rate 10−5, 10−5 Au 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='71m2 Dual constraint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='1 Cd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='08 Retrace steps 15 rw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='498 KL constraints ϵµ, ϵσ, ϵd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='1 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='015 αd, αc 10 zsr 350 γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='99 As a baseline, we also consider and simulate the same driving scenarios without the safe RL-assist agent in the loop (IDM only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The baseline powertrain control generates traction torque that compensates for resistances and fulfills the IDM driver’s requested acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' For gear decisions, an optimal gear with the lowest fuel rate is selected according to a scheme described in Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2020), which is model-based and has full knowledge of the engine fuel consumption map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Note that our RL agent has no such knowledge of the engine’s fuel consumption map or any of the modeled dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' EVALUATION RESULTS AND DISCUSSIONS The safety performance of the RL-ECBF assist is eval- uated during and after training following the preceding vehicle under ARTEMIS Urban drive cycle (Barlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=', 2009), which is different from the FTP cycle used for train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 3 illustrates how the RL-ECBF assist handles the worst case of training in which both the RL exploration and distracted driver are the sources of unsafe actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The distracted driver is modeled by IDM with a τ of 2seconds that only considers the approach term (zapp) for distance gaps less than 50m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Such a driver closes the initial 350m separation and collides with the preceding vehicle (red star on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' However, when RL-ECBF assist is introduced, no unsafe action is sent to the vehicle environment due to significant ECBF projections (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='D) and the fact that the RL becomes aware of the safety boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' We also looked at the performance of the RL-ECBF assist system with respect to meeting driver demand and improving fuel efficiency when paired with a conscientious driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' To evaluate this aspect, we model a relatively conscientious (good) driver via an IDM driver with the approach term (zapp) activated for distance gaps less than 100m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='4 and Table 2 show the performance comparison of the good IDM driver only case (with model-based gear and torque control) and when the same driver is assisted by a well-trained RL with ECBF safety filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' For the IDM driver only case, the root mean square error between the driver demand and the actual vehicle acceleration is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='38, and this value is improved to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='17 when the assist system is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In addition to enhancing driveability, the RL agent tends to operate at higher gears, resulting in MPG Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Evaluation of RL assist with ECBF filter after a few training episodes for a shortsighted IDM driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The red star in (A) shows collision for a case without the RL agent (IDM only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Evaluation of a well trained RL assist with ECBF filter for a conscientious driver improvement of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='34% over the baseline (IDM only with model-based powertrain control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The RL agent eventually learns to confine operations predominantly within the safe set defined by the ECBF filter, as the projection of unsafe actions usually produces suboptimal behavior (poor reward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' This fact is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='D in which the ECBF projections are relatively infrequent, small and limited to fast approaches in close proximities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Performance comparison between con- scious driver with and without RL-ECBF as- sist IDM without RL-ECBF assist IDM with RL-ECBF assist MPG 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='875(−) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='31(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='34%) arms(m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='17 Zmean(m) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content="7 Zmin(m) 2 2 IDM only RL with ECBF Preceding vehicle Ta collision Driver's desire A 10 200 300 305 310 315 0 B 20 ((s/ 10 0 c 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='5 yw (zs/u)e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content="5 2 D 1e4 Torque (Nm) 1 0 E 8 Gear 6 4 2 0 100 200 300 400 500 Time (s)RL with ECBF Driver's desire IDM only Preceding vehicle Ta A 10 0 300 310 0 B v(m/s)) 20 10 0 c 2 (zs/w)e 0 1e4 1e3 Torque (Nm) 1 0 5 580 600 620 8 Gear 6 JH 2 Hi 0 200 400 600 800 1000 Time (s)6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' CONCLUSION In this paper, a Driver-assist RL agent is formulated and demonstrated that can assist drivers in achieving better fuel economy and driveability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Safety is instilled into the RL agent by filtering unsafe actions using exponential control barrier functions (ECBF) both during training and actual operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The RL-ECBF assist system is trained to maximize a multi-objective reward structure that balances the fulfillment of the driver’s acceleration demands, fuel economy, smooth operation and power reserve objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' The acceleration request profile for a given driver is continuously adapted during the training of the RL agent to ensure enough acceleration potential is available for the particular driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Evaluations on a different drive cycle than the agent is trained on demonstrated that the RL- ECBF assist system successfully boosted fuel economy and driveability while ensuring safety, even when considering distracted drivers that would cause collisions without the driver assist system in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' In future works, we intend to use randomized 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' Predictive kinetic energy management for an add-on driver assistance eco-driving of heavy vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} +page_content=' IET Intelligent Transport Systems, 14(13), 1824–1834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQf_Po5/content/2301.00904v1.pdf'} diff --git a/jtAyT4oBgHgl3EQfx_nV/content/tmp_files/2301.00678v1.pdf.txt b/jtAyT4oBgHgl3EQfx_nV/content/tmp_files/2301.00678v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4c3919d0aac93b22afc055c5561feb102195650 --- /dev/null +++ b/jtAyT4oBgHgl3EQfx_nV/content/tmp_files/2301.00678v1.pdf.txt @@ -0,0 +1,4057 @@ +arXiv:2301.00678v1 [math.CA] 2 Jan 2023 +DPSU-22-3 +Another Type of Forward and Backward +Shift Relations for Orthogonal Polynomials +in the Askey Scheme +Satoru Odake +Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan +Abstract +The forward and backward shift relations are basic properties of the (basic) hyperge- +ometric orthogonal polynomials in the Askey scheme (Jacobi, Askey-Wilson, q-Racah, +big q-Jacobi etc.) and they are related to the factorization of the differential or differ- +ence operators. Based on other factorizations, we obtain another type of forward and +backward shift relations. +1 +Introduction +The (basic) hypergeometric orthogonal polynomials in the Askey scheme satisfy second or- +der differential or difference equations and the forward and backward shift relations are +their basic properties [1, 2]. The orthogonal polynomials in the Askey scheme provide us +with exactly solvable quantum mechanical models. Conversely, we can use the quantum +mechanical formulation as a tool to investigate orthogonal polynomials [3]. For example, the +forward and backward shift relations are a consequence of the shape invariance [4, 5, 3], and +the multi-indexed orthogonal polynomials ([6, 7, 8] etc.) are found by using the quantum +mechanical formulation. The Schr¨odinger equation is a second order differential equation +for ordinary quantum mechanics (oQM) and a second order difference equation for discrete +quantum mechanics (dQM). There are two types of dQM, dQM with pure imaginary shifts +(idQM) and dQM with real shifts (rdQM) [3]. The coordinate x for oQM and idQM is +continuous and that for rdQM is discrete. +The forward and backward shift relations are related to the factorization of the Hamil- +tonian. +Recently another factorization of the Hamiltonian was found in a study of the +state-adding Darboux transformations for the finite rdQM systems [9]. It gives another for- +ward and backward shift relations for the orthogonal polynomials appearing in the finite + +rdQM systems (q-Racah etc.), which were called the forward and backward x-shift rela- +tions [9]. In this paper, we investigate whether such new factorization and forward and +backward shift relations exist for other orthogonal polynomials. In addition to the finite +rdQM systems (q-Racah etc.), we examine the oQM systems (Jacobi etc.), the idQM sys- +tems (Askey-Wilson etc.), the semi-infinite rdQM systems (q-Meixner etc.) and the rdQM +systems with the Jackson integral type measure (big q-Jacobi etc.). We call the last category +rdQMJ. The quantum mechanical formulation of the rdQMJ systems needs two component +formalism [10]. We consider all the polynomials in chapter 9 and 14 of [2] and the dual +quantum q-Krawtchouk polynomial. +This paper is organized as follows. The orthogonal polynomials in the Askey scheme and +their second order differential or difference equations are recalled in section 2. The forward +and backward shift relations are reviewed in section 3. Section 4 is the main part of this paper +and new factorization and new forward and backward shift relations are presented. Section +5 is for a summary and comments. In Appendix A the data for the orthogonal polynomials +are given. We comment that the oQM systems described by the Bessel and pseudo Jacobi +polynomials are the Morse potential and the hyperbolic symmetric top II, respectively. We +also comment on an infinite sum orthogonality relations for the Stieltjes-Wigert polynomial. +2 +Orthogonal Polynomials in the Askey Scheme +In this section we fix the notation and recall the second order differential or difference +equations for the orthogonal polynomials in the Askey scheme. +In our quantum mechanical formulation [3], the orthogonal polynomials in the Askey +scheme are expressed as +ˇPn(x; λ) +def += Pn +� +η(x; λ); λ +� +: a polynomial of degree n in η(x; λ) +(2.1) +for n ∈ Z≥0 and ˇP−1(x; λ) +def += P−1(η(x; λ); λ) +def += 0. Here x is a coordinate of quantum me- +chanical system and η(x) is a sinusoidal coordinate [11], and λ = (λ1, λ2, . . .) are parameters, +whose dependence is expressed as f = f(λ) and f(x) = f(x; λ). The parameter q is 0 < q < 1 +and qλ stands for q(λ1,λ2,...) = (qλ1, qλ2, . . .), and we omit writing q-dependence. The data for +the orthogonal polynomials treated in this paper are given in Appendix A. We remark that +the polynomials ˇPn(x) in § A.3.1, whose orthogonality holds for n = 0, 1, . . . , N, are ill-defined +for n > N due to the normalization condition (A.34). So we should replace ˇPn(x) (n > N) +2 + +in § A.3.1 with the monic version ˇP monic +n +(x; λ) +def += cn(λ)−1 ˇPn(x; λ) (see (A.35)) in Theorem 1, +2.1 and 3.3 (with the replacements fn(λ) → f monic +n +(λ) = fn(λ)cn(λ)−1cn−1(λ + δ), etc.). +The Schr¨odinger equations of oQM and dQM systems are second order differential and +difference equations, respectively. By the similarity transformation in terms of the ground +state wavefunction, the similarity transformed Hamiltonian � +H(λ) is a second order differen- +tial or difference operator acting on the eigenpolynomials ˇPn(x; λ) [3], +oQM : +� +H(λ) +def += − d2 +dx2 − 2dw(x; λ) +dx +d +dx +� += −4c2(η) d2 +dη2 − 4c1(η; λ) d +dη +� +, +(2.2) +idQM : +� +H(λ) +def += V (x; λ)(eγp − 1) + V ∗(x; λ)(e−γp − 1), +(2.3) +rdQM : +� +H(λ) +def += B(x; λ)(1 − e∂) + D(x; λ)(1 − e−∂), +(2.4) +where the functions w(x), c1(η), c2(η), V (x), B(x) and D(x) are given in Appendix A. For +oQM, the coordinate x is a continuous variable and the Hamiltonian H(λ) is +H(λ) = − d2 +dx2 + U(x; λ), +U(x; λ) +def += +�dw(x; λ) +dx +�2 ++ d2w(x; λ) +dx2 +. +(2.5) +While the orthogonality relations of ˇPn(x) for B and pJ cases hold only for a finite number +of n (see (A.13)), we consider all n ∈ Z≥0, because we consider only differential equations +(or relations) in this paper. For idQM, the coordinate x is a continuous variable and the +momentum p is p = −i d +dx, and γ is a real constant (γ = 1 for non q-polynomial, γ = log q +for q-polynomial). The operator eαp (α: constant) is a shift operator, eαpf(x) = f(x − iα). +The ∗-operation on an analytic function f(x) = � +n anxn (an ∈ C) is defined by f ∗(x) = +� +n a∗ +nxn, in which a∗ +n is the complex conjugation of an. For rdQM, the Schr¨odinger equation +is a matrix eigenvalue problem. The similarity transformed Hamiltonian � +H = ( � +Hx,y) is a +matrix labeled by the coordinate x, which takes discrete values in {0, 1, . . . , N} or Z≥0. +In this paper, however, we treat x as a continuous variable x ∈ R, because we only deal +with difference equations (or relations). The operators e±∂ are shift operators e±∂ = e± d +dx, +e±∂f(x) = f(x ± 1). We consider ˇPn(x) with all n ∈ Z≥0 even for finite systems. +For rdQMJ (the rdQM system with Jackson integral type measure such as the big q- +Jacobi polynomial), its quantum mechanical formulation needs two component formalism +with two sinusoidal coordinates η(±)(x; λ) [10]. Since only difference equations (or relations) +are considered in this paper, we use η only (we do not use x) and treat η as a continuous +variable η ∈ R. The similarity transformed Hamiltonian � +HJ(λ) is a second order difference +3 + +operator acting on the eigenpolynomials Pn(η; λ) [10], +rdQMJ : +� +HJ(λ) +def += BJ(η; λ)(1 − qη d +dη ) + DJ(η; λ)(1 − q−η d +dη ), +(2.6) +where the functions BJ(η) and DJ(η) are given in § A.4. The operators q±η d +dη are q-shift +operators, q±η d +dη f(η) = f(q±1η). +The orthogonal polynomials in the Askey scheme studied in this paper have the following +property. +Theorem 1 [1, 2] The polynomials in Appendix A satisfy the second order differential or +difference equations for n ∈ Z≥0, +oQM, idQM, rdQM : +� +H(λ) ˇPn(x; λ) = En(λ) ˇPn(x; λ), +(2.7) +rdQMJ : � +HJ(λ)Pn(η; λ) = En(λ)Pn(η; λ). +(2.8) +We remark that the constant terms of � +H and � +HJ are chosen such that E0 = 0. For idQM, +the relation (2.7) is invariant under the ∗-operation. +3 +Forward and Backward Shift Relations +The similarity transformed Hamiltonians � +H(λ) (2.2)–(2.4) are factorized as +� +H(λ) = B(λ)F(λ), +(3.1) +where the forward and backward shift operators, F(λ) and B(λ), are defined by [4, 5], +oQM : F(λ) +def += cF +�dη(x) +dx +�−1 d +dx +� += cF +d +dη +� +, +(3.2) +B(λ) +def += −c−1 +F +�dη(x) +dx +d +dx + 4c1 +� +η(x); λ +�� � += −4c−1 +F +� +c2(η) d +dη + c1(η; λ) +�� +, +(3.3) +idQM : F(λ) +def += iϕ(x; λ)−1(e +γ +2 p − e− γ +2 p), +(3.4) +B(λ) +def += −i +� +V (x; λ)e +γ +2 p − V ∗(x; λ)e− γ +2 p� +ϕ(x; λ), +(3.5) +rdQM : F(λ) +def += B(0; λ)ϕ(x; λ)−1(1 − e∂), +(3.6) +B(λ) +def += +1 +B(0; λ) +� +B(x; λ) − D(x; λ)e−∂� +ϕ(x; λ), +(3.7) +and cF and ϕ(x; λ) are given in Appendix A. Since w(x), B(x), D(x) and V (x) satisfy +�dw(x; λ) +dx +�2 +− d2w(x; λ) +dx2 += +�dw(x; λ + δ) +dx +�2 ++ d2w(x; λ + δ) +dx2 ++ E1(λ), +(3.8) +4 + +V (x − iγ +2; λ) +V (x; λ + δ) = κ +ϕ(x; λ) +ϕ(x − iγ; λ), +(3.9) +V (x + iγ +2; λ) + V ∗(x − iγ +2; λ) = κ +� +V (x; λ + δ) + V ∗(x; λ + δ) +� +− E1(λ), +(3.10) +B(x + 1; λ) +B(x; λ + δ) = κ +ϕ(x; λ) +ϕ(x + 1; λ), +D(x; λ) +D(x; λ + δ) = κ +ϕ(x; λ) +ϕ(x − 1; λ), +(3.11) +B(x; λ) + D(x + 1; λ) = κ +� +B(x; λ + δ) + D(x; λ + δ) +� ++ E1(λ), +(3.12) +(κ and δ are given in Appendix A), we obtain +F(λ)B(λ) = κB(λ + δ)F(λ + δ) + E1(λ), +(3.13) +which is the (similarity transformed) shape invariance condition. The constants fn(λ) and +bn(λ) given in Appendix A satisfy +En(λ) = fn(λ)bn−1(λ) (n ∈ Z≥0), +(3.14) +and the energy eigenvalues En satisfy +En+1(λ) = κEn(λ + δ) + E1(λ) (n ∈ Z≥0). +(3.15) +Corresponding to the factorization (3.1), the shape invariance combined with the Crum’s +theorem give the following relations [4, 5, 3]. +Theorem 2.1 [2] For the polynomials in Appendix A.1, A.2 and A.3, the forward and back- +ward shift relations hold: +F(λ) ˇPn(x; λ) = fn(λ) ˇPn−1(x; λ + δ) (n ∈ Z≥0), +(3.16) +B(λ) ˇPn−1(x; λ + δ) = bn−1(λ) ˇPn(x; λ) (n ∈ Z≥1). +(3.17) +For idQM, the relations (3.16)–(3.17) are invariant under the ∗-operation. +The similarity transformed Hamiltonians � +HJ(λ) (2.6) are factorized as +� +HJ(λ) = BJ(λ)F J(λ). +(3.18) +Here the forward and backward shift operators, F J(λ) and BJ(λ), are defined by [10] +rdQMJ : F J(λ) +def += A +qη(1 − qη d +dη ), +(3.19) +BJ(λ) +def += +� +BJ(η; λ) − DJ(η; λ)q−η d +dη �qη +A , +(3.20) +5 + +where the constant A is given by +A = + + + + + + + +−DJ(1; λ) +: bqJ, bqL, qL, SW +−a +: ASCI +1 +: dqHeI +q +: dqHeII +. +(3.21) +We can show that BJ(η) and DJ(η) satisfy +qBJ(qr−1η; λ) = κBJ(η; λ + δ), +q−1DJ(r−1η; λ) = κDJ(η; λ + δ), +(3.22) +BJ(r−1η; λ) + DJ(qr−1η; λ) = κ +� +BJ(η; λ + δ) + DJ(η; λ + δ) +� ++ E1(λ), +(3.23) +where r is given by +r = + + + +q +: bqJ, bqL, dqHeII, qL +1 +: ASCI, dqHeI +q2 +: SW +. +(3.24) +Therefore we obtain +F J(λ)BJ(λ) +��� +η→r−1η= κBJ(λ + δ)F J(λ + δ) + E1(λ), +(3.25) +which is the (similarity transformed) shape invariance condition. The constants f J +n(λ) and +bJ +n(λ) given in § A.4 satisfy +En(λ) = f J +n(λ)bJ +n−1(λ) (n ∈ Z≥0), +(3.26) +and the energy eigenvalues En satisfy (3.15). Corresponding to the factorization (3.18), we +have the following relations [10]. +Theorem 2.2 [2] For the polynomials in Appendix A.4, the forward and backward shift re- +lations hold: +F J(λ)Pn(η; λ) = f J +n(λ)Pn−1(rη; λ + δ) (n ∈ Z≥0), +(3.27) +BJ(λ)Pn−1(rη; λ + δ) = bJ +n−1(λ)Pn(η; λ) (n ∈ Z≥1). +(3.28) +4 +New Forward and Backward Shift Relations +In this section, based on other factorizations of � +H and � +HJ, we present another type of forward +and backward shift relations. +6 + +4.1 +Polynomials in oQM systems +For the oQM systems described by the polynomials in § A.1 (except He and B), let us define +the operators ˜F(λ) and ˜B(λ) as follows: +L : (a) : +˜F(λ) +def += 1 +2x d +dx + g − 1 +2 +� += η d +dη + g − 1 +2 +� +, +(4.1) +˜B(λ) +def += −1 +2 +1 +x +d +dx + 1 +� += − d +dη + 1 +� +, +(4.2) +(b) : +˜F(λ) +def += −1 +2 +1 +x +d +dx + 1 +� += − d +dη + 1 +� +, +(4.3) +˜B(λ) +def += 1 +2x d +dx + g + 1 +2 +� += η d +dη + g + 1 +2 +� +, +(4.4) +J : (a) : +˜F(λ) +def += 1 +2 tan x d +dx + g − 1 +2 +� += −(1 − η) d +dη + g − 1 +2 +� +, +(4.5) +˜B(λ) +def += −1 +2 cot x d +dx + h + 1 +2 +� += (1 + η) d +dη + h + 1 +2 +� +, +(4.6) +(b) : +˜F(λ) +def += −1 +2 cot x d +dx + h − 1 +2 +� += (1 + η) d +dη + h − 1 +2 +� +, +(4.7) +˜B(λ) +def += 1 +2 tan x d +dx + g + 1 +2 +� += −(1 − η) d +dη + g + 1 +2 +� +, +(4.8) +pJ : (a) : +˜F(λ) +def += +� +tanh x + +i +cosh x +� d +dx − h − 1 +2 − iµ +� += (η + i) d +dη − h − 1 +2 − iµ +� +, (4.9) +˜B(λ) +def += +� +tanh x − +i +cosh x +� d +dx − h + 1 +2 + iµ +� += (η − i) d +dη − h + 1 +2 + iµ +� +, (4.10) +(b) : +˜F(λ) +def += +� +tanh x − +i +cosh x +� d +dx − h − 1 +2 + iµ +� += (η − i) d +dη − h − 1 +2 + iµ +� +, (4.11) +˜B(λ) +def += +� +tanh x + +i +cosh x +� d +dx − h + 1 +2 − iµ +� += (η + i) d +dη − h + 1 +2 − iµ +� +. (4.12) +Let us define the constants ˜fn(λ), ˜bn(λ) and ¯δ as follows: +L : (a) : +˜fn(λ) = n + g − 1 +2, ˜bn(λ) = 1, +¯δ = 1, +(4.13) +(b) : +˜fn(λ) = 1, ˜bn(λ) = n + g + 1 +2, +¯δ = −1, +(4.14) +J : (a) : +˜fn(λ) = n + g − 1 +2, ˜bn(λ) = n + h + 1 +2, +¯δ = (1, −1), +(4.15) +(b) : +˜fn(λ) = n + h − 1 +2, ˜bn(λ) = n + g + 1 +2, +¯δ = (−1, 1), +(4.16) +pJ : (a) : +˜fn(λ) = n − h − 1 +2 − iµ, ˜bn(λ) = n − h + 1 +2 + iµ, +¯δ = (0, i), +(4.17) +(b) : +˜fn(λ) = n − h − 1 +2 + iµ, ˜bn(λ) = n − h + 1 +2 − iµ, +¯δ = (0. − i). +(4.18) +We remark that the second components of ¯δ for pJ are unphysical values. +7 + +Then we can show that +� +H(λ) = 4 +� ˜B(λ) ˜F(λ) − ˜f0(λ)˜b0(λ) +� +, +(4.19) +En(λ) = 4 +� ˜fn(λ)˜bn(λ) − ˜f0(λ)˜b0(λ) +� +(n ∈ Z≥0). +(4.20) +Corresponding to this factorization (4.19), the following relations are obtained by direct +calculation. +Theorem 3.1 For the polynomials in § A.1 (except He and B), the following forward and +backward shift relations hold for n ∈ Z≥0, +˜F(λ) ˇPn(x; λ) = ˜fn(λ) ˇPn(x; λ − ¯δ), +(4.21) +˜B(λ) ˇPn(x; λ − ¯δ) = ˜bn(λ) ˇPn(x; λ). +(4.22) +We think that these identities (4.21)–(4.22) may be known formulas but this interpretation +is new. +Remark 1.1 +Two formulas with ¯δ and −¯δ are equivalent by interchanging ˜F and ˜B, +e.g. (4.21) and (4.22) for L (b) agree with (4.22) and (4.21) for L (a) with the replacement +g → g + 1, respectively. For He and B, we do not have new factorization (4.19) and new +forward and backward shift relations (4.21)–(4.22). +4.2 +Polynomials in idQM systems +For the idQM systems described by the polynomials in § A.2, let us define the operators +˜F(λ) and ˜B(λ) as follows: +˜F(λ) +def += V1(x + iγ +2; λ)e +γ +2 p + V ∗ +1 (x − iγ +2; λ)e− γ +2 p, +(4.23) +˜B(λ) +def += V2(x; λ)e +γ +2 p + V ∗ +2 (x; λ)e− γ +2 p. +(4.24) +The potential functions V1(x) and V2(x) satisfy +V (x; λ) = V1(x; λ)V2(x; λ), +(4.25) +and their explicit forms are given by +cH : (a) : V1(x; λ) = a1 + ix, +V2(x; λ) = a2 + ix, +(4.26) +(b) : V1(x; λ) = a2 + ix, +V2(x; λ) = a1 + ix, +(4.27) +8 + +MP : (a) : V1(x; λ) = a + ix, +V2(x; λ) = ei( π +2 −φ), +(4.28) +(b) : V1(x; λ) = ei( π +2 −φ), +V2(x; λ) = a + ix, +(4.29) +W : Assume {a∗ +j, a∗ +k} = {aj, ak} (as a set) and set {l, m} = {1, 2, 3, 4}\{j, k}, +Vi(x) = V (j,k) +i +(x) (i = 1, 2), +V1(x; λ) = (aj + ix)(ak + ix) +2ix + 1 +, +V2(x; λ) = (al + ix)(am + ix) +2ix +, +(4.30) +cdH : Assume {a∗ +j, a∗ +k} = {aj, ak} (as a set) and set {l} = {1, 2, 3}\{j, k}, +Vi(x) = V (j,k) +i +(x) (i = 1, 2), +(a) : V1(x; λ) = (aj + ix)(ak + ix) +2ix + 1 +, +V2(x; λ) = al + ix +2ix +, +(4.31) +(b) : V1(x; λ) = al + ix +2ix + 1, +V2(x; λ) = (aj + ix)(ak + ix) +2ix +, +(4.32) +AW : Assume {a∗ +j, a∗ +k} = {aj, ak} (as a set) and set {l, m} = {1, 2, 3, 4}\{j, k}, +Vi(x) = V (j,k) +i +(x) (i = 1, 2), +V1(x; λ) = (1 − ajeix)(1 − akeix) +1 − qe2ix +, +V2(x; λ) = (1 − aleix)(1 − ameix) +1 − e2ix +, +(4.33) +cdqH : Assume {a∗ +j, a∗ +k} = {aj, ak} (as a set) and set {l} = {1, 2, 3}\{j, k}, +Vi(x) = V (j,k) +i +(x) (i = 1, 2), +(a) : V1(x; λ) = (1 − ajeix)(1 − akeix) +1 − qe2ix +, +V2(x; λ) = 1 − aleix +1 − e2ix , +(4.34) +(b) : V1(x; λ) = 1 − aleix +1 − qe2ix, +V2(x; λ) = (1 − ajeix)(1 − akeix) +1 − e2ix +, +(4.35) +ASC : Assume a1, a2 ∈ R for (b) and (c), +(a) : V1(x; λ) = (1 − a1eix)(1 − a2eix) +1 − qe2ix +, +V2(x; λ) = +1 +1 − e2ix, +(4.36) +(b) : V1(x; λ) = 1 − a1eix +1 − qe2ix , +V2(x; λ) = 1 − a2eix +1 − e2ix , +(4.37) +(c) : V1(x; λ) = 1 − a2eix +1 − qe2ix , +V2(x; λ) = 1 − a1eix +1 − e2ix , +(4.38) +(d) : V1(x; λ) = +1 +1 − qe2ix, +V2(x; λ) = (1 − a1eix)(1 − a2eix) +1 − e2ix +, +(4.39) +cbqHe : (a) : V1(x; λ) = 1 − aeix +1 − qe2ix, +V2(x; λ) = +1 +1 − e2ix, +(4.40) +(b) : V1(x; λ) = +1 +1 − qe2ix, +V2(x; λ) = 1 − aeix +1 − e2ix , +(4.41) +cqHe : V1(x; λ) = +1 +1 − qe2ix, +V2(x; λ) = +1 +1 − e2ix, +(4.42) +9 + +cqJ : (a) : V1(x; λ) = (1 − q +1 +2(α+ 1 +2)eix)(1 − q +1 +2 (α+ 3 +2 )eix) +1 − qe2ix +, +V2(x; λ) = (1 + q +1 +2(β+ 1 +2)eix)(1 + q +1 +2 (β+ 3 +2 )eix) +1 − e2ix +, +(4.43) +(b) : V1(x; λ) = (1 + q +1 +2(β+ 1 +2)eix)(1 + q +1 +2 (β+ 3 +2)eix) +1 − qe2ix +, +V2(x; λ) = (1 − q +1 +2(α+ 1 +2)eix)(1 − q +1 +2 (α+ 3 +2 )eix) +1 − e2ix +, +(4.44) +cqL : (a) : V1(x; λ) = (1 − q +1 +2(α+ 1 +2)eix)(1 − q +1 +2 (α+ 3 +2 )eix) +1 − qe2ix +, +V2(x; λ) = +1 +1 − e2ix, +(4.45) +(b) : V1(x; λ) = +1 +1 − qe2ix, +V2(x; λ) = (1 − q +1 +2 (α+ 1 +2 )eix)(1 − q +1 +2(α+ 3 +2)eix) +1 − e2ix +, +(4.46) +cqH : (a) : V1(x; λ) = (1 − a1e2iφeix)(1 − a∗ +1eix) +1 − qe2iφe2ix +, V2(x; λ) = (1 − a2e2iφeix)(1 − a∗ +2eix) +1 − e2iφe2ix +, +(4.47) +(b) : V1(x; λ) = (1 − a2e2iφeix)(1 − a∗ +2eix) +1 − qe2iφe2ix +, V2(x; λ) = (1 − a1e2iφeix)(1 − a∗ +1eix) +1 − e2iφe2ix +, +(4.48) +qMP : (a) : V1(x; λ) = (1 − ae2iφeix)(1 − aeix) +1 − qe2iφe2ix +, +V2(x; λ) = +1 +1 − e2iφe2ix, +(4.49) +(b) : V1(x; λ) = +1 +1 − qe2iφe2ix , +V2(x; λ) = (1 − ae2iφeix)(1 − aeix) +1 − e2iφe2ix +. +(4.50) +Let us define the constants ˜fn(λ), ˜bn(λ) and ¯δ as follows: +cH : (a) : +˜fn(λ) = a1 + a∗ +1 + n − 1, ˜bn(λ) = a2 + a∗ +2 + n, +¯δ = ( 1 +2, −1 +2), +(4.51) +(b) : +˜fn(λ) = a2 + a∗ +2 + n − 1, ˜bn(λ) = a1 + a∗ +1 + n, +¯δ = (−1 +2, 1 +2), +(4.52) +MP : (a) : +˜fn(λ) = 2a + n − 1, ˜bn(λ) = 2 sin φ, +¯δ = ( 1 +2, 0), +(4.53) +(b) : +˜fn(λ) = 2 sin φ, ˜bn(λ) = 2a + n, +¯δ = (−1 +2, 0), +(4.54) +W : +˜fn(λ) = aj + ak + n − 1, ˜bn(λ) = al + am + n, +(¯δ)j = (¯δ)k = 1 +2, +(¯δ)l = (¯δ)m = −1 +2, +(4.55) +cdH : (a) : +˜fn(λ) = aj + ak + n − 1, ˜bn(λ) = 1, +(¯δ)j = (¯δ)k = 1 +2, +(¯δ)l = −1 +2, +(4.56) +(b) : +˜fn(λ) = 1, ˜bn(λ) = aj + ak + n, +(¯δ)l = 1 +2, +(¯δ)j = (¯δ)k = −1 +2, +(4.57) +AW : +˜fn(λ) = q− n +2 (1 − ajakqn−1), ˜bn(λ) = q− n +2 (1 − alamqn), +(¯δ)j = (¯δ)k = 1 +2, +(¯δ)l = (¯δ)m = −1 +2, +(4.58) +cdqH : (a) : ˜fn(λ) = q− n +2 (1 − ajakqn−1), ˜bn(λ) = q− n +2 , (¯δ)j = (¯δ)k = 1 +2, (¯δ)l = −1 +2, (4.59) +10 + +(b) : ˜fn(λ) = q− n +2 , ˜bn(λ) = q− n +2 (1 − ajakqn), (¯δ)l = 1 +2, (¯δ)j = (¯δ)k = −1 +2, +(4.60) +ASC : (a) : +˜fn(λ) = q− n +2 (1 − a1a2qn−1), ˜bn(λ) = q− n +2 , +¯δ = ( 1 +2, 1 +2), +(4.61) +(b) : +˜fn(λ) = q− n +2 , ˜bn(λ) = q− n +2 , +¯δ = ( 1 +2, −1 +2), +(4.62) +(c) : +˜fn(λ) = q− n +2 , ˜bn(λ) = q− n +2 , +¯δ = (−1 +2, 1 +2), +(4.63) +(d) : +˜fn(λ) = q− n +2 , ˜bn(λ) = q− n +2 (1 − a1a2qn), +¯δ = (−1 +2, −1 +2), +(4.64) +cbqHe : (a) : +˜fn(λ) = q− n +2 , ˜bn(λ) = q− n +2 , +¯δ = 1 +2, +(4.65) +(b) : +˜fn(λ) = q− n +2 , ˜bn(λ) = q− n +2 , +¯δ = −1 +2, +(4.66) +cqHe : +˜fn(λ) = q− n +2 , ˜bn(λ) = q− n +2 , +¯δ : none, +(4.67) +cqJ : (a) : +˜fn(λ) = 1 − qα+n, ˜bn(λ) = q−n(1 − qβ+n+1), +¯δ = (1, −1), +(4.68) +(b) : +˜fn(λ) = q−n(1 − qβ+n), ˜bn(λ) = 1 − qα+n+1, +¯δ = (−1, 1), +(4.69) +cqL : (a) : +˜fn(λ) = 1 − qα+n, ˜bn(λ) = q−n, +¯δ = 1, +(4.70) +(b) : +˜fn(λ) = q−n, ˜bn(λ) = 1 − qα+n+1, +¯δ = −1, +(4.71) +cqH : (a) : +˜fn(λ) = q− n +2 (1 − a1a∗ +1qn−1), ˜bn(λ) = q− n +2 (1 − a2a∗ +2qn), ¯δ = ( 1 +2, −1 +2, 0), (4.72) +(b) : +˜fn(λ) = q− n +2 (1 − a2a∗ +2qn−1), ˜bn(λ) = q− n +2 (1 − a1a∗ +1qn), ¯δ = (−1 +2, 1 +2, 0), (4.73) +qMP : (a) : +˜fn(λ) = q− n +2 (1 − a2qn−1), ˜bn(λ) = q− n +2 , +¯δ = ( 1 +2, 0), +(4.74) +(b) : +˜fn(λ) = q− n +2 , ˜bn(λ) = q− n +2 (1 − a2qn), +¯δ = (−1 +2, 0). +(4.75) +Then we can show that V1(x) and V2(x) satisfy +V1(x + iγ; λ)V ∗ +2 (x; λ) + V ∗ +1 (x − iγ; λ)V2(x; λ) − ˜f0(λ)˜b0(λ) = −V (x; λ) − V ∗(x; λ), (4.76) +and the constants ˜fn and ˜bn satisfy +En(λ) = ˜fn(λ)˜bn(λ) − ˜f0(λ)˜b0(λ) (n ∈ Z≥0). +(4.77) +The relation (4.76) gives other factorizations of � +H(λ) (2.3), +� +H(λ) = ˜B(λ) ˜F(λ) − ˜f0(λ)˜b0(λ). +(4.78) +Corresponding to this factorization (4.78), we obtain the following relations. +Theorem 3.2 For the polynomials in § A.2, the following forward and backward shift rela- +tions hold for n ∈ Z≥0, +˜F(λ) ˇPn(x; λ) = ˜fn(λ) ˇPn(x; λ − ¯δ), +(4.79) +˜B(λ) ˇPn(x; λ − ¯δ) = ˜bn(λ) ˇPn(x; λ). +(4.80) +11 + +Proof: It is sufficient to show (4.79), because (2.7) and (4.77)–(4.79) imply (4.80). Taking +AW (4.33) with (j, k) = (1, 2) as an example, let us prove (4.79). It is shown by direct +calculation: +˜F(λ) ˇPn(x; λ) += (1 − a1q− 1 +2eix)(1 − a2q− 1 +2eix) +1 − e2ix +(a1a2, a1a3, a1a4 ; q)n +an +1 +× 4φ3 +�q−n, a1a2a3a4qn−1, a1q +1 +2eix, a1q− 1 +2e−ix +a1a2, a1a3, a1a4 +��� q ; q +� ++ (1 − a1q− 1 +2e−ix)(1 − a2q− 1 +2e−ix) +1 − e−2ix +(a1a2, a1a3, a1a4 ; q)n +an +1 +× 4φ3 +�q−n, a1a2a3a4qn−1, a1q− 1 +2eix, a1q +1 +2e−ix +a1a2, a1a3, a1a4 +��� q ; q +� += (a1a2, a1a3, a1a4 ; q)n +an +1(1 − e2ix) +n +� +k=0 +(q−n, a1a2a3a4qn−1, a1q− 1 +2eix, a1q− 1 +2e−ix ; q)k +(a1a2, a1a3, a1a4 ; q)k +qk +(q ; q)k +× +� +(1 − a1eixqk− 1 +2)(1 − a2q− 1 +2eix) − e2ix(1 − a1e−ixqk− 1 +2)(1 − a2q− 1 +2e−ix) +� += (a1a2, a1a3, a1a4 ; q)n +an +1(1 − e2ix) +n +� +k=0 +(q−n, a1a2a3a4qn−1, a1q− 1 +2eix, a1q− 1 +2e−ix ; q)k +(a1a2, a1a3, a1a4 ; q)k +qk +(q ; q)k +× (1 − a1a2qk−1)(1 − e2ix) += (a1a2, a1a3, a1a4 ; q)n +an +1 +n +� +k=0 +(1 − a1a2q−1)(q−n, a1a2a3a4qn−1, a1q− 1 +2eix, a1q− 1 +2e−ix ; q)k +(a1a2q−1, a1a3, a1a4 ; q)k +qk +(q ; q)k += q− n +2 (1 − a1a2qn−1) +× (a1a2q−1, a1a3, a1a4 ; q)n +(a1q− 1 +2)n +n +� +k=0 +(q−n, a1a2a3a4qn−1, a1q− 1 +2eix, a1q− 1 +2e−ix ; q)k +(a1a2q−1, a1a3, a1a4 ; q)k +qk +(q ; q)k += ˜fn(λ) ˇPn(x; λ − ¯δ). +The other cases are proved in the same way. +Remark 2.1 Two formulas with ¯δ and −¯δ are equivalent by interchanging ˜F and ˜B, e.g. +(4.79) and (4.80) for cH (b) agree with (4.80) and (4.79) for cH (a) with the replacements +a1 → a1 + 1 +2 and a2 → a2 − 1 +2, respectively. +Remark 2.2 The relations (4.79)–(4.80) are invariant under the ∗-operation. In contrast +to the x-shift relations studied in [9] (see Theorem 3.3), the coordinate x is not shifted, and +only the parameters λ are shifted. We choose the operators ˜F(λ) and ˜B(λ) (4.23)–(4.24) to +12 + +respect this ∗-operation invariance. See also Remark 3.3. +Remark 2.3 We can show that +V1(x + iγ +2; λ)V2(x − iγ +2; λ) = V1(x; λ − ¯δ)V2(x; λ − ¯δ), +V1(x + iγ +2; λ)V ∗ +2 (x − iγ +2; λ) + V ∗ +1 (x − iγ +2; λ)V2(x + iγ +2; λ) − ˜f0(λ)˜b0(λ) +(4.81) += V1(x + iγ; λ − ¯δ)V ∗ +2 (x; λ − ¯δ) + V ∗ +1 (x − iγ; λ − ¯δ)V2(x; λ − ¯δ) − ˜f0(λ − ¯δ)˜b0(λ − ¯δ), +which imply +˜F(λ) ˜B(λ) − ˜f0(λ)˜b0(λ) = ˜B(λ − ¯δ) ˜F(λ − ¯δ) − ˜f0(λ − ¯δ)˜b0(λ − ¯δ). +(4.82) +4.3 +Polynomials in rdQM systems +For the rdQM systems described by the polynomials in § A.3 (except C and qB), let us define +the operators ˜F(λ) and ˜B(λ) as follows: +˜F(λ) +def += D1(x + 1; λ) + B1(x; λ)e∂, +(4.83) +˜B(λ) +def += B2(x; λ) + D2(x; λ)e−∂. +(4.84) +The potential functions B1(x), B2(x), D1(x) and D2(x) satisfy +B(x; λ) = B1(x; λ)B2(x; λ), +D(x; λ) = D1(x; λ)D2(x; λ), +(4.85) +and their explicit forms are given by +H : (a) : B1(x; λ) = N − x +N + 1, +B2(x; λ) = (N + 1)(x + a), +D1(x; λ) = +x +N + 1, +D2(x; λ) = (N + 1)(b + N − x), +(4.86) +(b) : B1(x; λ) = x + a +a − 1, +B2(x; λ) = (a − 1)(N − x), +D1(x; λ) = +x +1 − a, +D2(x; λ) = (1 − a)(b + N − x), +(4.87) +(c) : B1(x; λ) = a(N − x), +B2(x; λ) = x + a +a +, +D1(x; λ) = −a(b + N − x), +D2(x; λ) = −x +a, +(4.88) +(d) : B1(x; λ) = N(x + a), +B2(x; λ) = N − x +N +, +D1(x; λ) = N(b + N − x), +D2(x; λ) = x +N , +(4.89) +K : (a) : B1(x; λ) = N − x +N + 1, +B2(x; λ) = (N + 1)p, +13 + +D1(x; λ) = +x +N + 1, +D2(x; λ) = (N + 1)(1 − p), +(4.90) +(b) : B1(x; λ) = Np, +B2(x; λ) = N − x +N +, +D1(x; λ) = N(1 − p), +D2(x; λ) = x +N , +(4.91) +R : (a) : B1(x; λ) = − +(x − N)(x + d) +(N + 1)(2x + 1 + d), +B2(x; λ) = (N + 1)(x + b)(x + c) +2x + d +, +D1(x; λ) = +(x + d + N)x +(N + 1)(2x − 1 + d), D2(x; λ) = −(N + 1)(x + d − b)(x + d − c) +2x + d +, +(4.92) +(b) : B1(x; λ) = +(x + b)(x + d) +(b − 1)(2x + 1 + d), +B2(x; λ) = −(b − 1)(x − N)(x + c) +2x + d +, +D1(x; λ) = − +(x + d − b)x +(b − 1)(2x − 1 + d), D2(x; λ) = (b − 1)(x + d + N)(x + d − c) +2x + d +, +(4.93) +(c) : B1(x; λ) = +(x + c)(x + d) +(c − 1)(2x + 1 + d), +B2(x; λ) = −(c − 1)(x − N)(x + b) +2x + d +, +D1(x; λ) = − +(x + d − c)x +(c − 1)(2x − 1 + d), D2(x; λ) = (c − 1)(x + d + N)(x + d − b) +2x + d +, +(4.94) +(d) : B1(x; λ) = −c(x − N)(x + b) +2x + 1 + d +, +B2(x; λ) = (x + c)(x + d) +c(2x + d) +, +D1(x; λ) = c(x + d + N)(x + d − b) +2x − 1 + d +, +D2(x; λ) = −(x + d − c)x +c(2x + d) , +(4.95) +(e) : B1(x; λ) = −b(x − N)(x + c) +2x + 1 + d +, +B2(x; λ) = (x + b)(x + d) +b(2x + d) +, +D1(x; λ) = b(x + d + N)(x + d − c) +2x − 1 + d +, +D2(x; λ) = −(x + d − b)x +b(2x + d) , +(4.96) +(f) : B1(x; λ) = N(x + b)(x + c) +2x + 1 + d +, +B2(x; λ) = −(x − N)(x + d) +N(2x + d) +, +D1(x; λ) = −N(x + d − b)(x + d − c) +2x − 1 + d +, +D2(x; λ) = (x + d + N)x +N(2x + d) , +(4.97) +dH : (a) : B1(x; λ) = (x + a + b − 1)(N − x) +(N + 1)(2x + a + b) , +B2(x; λ) = (N + 1)(x + a) +2x − 1 + a + b , +D1(x; λ) = +x(x + a + b + N − 1) +(N + 1)(2x − 2 + a + b), D2(x; λ) = (N + 1)(x + b − 1) +2x − 1 + a + b +, (4.98) +(b) : B1(x; λ) = (x + a)(x + a + b − 1) +(a − 1)(2x + a + b) , +B2(x; λ) = (a − 1)(N − x) +2x − 1 + a + b , +14 + +D1(x; λ) = +x(x + b − 1) +(1 − a)(2x − 2 + a + b), D2(x; λ) = (1 − a)(x + a + b + N − 1) +2x − 1 + a + b +, +(4.99) +(c) : B1(x; λ) = x + a + b − 1 +2x + a + b , +B2(x; λ) = (x + a)(N − x) +2x − 1 + a + b , +D1(x; λ) = +x +2x − 2 + a + b, D2(x; λ) = (x + b − 1)(x + a + b + N − 1) +2x − 1 + a + b +, +(4.100) +(d) : B1(x; λ) = (x + a)(N − x) +2x + a + b +, +B2(x; λ) = x + a + b − 1 +2x − 1 + a + b, +D1(x; λ) = (x + b − 1)(x + a + b + N − 1) +2x − 2 + a + b +, D2(x; λ) = +x +2x − 1 + a + b, +(4.101) +(e) : B1(x; λ) = a(N − x) +2x + a + b, +B2(x; λ) = (x + a)(x + a + b − 1) +a(2x − 1 + a + b) +, +D1(x; λ) = −a(x + a + b + N − 1) +2x − 2 + a + b +, D2(x; λ) = − +x(x + b − 1) +a(2x − 1 + a + b), (4.102) +(f) : B1(x; λ) = N(x + a) +2x + a + b, +B2(x; λ) = (x + a + b − 1)(N − x) +N(2x − 1 + a + b) +, +D1(x; λ) = N(x + b − 1) +2x − 2 + a + b, +D2(x; λ) = x(x + a + b + N − 1) +N(2x − 1 + a + b) , +(4.103) +dqqK : (a) : B1(x; λ) = q−N−1(1 − qN−x) +q−N−1 − 1 +, +B2(x; λ) = (q−N−1 − 1)p−1q−x, +D1(x; λ) = +q−x − 1 +q−N−1 − 1, +D2(x; λ) = (q−N−1 − 1)(1 − p−1q−x), +(4.104) +(b) : B1(x; λ) = q−x−1, +B2(x; λ) = p−1q−N(1 − qN−x), +D1(x; λ) = −(q−x − 1), +D2(x; λ) = −(1 − p−1q−x), +(4.105) +(c) : B1(x; λ) = p−1q−N−1(1 − qN−x), +B2(x; λ) = q−x, +D1(x; λ) = −(1 − p−1q−x), +D2(x; λ) = −(q−x − 1), +(4.106) +(d) : B1(x; λ) = (1 − qN)p−1q−x−N−1, +B2(x; λ) = 1 − qN−x +1 − qN , +D1(x; λ) = (q−N − 1)(1 − p−1q−x), +D2(x; λ) = q−x − 1 +q−N − 1, +(4.107) +qH : (a) : B1(x; λ) = qx−N − 1 +q−N−1 − 1, +B2(x; λ) = (q−N−1 − 1)(1 − aqx), +D1(x; λ) = +1 − qx +1 − qN+1, +D2(x; λ) = (1 − qN+1)aq−1(qx−N − b), +(4.108) +(b) : B1(x; λ) = 1 − aqx +1 − aq−1, +B2(x; λ) = (1 − aq−1)(qx−N − 1), +15 + +D1(x; λ) = aq−1(1 − qx) +aq−1 − 1 +, +D2(x; λ) = (aq−1 − 1)(qx−N − b), +(4.109) +(c) : B1(x; λ) = (1 − a)(qx−N − 1), +B2(x; λ) = 1 − aqx +1 − a , +D1(x; λ) = (a − 1)q−1(qx−N − b), +D2(x; λ) = a(1 − qx) +a − 1 +, +(4.110) +(d) : B1(x; λ) = (q−N − 1)(1 − aqx), +B2(x; λ) = qx−N − 1 +q−N − 1 , +D1(x; λ) = (1 − qN)aq−1(qx−N − b), +D2(x; λ) = 1 − qx +1 − qN , +(4.111) +qK : (a) : B1(x; λ) = qx−N − 1 +q−N−1 − 1, +B2(x; λ) = q−N−1 − 1, +D1(x; λ) = +1 − qx +1 − qN+1, +D2(x; λ) = (1 − qN+1)p, +(4.112) +(b) : B1(x; λ) = q−N − 1, +B2(x; λ) = qx−N − 1 +q−N − 1 , +D1(x; λ) = (1 − qN)p, +D2(x; λ) = 1 − qx +1 − qN , +(4.113) +qqK : (a) : B1(x; λ) = qx−N − 1 +q−N−1 − 1, +B2(x; λ) = (q−N−1 − 1)p−1qx, +D1(x; λ) = +1 − qx +1 − qN+1, +D2(x; λ) = (1 − qN+1)(1 − p−1qx−N−1), +(4.114) +(b) : B1(x; λ) = qqx, +B2(x; λ) = q−1p−1(qx−N − 1), +D1(x; λ) = 1 − qx, +D2(x; λ) = 1 − p−1qx−N−1, +(4.115) +(c) : B1(x; λ) = p−1(qx−N − 1), +B2(x; λ) = qx, +D1(x; λ) = 1 − p−1qx−N−1, +D2(x; λ) = 1 − qx, +(4.116) +(d) : B1(x; λ) = (q−N − 1)p−1qx, +B2(x; λ) = qx−N − 1 +q−N − 1 , +D1(x; λ) = (1 − qN)(1 − p−1qx−N−1), +D2(x; λ) = 1 − qx +1 − qN , +(4.117) +aqK : (a) : B1(x; λ) = qx−N − 1 +q−N−1 − 1, +B2(x; λ) = (q−N−1 − 1)(1 − pqx+1), +D1(x; λ) = +1 − qx +1 − qN+1, +D2(x; λ) = (1 − qN+1)pqx−N, +(4.118) +(b) : B1(x; λ) = 1 − pqx+1 +1 − p +, +B2(x; λ) = (1 − p)(qx−N − 1), +D1(x; λ) = p(1 − qx) +p − 1 +, +D2(x; λ) = (p − 1)qx−N, +(4.119) +16 + +(c) : B1(x; λ) = (1 − pq)(qx−N − 1), +B2(x; λ) = 1 − pqx+1 +1 − pq +, +D1(x; λ) = (pq − 1)qx−N−1, +D2(x; λ) = pq(1 − qx) +pq − 1 +, +(4.120) +(d) : B1(x; λ) = (q−N − 1)(1 − pqx+1), +B2(x; λ) = qx−N − 1 +q−N − 1 , +D1(x; λ) = (q−N − 1)pqx, +D2(x; λ) = q−N(1 − qx) +q−N − 1 +, +(4.121) +qR : (a) : B1(x; λ) = − +(1 − qx−N)(1 − dqx) +(q−N−1 − 1)(1 − dq2x+1), +B2(x; λ) = (q−N−1 − 1)(1 − bqx)(1 − cqx) +1 − dq2x +, +D1(x; λ) = +(1 − dqx+N)(1 − qx) +(1 − qN+1)(1 − dq2x−1), +D2(x; λ) = − ˜d (1 − qN+1)(1 − b−1dqx)(1 − c−1dqx) +1 − dq2x +, +(4.122) +(b) : B1(x; λ) = +(1 − bqx)(1 − dqx) +(1 − bq−1)(1 − dq2x+1), +B2(x; λ) = −(1 − bq−1)(1 − qx−N)(1 − cqx) +1 − dq2x +, +D1(x; λ) = (1 − b−1dqx)(1 − qx) +(1 − b−1q)(1 − dq2x−1), +D2(x; λ) = − ˜d (1 − b−1q)(1 − dqx+N)(1 − c−1dqx) +1 − dq2x +, +(4.123) +(c) : B1(x; λ) = +(1 − cqx)(1 − dqx) +(1 − cq−1)(1 − dq2x+1), +B2(x; λ) = −(1 − cq−1)(1 − qx−N)(1 − bqx) +1 − dq2x +, +D1(x; λ) = (1 − c−1dqx)(1 − qx) +(1 − c−1q)(1 − dq2x−1), +D2(x; λ) = − ˜d (1 − c−1q)(1 − dqx+N)(1 − b−1dqx) +1 − dq2x +, +(4.124) +(d) : B1(x; λ) = −(1 − c)(1 − qx−N)(1 − bqx) +1 − dq2x+1 +, +B2(x; λ) = (1 − cqx)(1 − dqx) +(1 − c)(1 − dq2x) , +D1(x; λ) = − ˜d (1 − c−1)(1 − dqx+N)(1 − b−1dqx) +1 − dq2x−1 +, +D2(x; λ) = (1 − c−1dqx)(1 − qx) +(1 − c−1)(1 − dq2x) , +(4.125) +(e) : B1(x; λ) = −(1 − b)(1 − qx−N)(1 − cqx) +1 − dq2x+1 +, +B2(x; λ) = (1 − bqx)(1 − dqx) +(1 − b)(1 − dq2x) , +17 + +D1(x; λ) = − ˜d (1 − b−1)(1 − dqx+N)(1 − c−1dqx) +1 − dq2x−1 +, +D2(x; λ) = (1 − b−1dqx)(1 − qx) +(1 − b−1)(1 − dq2x) , +(4.126) +(f) : B1(x; λ) = (q−N − 1)(1 − bqx)(1 − cqx) +1 − dq2x+1 +, +B2(x; λ) = −(1 − qx−N)(1 − dqx) +(q−N − 1)(1 − dq2x), +D1(x; λ) = − ˜d (1 − qN)(1 − b−1dqx)(1 − c−1dqx) +1 − dq2x−1 +, +D2(x; λ) = (1 − dqx+N)(1 − qx) +(1 − qN)(1 − dq2x) , +(4.127) +dqH : (a) : B1(x; λ) = (qx−N − 1)(1 − abqx−1) +(q−N−1 − 1)(1 − abq2x), +B2(x; λ) = (q−N−1 − 1)(1 − aqx) +1 − abq2x−1 +, +D1(x; λ) = (1 − qx)(1 − abqx+N−1) +(1 − qN+1)(1 − abq2x−2), +D2(x; λ) = (1 − qN+1)aqx−N−1(1 − bqx−1) +1 − abq2x−1 +, +(4.128) +(b) : B1(x; λ) = (1 − aqx)(1 − abqx−1) +(1 − aq−1)(1 − abq2x), +B2(x; λ) = (1 − aq−1)(qx−N − 1) +1 − abq2x−1 +, +D1(x; λ) = +(1 − qx)(1 − bqx−1) +(1 − a−1q)(1 − abq2x−2), +D2(x; λ) = (1 − a−1q)aqx−N−1(1 − abqx+N−1) +1 − abq2x−1 +, +(4.129) +(c) : B1(x; λ) = 1 − abqx−1 +1 − abq2x , +B2(x; λ) = (qx−N − 1)(1 − aqx) +1 − abq2x−1 +, +D1(x; λ) = bq−2aqx(1 − qx) +1 − abq2x−2 +, D2(x; λ) = q−N−1(1 − abqx+N−1)(1 − bqx−1) +bq−2(1 − abq2x−1) +, +(4.130) +(d) : B1(x; λ) = (qx−N − 1)(1 − aqx) +1 − abq2x +, +B2(x; λ) = 1 − abqx−1 +1 − abq2x−1, +D1(x; λ) = q−N(1 − abqx+N−1)(1 − bqx−1) +b(1 − abq2x−2) +, D2(x; λ) = baqx−1(1 − qx) +1 − abq2x−1 +, +(4.131) +(e) : B1(x; λ) = (1 − a)(qx−N − 1) +1 − abq2x +, +B2(x; λ) = (1 − aqx)(1 − abqx−1) +(1 − a)(1 − abq2x−1) , +D1(x; λ) = (a − 1)qx−N−1(1 − abqx+N−1) +1 − abq2x−2 +, D2(x; λ) = a(1 − qx)(1 − bqx−1) +(a − 1)(1 − abq2x−1), +(4.132) +(f) : B1(x; λ) = (q−N − 1)(1 − aqx) +1 − abq2x +, +B2(x; λ) = (qx−N − 1)(1 − abqx−1) +(q−N − 1)(1 − abq2x−1), +18 + +D1(x; λ) = (1 − qN)aqx−N−1(1 − bqx−1) +1 − abq2x−2 +, D2(x; λ) = (1 − qx)(1 − abqx+N−1) +(1 − qN)(1 − abq2x−1) , +(4.133) +dqK : (a) : B1(x; λ) = +(qx−N − 1)(1 + pqx) +(q−N−1 − 1)(1 + pq2x+1), +B2(x; λ) = q−N−1 − 1 +1 + pq2x , +D1(x; λ) = +(1 − qx)(1 + pqx+N) +(1 − qN+1)(1 + pq2x−1), D2(x; λ) = (1 − qN+1)pq2x−N−1 +1 + pq2x +, (4.134) +(b) : B1(x; λ) = +1 + pqx +1 + pq2x+1, +B2(x; λ) = qx−N − 1 +1 + pq2x , +D1(x; λ) = −pqx−1(1 − qx) +1 + pq2x−1 +, +D2(x; λ) = −qx−N(1 + pqx+N) +1 + pq2x +, +(4.135) +(c) : B1(x; λ) = qx−N − 1 +1 + pq2x+1, +B2(x; λ) = 1 + pqx +1 + pq2x, +D1(x; λ) = −qx−N−1(1 + pqx+N) +1 + pq2x−1 +, +D2(x; λ) = −pqx(1 − qx) +1 + pq2x +, +(4.136) +(d) : B1(x; λ) = +q−N − 1 +1 + pq2x+1, +B2(x; λ) = (qx−N − 1)(1 + pqx) +(q−N − 1)(1 + pq2x), +D1(x; λ) = (1 − qN)pq2x−N−1 +1 + pq2x−1 +, +D2(x; λ) = (1 − qx)(1 + pqx+N) +(1 − qN)(1 + pq2x) , +(4.137) +M : (a) : B1(x; λ) = x + β +β − 1, +B2(x; λ) = (β − 1)c +1 − c , +D1(x; λ) = +x +1 − β, +D2(x; λ) = 1 − β +1 − c , +(4.138) +(b) : B1(x; λ) = +βc +1 − c, +B2(x; λ) = x + β +β +, +D1(x; λ) = − +β +1 − c, +D2(x; λ) = −x +β , +(4.139) +lqJ : (a) : B1(x; λ) = q−1(q−x − b) +1 − bq−1 +, +B2(x; λ) = (1 − bq−1)a, +D1(x; λ) = q−x − 1 +bq−1 − 1, +D2(x; λ) = bq−1 − 1, +(4.140) +(b) : B1(x; λ) = (1 − b)aq−1, +B2(x; λ) = q−x − b +1 − b , +D1(x; λ) = b − 1, +D2(x; λ) = q−x − 1 +b − 1 , +(4.141) +lqL : (a) : B1(x; λ) = q−x−1, +B2(x; λ) = a, +D1(x; λ) = −(q−x − 1), +D2(x; λ) = −1, +(4.142) +(b) : B1(x; λ) = aq−1, +B2(x; λ) = q−x, +D1(x; λ) = −1, +D2(x; λ) = −(q−x − 1), +(4.143) +19 + +qM : (a) : B1(x; λ) = qx+1, +B2(x; λ) = cq−1(1 − bqx+1), +D1(x; λ) = 1 − qx, +D2(x; λ) = 1 + bcqx, +(4.144) +(b) : B1(x; λ) = 1 − bqx+1 +1 − b +, +B2(x; λ) = (1 − b)cqx, +D1(x; λ) = 1 − qx +1 − b−1, +D2(x; λ) = (1 − b−1)(1 + bcqx), +(4.145) +(c) : B1(x; λ) = (1 − bq)cqx, +B2(x; λ) = 1 − bqx+1 +1 − bq +, +D1(x; λ) = (1 − b−1q−1)(1 + bcqx), +D2(x; λ) = +1 − qx +1 − b−1q−1, +(4.146) +(d) : B1(x; λ) = c(1 − bqx+1), +B2(x; λ) = qx, +D1(x; λ) = 1 + bcqx, +D2(x; λ) = 1 − qx, +(4.147) +ASCII : (a) : B1(x; λ) = qx+1, +B2(x; λ) = aqx, +D1(x; λ) = 1 − qx, +D2(x; λ) = 1 − aqx, +(4.148) +(b) : B1(x; λ) = aqx+1, +B2(x; λ) = qx, +D1(x; λ) = 1 − aqx, +D2(x; λ) = 1 − qx, +(4.149) +qC : (a) : B1(x; λ) = qx+1, +B2(x; λ) = aq−1, +D1(x; λ) = 1 − qx, +D2(x; λ) = 1, +(4.150) +(b) : B1(x; λ) = a, +B2(x; λ) = qx, +D1(x; λ) = 1, +D2(x; λ) = 1 − qx. +(4.151) +Let us define the constants ˜fn(λ), ˜bn(λ) and ¯δ as follows: +H : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (0, 0, −1), +(4.152) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −(n + a − 1)(n + b), +¯δ = (1, −1, 0), +(4.153) +(c) : +˜fn(λ) = −(n + a)(n + b − 1), ˜bn(λ) = 1, +¯δ = (−1, 1, 0), +(4.154) +(d) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (0, 0, 1), +(4.155) +K : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (0, −1), +(4.156) +(b) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (0, 1), +(4.157) +R : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (1, 0, 0, 1), +(4.158) +(b) : ˜fn(λ) = 1, ˜bn(λ) = −(n + b − 1)(n + c − d − N), ¯δ = (0, 1, 0, 1), +(4.159) +(c) : ˜fn(λ) = 1, ˜bn(λ) = −(n + c − 1)(n + b − d − N), ¯δ = (0, 0, 1, 1), +(4.160) +(d) : ˜fn(λ) = −(n + c)(n + b − d − N − 1), ˜bn(λ) = 1, ¯δ = (0, 0, −1, −1), (4.161) +20 + +(e) : ˜fn(λ) = −(n + b)(n + c − d − N − 1), ˜bn(λ) = 1, ¯δ = (0, −1, 0, −1), (4.162) +(f) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (−1, 0, 0, −1), +(4.163) +dH : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (0, 1, −1), +(4.164) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −(n + a − 1), +¯δ = (1, 0, 0), +(4.165) +(c) : +˜fn(λ) = 1, ˜bn(λ) = −(n − b − N + 1), +¯δ = (0, 1, 0), +(4.166) +(d) : +˜fn(λ) = −(n − b − N), ˜bn(λ) = 1, +¯δ = (0, −1, 0), +(4.167) +(e) : +˜fn(λ) = −(n + a), ˜bn(λ) = 1, +¯δ = (−1, 0, 0), +(4.168) +(f) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (0, −1, 1), +(4.169) +dqqK : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (1, −1), +(4.170) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − p−1qn−N), +¯δ = (1, 0), +(4.171) +(c) : +˜fn(λ) = −q−n(1 − p−1qn−N−1), ˜bn(λ) = 1, +¯δ = (−1, 0), +(4.172) +(d) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (−1, 1), +(4.173) +qH : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (0, 0, −1), +(4.174) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − aqn−1)(1 − bqn), +¯δ = (1, −1, 0), +(4.175) +(c) : +˜fn(λ) = −q−n(1 − aqn)(1 − bqn−1), ˜bn(λ) = 1, +¯δ = (−1, 1, 0), +(4.176) +(d) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (0, 0, 1), +(4.177) +qK : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (0, −1), +(4.178) +(b) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (0, 1), +(4.179) +qqK : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (0, −1), +(4.180) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −p−1q−1(1 − pqn+1), +¯δ = (−1, 0), +(4.181) +(c) : +˜fn(λ) = −p−1(1 − pqn) ˜bn(λ) = 1, +¯δ = (1, 0), +(4.182) +(d) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (0, 1), +(4.183) +aqK : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (0, −1), +(4.184) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − pqn), +¯δ = (1, 0), +(4.185) +(c) : +˜fn(λ) = −q−n(1 − pqn+1), ˜bn(λ) = 1, +¯δ = (−1, 0), +(4.186) +(d) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (0, 1), +(4.187) +qR : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (1, 0, 0, 1), +(4.188) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − bqn−1)(1 − cd−1qn−N), ¯δ = (0, 1, 0, 1), (4.189) +21 + +(c) : +˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − cqn−1)(1 − bd−1qn−N), ¯δ = (0, 0, 1, 1), (4.190) +(d) : +˜fn(λ) = −q−n(1 − cqn)(1 − bd−1qn−N−1), ˜bn(λ) = 1, ¯δ = (0, 0, −1, −1), +(4.191) +(e) : +˜fn(λ) = −q−n(1 − bqn)(1 − cd−1qn−N−1), ˜bn(λ) = 1, ¯δ = (0, −1, 0, −1), +(4.192) +(f) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (−1, 0, 0, −1), +(4.193) +dqH : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (0, 1, −1), +(4.194) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − aqn−1), +¯δ = (1, 0, 0), +(4.195) +(c) : +˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − b−1qn−N+1), +¯δ = (0, 1, 0), +(4.196) +(d) : +˜fn(λ) = −q−n(1 − b−1qn−N), ˜bn(λ) = 1, +¯δ = (0, −1, 0), +(4.197) +(e) : +˜fn(λ) = −q−n(1 − aqn), ˜bn(λ) = 1, +¯δ = (−1, 0, 0), +(4.198) +(f) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (0, −1, 1), +(4.199) +dqK : (a) : +˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), +¯δ = (1, −1), +(4.200) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −q−n, +¯δ = (1, 0), +(4.201) +(c) : +˜fn(λ) = −q−n, ˜bn(λ) = 1, +¯δ = (−1, 0), +(4.202) +(d) : +˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, +¯δ = (−1, 1), +(4.203) +M : (a) : +˜fn(λ) = 1, ˜bn(λ) = −(n + β − 1), +¯δ = (1, 0), +(4.204) +(b) : +˜fn(λ) = −(n + β), ˜bn(λ) = 1, +¯δ = (−1, 0), +(4.205) +lqJ : (a) : +˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − aqn)(1 − bqn−1), +¯δ = (−1, 1), +(4.206) +(b) : +˜fn(λ) = −q−n(1 − aqn−1)(1 − bqn), ˜bn(λ) = 1, +¯δ = (1, −1), +(4.207) +lqL : (a) : +˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − aqn), +¯δ = −1, +(4.208) +(b) : +˜fn(λ) = −q−n(1 − aqn−1), ˜bn(λ) = 1, +¯δ = 1, +(4.209) +qM : (a) : +˜fn(λ) = 1, ˜bn(λ) = qn + cq−1, +¯δ = (0, 1), +(4.210) +(b) : +˜fn(λ) = 1, ˜bn(λ) = −b−1(1 − bqn), +¯δ = (1, 0), +(4.211) +(c) : +˜fn(λ) = −b−1q−1(1 − bqn+1), ˜bn(λ) = 1, +¯δ = (−1, 0), +(4.212) +(d) : +˜fn(λ) = qn + c, ˜bn(λ) = 1, +¯δ = (0, −1), +(4.213) +ASCII : (a) : +˜fn(λ) = 1, ˜bn(λ) = qn, +¯δ = 1, +(4.214) +(b) : +˜fn(λ) = qn, ˜bn(λ) = 1, +¯δ = −1, +(4.215) +qC : (a) : +˜fn(λ) = 1, ˜bn(λ) = qn + aq−1, +¯δ = 1, +(4.216) +22 + +(b) : +˜fn(λ) = qn + a, ˜bn(λ) = 1, +¯δ = −1. +(4.217) +Then we can show that B1(x), B2(x), D1(x) and D2(x) satisfy +B1(x − 1; λ)D2(x; λ) + D1(x + 1; λ)B2(x; λ) − ˜f0(λ)˜b0(λ) = −B(x; λ) − D(x; λ), (4.218) +and the constants ˜fn and ˜bn satisfy +En(λ) = ˜f0(λ)˜b0(λ) − ˜fn(λ)˜bn(λ) (n ∈ Z≥0). +(4.219) +The relation (4.218) gives another factorization of � +H(λ) (2.4), +� +H(λ) = − ˜B(λ) ˜F(λ) + ˜f0(λ)˜b0(λ). +(4.220) +Corresponding to this factorization (4.220), we obtain the following relations. +Theorem 3.3 For the polynomials in § A.3 (except C and qB), the following forward and +backward shift relations hold for n ∈ Z≥0, +˜F(λ) ˇPn(x; λ) = ˜fn(λ) ˇPn(x + s; λ − ¯δ), +(4.221) +˜B(λ) ˇPn(x + s; λ − ¯δ) = ˜bn(λ) ˇPn(x; λ), +(4.222) +where s is given by +s = + + + + + + + +1 +: H (a)(b), K (a), R (a)(b)(c), dH (a)(b)(c), dqqK (a)(b), qH (a)(b), +qK (a), qqK (a)(b), aqK (a)(b), qR (a)(b)(c), dqH (a)(b)(c), +dqK (a)(b), M (a), lqJ (a), lqL (a), qM (a)(b), ASCII (a), qC (a) +0 +: others +. +(4.223) +Proof: It is sufficient to show (4.221), because (2.7) and (4.219)–(4.221) imply (4.222). +Taking qR (a) (4.122) as an example, let us prove (4.221). It is shown by direct calculation: +˜F(λ) ˇPn(x; λ) += (1 − a−1dqx+1)(1 − qx+1) +(1 − a−1q)(1 − dq2x+1) +4φ3 +�q−n, abcd−1qn−1, q−x, dqx +a, b, c +��� q ; q +� +− +(1 − aqx)(1 − dqx) +(aq−1 − 1)(1 − dq2x+1) 4φ3 +�q−n, abcd−1qn−1, q−x−1, dqx+1 +a, b, c +��� q ; q +� += +1 +(1 − aq−1)(1 − dq2x+1) +n +� +k=0 +(q−n, abcd−1qn−1, q−x−1, dqx ; q)k +(a, b, c ; q)k +qk +(q ; q)k +× +� +−aq−1(1 − a−1dqx+1)(−qx+1)(1 − q−x+k−1) + (1 − aqx)(1 − dqx+k) +� +23 + += +1 +(1 − aq−1)(1 − dq2x+1) +n +� +k=0 +(q−n, abcd−1qn−1, q−x−1, dqx ; q)k +(a, b, c ; q)k +qk +(q ; q)k +(1 − aqk−1)(1 − dq2x+1) += +n +� +k=0 +(q−n, abcd−1qn−1, q−x−1, dqx ; q)k +(aq−1, b, c ; q)k +qk +(q ; q)k += ˜fn(λ) ˇPn(x + s; λ − ¯δ). +The other cases are proved in the same way. +Remark 3.1 Two formulas with ¯δ and −¯δ are equivalent by interchanging ˜F and ˜B, e.g. +(4.221) and (4.222) for H (c) agree with (4.222) and (4.221) for H (b) with the replacements +a → a + 1 and b → b − 1, respectively. For C and qB, we do not have new factorization +(4.220) and new forward and backward shift relations (4.221)–(4.222). +Remark 3.2 +The relations (4.221)–(4.222) for twelve cases ((a) of H, K, R, dH, dqqK, +qH, qK, qqK, aqK, qR, dqH, dqK, which have ˜fn = 1, ˜bn(λ) = EN+1(λ) − En(λ), s = 1 +and D1(0; λ) = B1(N; λ) = 0) were given in [9] and they were called forward and backward +x-shift relations. By considering e−∂ ˜F(λ) and ˜B(λ)e∂, the above results (4.220) and (4.221)– +(4.222) with s = 1 are rewritten as +� +H(λ) = − +� ˜B(λ)e∂�� +e−∂ ˜F(λ) +� ++ ˜f0(λ)˜b0(λ), +(4.224) +� +e−∂ ˜F(λ) +� ˇPn(x; λ) = ˜fn(λ) ˇPn(x; λ − ¯δ), +(4.225) +� ˜B(λ)e∂� ˇPn(x; λ − ¯δ) = ˜bn(λ) ˇPn(x; λ). +(4.226) +That is, x is not shifted. As an identity of polynomial, the x-shift is not essential. However, +this x-shift has important implications in the state-adding Darboux transformation for the +finite rdQM systems [9, 12]. +Remark 3.3 AW and qR polynomials are related as [2, 13] +eixAW = d +1 +2qxqR, +(a1, a2, a3, a4) = (ad− 1 +2, bd− 1 +2, cd− 1 +2, d +1 +2), +ˇP AW +n +(xAW; λAW) = d− n +2 (a, b, c ; q)n ˇP qR +n (xqR; λqR). +(4.227) +For the (j, k) = (1, 4) case in (4.33), the operators ˜F and ˜B for AW are related to those for +qR (a) (4.122) as +e +γ +2 p ˜F AW(λAW) = −(q−N−1 − 1) ˜F qR(λqR), +˜BAW(λAW)e− γ +2 p = (q−N−1 − 1)−1 ˜BqR(λqR). +(4.228) +24 + +These extra factors e± γ +2 p give the property in Remark 2.2. Similarly AW with (j, k) = (2, 4), +(3, 4), (1, 2), (1, 3) and (2, 4) cases correspond to qR (b), (c), (d), (e) and (f), respectively. +Remark 3.4 We can show that +B1(x − s; λ)B2(x − s + 1; λ) = B(x; λ − ¯δ), +D1(x − s + 1; λ)D2(x − s; λ) = D(x; λ − ¯δ), +B1(x − s; λ)D2(x − s + 1; λ) + D1(x − s + 1; λ)B2(x − s; λ) − ˜f0(λ)˜b0(λ) +(4.229) += B1(x − 1; λ − ¯δ)D2(x; λ − ¯δ) + D1(x + 1; λ − ¯δ)B2(x; λ − ¯δ) − ˜f0(λ − ¯δ)˜b0(λ − ¯δ), +which imply +˜F(λ) ˜B(λ) +��� +x→x−s − ˜f0(λ)˜b0(λ) = ˜B(λ − ¯δ) ˜F(λ − ¯δ) − ˜f0(λ − ¯δ)˜b0(λ − ¯δ). +(4.230) +4.4 +Polynomials in rdQMJ systems +For the rdQMJ systems described by the polynomials in § A.4 (except dqHeI, dqHeII and +SW), let us define the operators ˜F J(λ) and ˜BJ(λ) as follows: +˜F J(λ) +def += DJ +1(qη; λ) + BJ +1(η; λ)qη d +dη , +(4.231) +˜BJ(λ) +def += BJ +2(η; λ) + DJ +2(η; λ)q−η d +dη . +(4.232) +The potential functions BJ +1(η), BJ +2(η), DJ +1(η) and DJ +2(η) satisfy +BJ(η; λ) = BJ +1(η; λ)BJ +2(η; λ), +DJ(η; λ) = DJ +1(η; λ)DJ +2(η; λ), +(4.233) +and their explicit forms are given by +bqJ : (a) : BJ +1(η; λ) = η−1a(1 − η) +1 − a +, +BJ +2(η; λ) = (1 − a)η−1q(bη − c), +DJ +1(η; λ) = η−1(aq − η) +a − 1 +, +DJ +2(η; λ) = (a − 1)η−1(η − cq), +(4.234) +(b) : BJ +1(η; λ) = η−1(1 − η) +c−1 − 1 , +BJ +2(η; λ) = (c−1 − 1)η−1aq(bη − c), +DJ +1(η; λ) = η−1(η − cq) +1 − c +, +DJ +2(η; λ) = (1 − c)η−1(aq − η), +(4.235) +(c) : BJ +1(η; λ) = (c−1q−1 − 1)η−1aq(bη − c), +BJ +2(η; λ) = η−1(1 − η) +c−1q−1 − 1, +DJ +1(η; λ) = (1 − cq)η−1(aq − η), +DJ +2(η; λ) = η−1(η − cq) +1 − cq +, +(4.236) +(d) : BJ +1(η; λ) = (1 − aq)η−1(bη − c), +BJ +2(η; λ) = η−1aq(1 − η) +1 − aq +, +25 + +DJ +1(η; λ) = (aq − 1)η−1(η − cq), +DJ +2(η; λ) = η−1(aq − η) +aq − 1 +, +(4.237) +bqL : (a) : BJ +1(η; λ) = η−1a(1 − η) +1 − a +, +BJ +2(η; λ) = −(1 − a)η−1bq, +DJ +1(η; λ) = η−1(aq − η) +a − 1 +, +DJ +2(η; λ) = (a − 1)η−1(η − bq), +(4.238) +(b) : BJ +1(η; λ) = η−1b(1 − η) +1 − b +, +BJ +2(η; λ) = −(1 − b)η−1aq, +DJ +1(η; λ) = η−1(η − bq) +1 − b +, +DJ +2(η; λ) = (1 − b)η−1(aq − η), +(4.239) +(c) : BJ +1(η; λ) = (bq − 1)η−1a, +BJ +2(η; λ) = −η−1bq(1 − η) +bq − 1 +, +DJ +1(η; λ) = (1 − bq)η−1(aq − η), +DJ +2(η; λ) = η−1(η − bq) +1 − bq +, +(4.240) +(d) : BJ +1(η; λ) = (aq − 1)η−1b, +BJ +2(η; λ) = −η−1aq(1 − η) +aq − 1 +, +DJ +1(η; λ) = (aq − 1)η−1(η − bq), +DJ +2(η; λ) = η−1(aq − η) +aq − 1 +, +(4.241) +ASCI : (a) : BJ +1(η; λ) = η−1q−1, +BJ +2(η; λ) = −η−1a, +DJ +1(η; λ) = −η−1(1 − η), +DJ +2(η; λ) = −η−1(η − a), +(4.242) +(b) : BJ +1(η; λ) = −η−1aq−1, +BJ +2(η; λ) = η−1, +DJ +1(η; λ) = −η−1(η − a), +DJ +2(η; λ) = −η−1(1 − η), +(4.243) +qL : (a) : BJ +1(η; λ) = η−1(1 + η), +BJ +2(η; λ) = 1, +DJ +1(η; λ) = −η−1q, +DJ +2(η; λ) = −a−1q−1, +(4.244) +(b) : BJ +1(η; λ) = 1, +BJ +2(η; λ) = η−1(1 + η), +DJ +1(η; λ) = −a−1, +DJ +2(η; λ) = −η−1. +(4.245) +Let us define the constants ˜f J +n(λ), ˜bJ +n(λ) and ¯δ as follows: +bqJ : (a) : +˜f J +n(λ) = 1, ˜bJ +n(λ) = −q−n(1 − aqn)(1 − bqn+1), +¯δ = (1, −1, 0), +(4.246) +(b) : +˜f J +n(λ) = 1, ˜bJ +n(λ) = −q−n(1 − cqn)(1 − abc−1qn+1), +¯δ = (0, 0, 1), +(4.247) +(c) : +˜f J +n(λ) = −q−n(1 − cqn+1)(1 − abc−1qn), ˜bJ +n(λ) = 1, +¯δ = (0, 0, −1), +(4.248) +(d) : +˜f J +n(λ) = −q−n(1 − aqn+1)(1 − bqn), ˜bJ +n(λ) = 1, +¯δ = (−1, 1, 0), +(4.249) +bqL : (a) : +˜f J +n(λ) = 1, ˜bJ +n(λ) = −q−n(1 − aqn), +¯δ = (1, 0), +(4.250) +(b) : +˜f J +n(λ) = 1, ˜bJ +n(λ) = −q−n(1 − bqn), +¯δ = (0, 1), +(4.251) +26 + +(c) : +˜f J +n(λ) = −q−n(1 − bqn+1), ˜bJ +n(λ) = 1, +¯δ = (0, −1), +(4.252) +(d) : +˜f J +n(λ) = −q−n(1 − aqn+1), ˜bJ +n(λ) = 1, +¯δ = (−1, 0), +(4.253) +ASCI : (a) : +˜f J +n(λ) = 1, ˜bJ +n(λ) = −q−n, +¯δ = −1, +(4.254) +(b) : +˜f J +n(λ) = −q−n, ˜bJ +n(λ) = 1, +¯δ = 1, +(4.255) +qL : (a) : +˜f J +n(λ) = 1, ˜bJ +n(λ) = −a−1q−1(1 − aqn+1), +¯δ = −1, +(4.256) +(b) : +˜f J +n(λ) = −a−1(1 − aqn), ˜bJ +n(λ) = 1, +¯δ = 1. +(4.257) +Then we can show that BJ +1(η), BJ +2(η), DJ +1(η) and DJ +2(η) satisfy +BJ +1(q−1η; λ)DJ +2(η; λ) + DJ +1(qη; λ)BJ +2(η; λ) − ˜f J +0 (λ)˜bJ +0(λ) = −BJ(η; λ) − DJ(η; λ), +(4.258) +and the constants ˜f J +n and ˜bJ +n satisfy +En(λ) = ˜f J +0 (λ)˜bJ +0(λ) − ˜f J +n(λ)˜bJ +n(λ) (n ∈ Z≥0). +(4.259) +The relations (4.258) give other factorizations of � +H(λ) (2.6), +� +HJ(λ) = − ˜BJ(λ) ˜F J(λ) + ˜f J +0 (λ)˜bJ +0(λ). +(4.260) +Corresponding to this factorization (4.260), we obtain the following relations. +Theorem 3.4 For the polynomials in § A.4 (except dqHeI, dqHeII and SW), the following +forward and backward shift relations hold for n ∈ Z≥0, +˜F J(λ)Pn(η; λ) = ˜f J +n(λ)Pn(r′η; λ − ¯δ), +(4.261) +˜BJ(λ)Pn(r′η; λ − ¯δ) = ˜bJ +n(λ)Pn(η; λ), +(4.262) +where r′ is given by +r′ = +� q +: bqJ (c)(d), bqL (c)(d), ASCI (a), qL (b) +1 +: bqJ (a)(b), bqL (a)(b), ASCI (b), qL (a) . +(4.263) +Proof: It is sufficient to show (4.261), because (2.8) and (4.259)–(4.261) imply (4.262). +Taking bqJ (a) (4.234) as an example, let us prove (4.261). It is shown by direct calculation: +˜F J(λ)Pn(η; λ) += η−1(a − η) +a − 1 +3φ2 +�q−n, abqn+1, η +aq, cq +��� q ; q +� ++ η−1a(1 − η) +1 − a +3φ2 +�q−n, abqn+1, qη +aq, cq +��� q ; q +� +27 + += η−1 +1 − a +n +� +k=0 +(q−n, abqn+1, η ; q)k +(aq, cq ; q)k +qk +(q ; q)k +� +−(a − η) + a(1 − ηqk) +� += η−1 +1 − a +n +� +k=0 +(q−n, abqn+1, η ; q)k +(aq, cq ; q)k +qk +(q ; q)k +(1 − aqk)η += +n +� +k=0 +(q−n, abqn+1, η ; q)k +(a, cq ; q)k +qk +(q ; q)k += ˜f J +n(λ)Pn(r′η; λ − ¯δ). +The other cases are proved in the same way. +Remark 4.1 +Two formulas with ¯δ and −¯δ are equivalent by interchanging ˜F J and ˜BJ, +e.g. +(4.261) and (4.262) for bqJ (d) agree with (4.262) and (4.261) for bqJ (a) with the +replacements a → aq and b → bq−1, respectively. For dqHeI, dqHeII and SW, we do not +have new factorization (4.260) and new forward and backward shift relations (4.261)–(4.262). +Remark 4.2 As in Remark 3.2, by considering q−η d +dη ˜F J(λ) and ˜BJ(λ)qη d +dη , the above results +(4.260) and (4.261)–(4.262) with r′ = q are rewritten as +� +HJ(λ) = − +� ˜BJ(λ)qη d +dη �� +q−η d +dη ˜F J(λ) +� ++ ˜f J +0 (λ)˜bJ +0(λ), +(4.264) +� +q−η d +dη ˜F J(λ) +� +Pn(η; λ) = ˜f J +n(λ)Pn(η; λ − ¯δ), +(4.265) +� ˜BJ(λ)qη d +dη � +Pn(η; λ − ¯δ) = ˜bJ +n(λ)Pn(η; λ). +(4.266) +That is, η is not q-shifted. As an identity of polynomial, the q-shift of η is not essential. +Remark 4.3 We can show that +BJ +1(r′ −1η; λ)BJ +2(qr′ −1η; λ) = BJ(η; λ − ¯δ), +DJ +1(qr′ −1η; λ)DJ +2(r′ −1η; λ) = DJ(η; λ − ¯δ), +BJ +1(r′ −1η; λ)DJ +2(qr′ −1η; λ) + DJ +1(qr′ −1η; λ)BJ +2(r′ −1η; λ) − ˜f J +0 (λ)˜bJ +0(λ) +(4.267) += BJ +1(q−1η; λ − ¯δ)DJ +2(η; λ − ¯δ) + DJ +1(qη; λ − ¯δ)BJ +2(η; λ − ¯δ) − ˜f J +0 (λ − ¯δ)˜bJ +0(λ − ¯δ), +which imply +˜F J(λ) ˜BJ(λ) +��� +η→r′ −1η − ˜f J +0 (λ)˜bJ +0(λ) = ˜BJ(λ − ¯δ) ˜F J(λ − ¯δ) − ˜f J +0 (λ − ¯δ)˜bJ +0(λ − ¯δ). +(4.268) +5 +Summary and Comments +The orthogonal polynomials in the Askey scheme satisfy second order differential or difference +equations (Theorem 1) and we study them by using quantum mechanical formulation (oQM, +28 + +idQM, rdQM, rdQMJ). The forward and backward shift relations are their basic properties +(Theorem 2.1, 2.2), in which the degree n and the parameters λ are shifted. +They are +based on the factorizations of the differential or difference operators � +H (3.1) and � +HJ (3.18). +Motivated by the recently found forward and backward x-shift relations [9], in which the +coordinate x and parameters λ are shifted, we have tried to find new forward and backward +relations. We have found new factorizations of � +H (4.19), (4.78), (4.220) and � +HJ (4.260), +and based on them, we have obtained another type of forward and backward shift relations +(Theorem 3.1, 3.2, 3.3, 3.4). In these new forward and backward shift relations except for +some cases of rdQM and rdQMJ, only the parameters λ are shifted. +As an identity of +polynomial, the x-shift (or q-shift of η) is not essential (Remark 3.2, 4.2). +The forward and backward shift relations are related to the shape invariance property +of quantum mechanical systems [4, 5, 3, 10]. It is an interesting problem to investigate the +quantum mechanical implications of the new forward and backward shift relations obtained +in this paper (cf. Remark 2.3, 3.4, 4.3). Especially the twelve finite rdQM cases in Remark 3.2 +are interesting. In these cases, the x-shift has important implications related to the state- +adding Darboux transformations [9, 12]. We will report this topic elsewhere. +The case-(1) multi-indexed orthogonal polynomials are constructed for R and qR [7], W +and AW [8], M, lqJ and lqL [14], cH and MP [15], and they have shape invariant property, +namely, satisfy the forward and backward shift relations like Theorem 2.1. It is an interesting +problem to investigate whether these multi-indexed polynomials satisfy new forward and +backward shift relations such as Theorem 3.2 and 3.3. +Acknowledgements +This work is supported by JSPS KAKENHI Grant Number JP19K03667. +A +Data for Orthogonal Polynomials +We give the data for the orthogonal polynomials (2.1) in the Askey scheme (all the poly- +nomials in chapters 9 and 14 of [2] and the dual quantum q-Krawtchouk), which satisfy +second order differential or difference equations. The parametrization of some polynomials +are different from the conventional ones. Since we do not consider orthogonality relations in +29 + +this paper, we do not care about concrete ranges of parameters. The Pochhammer symbol +(shifted factorial), the hypergeometric series and their q-versions are defined by +(a)n +def += +n−1 +� +j=0 +(a + j), +(a1, . . . , ar)n +def += +r� +k=1 +(ak)n, +(A.1) +(a ; q)n +def += +n−1 +� +j=0 +(1 − aqj), +(a1, . . . , ar ; q)n +def += +r� +k=1 +(ak ; q)n, +(A.2) +rFs +�a1, . . . , ar +b1, . . . , bs +��� x +� +def += +∞ +� +k=0 +(a1, . . . , ar)k +(b1, . . . , bs)k +xk +k! , +(A.3) +rφs +�a1, . . . , ar +b1, . . . , bs +��� q ; z +� +def += +∞ +� +k=0 +(a1, . . . , ar ; q)k +(b1, . . . , bs ; q)k +(−1)(1+s−r)kq(1+s−r)(k +2) +zk +(q ; q)k +, +(A.4) +with the conventions �n−1 +j=n ∗ = 0 and �n−1 +j=n ∗ = 1. +A.1 +Polynomials in oQM +We consider the following five polynomials [2, 3, 16]: Hermite (He), Laguerre (L), Jacobi +(J), Bessel (B) and pseudo Jacobi (pJ). The sinusoidal coordinates η(x) are +(i) : +η(x) = x +: He, +(ii) : +η(x) = x2 +: L, +(iii) : +η(x) = cos 2x +: J, +(iv) : +η(x) = ex +: B, +(v) : +η(x) = sinh x +: pJ. +(A.5) +The functions c1(η; λ) and c2(η) are defined by +4c1(η; λ) +def += d2η(x) +dx2 ++ 2dw(x; λ) +dx +dη(x) +dx , +4c2(η) +def += +�dη(x) +dx +�2 +, +(A.6) +and the potential U(x; λ) is given by (2.5). +The parameters λ are real. +The standard +parametrizations [2] are as follows: +L : αstandard = g − 1 +2, +J : (α, β)standard = (g − 1 +2, h − 1 +2), +B : astandard = −2h − 1, +pJ : (N, ν)standard = (h − 1 +2, −µ), +(A.7) +and h of pJ is a continuous parameter. +30 + +A.1.1 +(i) η(x) = x +He : λ : none, +δ : none, +κ = 1, +En(λ) = 2n, +fn(λ) = 2n, +bn(λ) = 1, +c1(η; λ) = −1 +2η, +c2(η) = 1 +4, +cF = 1, +w(x; λ) = −1 +2x2, +U(x; λ) = x2 − 1, +Pn(η; λ) = (2η)n +2F0 +�−n +2, n−1 +2 +− +��� − 1 +η2 +� += n! +[ n +2 ] +� +k=0 +(−1)k(2η)n−2k +k! (n − 2k)! , +(A.8) +where [x] denotes the greatest integer not exceeding x. +A.1.2 +(ii) η(x) = x2 +L : λ = g, +δ = 1, +κ = 1, +En(λ) = 4n, +fn(λ) = −2, +bn(λ) = −2(n + 1), +c1(η; λ) = g + 1 +2 − η, +c2(η) = η, +cF = 2, +w(x; λ) = −1 +2x2 + g log x, +U(x; λ) = x2 + g(g − 1) +x2 +− 2g − 1, +Pn(η; λ) = (g + 1 +2)n +n! +1F1 +� −n +g + 1 +2 +��� η +� += L +(g− 1 +2) +n +(η). +(A.9) +A.1.3 +(iii) η(x) = cos 2x +J : λ = (g, h), +δ = (1, 1), +κ = 1, +En(λ) = 4n(n + g + h), +fn(λ) = −2(n + g + h), +bn(λ) = −2(n + 1), +c1(η; λ) = h − g − (g + h + 1)η, +c2(η) = 1 − η2, +cF = −4, +w(x; λ) = g log sin x + h log cos x, +U(x; λ) = g(g − 1) +sin x2 ++ h(h − 1) +cos x2 +− (g + h)2, +Pn(η; λ) = (g + 1 +2)n +n! +2F1 +�−n, n + g + h +g + 1 +2 +��� 1 − η +2 +� += P +(g− 1 +2 ,h− 1 +2) +n +(η). +(A.10) +A.1.4 +(iv) η(x) = ex +B : λ = h, +δ = −1, +κ = 1, +En(λ) = n(2h − n), +fn(λ) = −1 +2n(2h − n), +bn(λ) = −2, +4c1(η; λ) = 2 + (1 − 2h)η, +4c2(η) = η2, +cF = 1, +w(x; λ) = −hx − e−x, +U(x; λ) = e−2x − (2h + 1)e−x + h2, +31 + +Pn(η; λ) = 2F0 +�−n, n − 2h +− +��� −η +2 +� += (n − 2h)n +�η +2 +�n +1F0 +� +−n +2h + 1 − 2n +��� 2 +η +� += (−1)nn! +�η +2 +�n +L(2h−2n) +n +(2η−1). +(A.11) +A.1.5 +(v) η(x) = sinh x +pJ : λ = (h, µ), +δ = (−1, 0), +κ = 1, +En(λ) = n(2h − n), +fn(λ) = n, +bn(λ) = 2h − n − 1, +4c1(η; λ) = (1 − 2h)η − 2µ, +4c2(η) = 1 + η2, +cF = 1, +w(x; λ) = −h log cosh x − µ tan−1 sinh x, +U(x; λ) = −h(h + 1) + µ2 + µ(2h + 1) sinh x +cosh2 x ++ h2, +Pn(η; λ) = (−2i)n(−h + 1 +2 − iµ)n +(n − 2h)n +2F1 +� −n, n − 2h +−h + 1 +2 − iµ +��� 1 − iη +2 +� += (η + i)n +2F1 +�−n, h + 1 +2 + iµ − n +2h + 1 − 2n +��� +2 +1 − iη +� += (−2i)nn! +(n − 2h)n +P +(−h− 1 +2−iµ,−h− 1 +2 +iµ) +n +(iη). +(A.12) +The Bessel and pseudo Jacobi polynomials have not been treated in our previous papers. +Their oQM systems are the Morse potential (§ 4.1 of [16] with the replacement x → −x. +The parameter µ can be taken as µ = 1 by shifting x.) and the hyperbolic symmetric top +II, respectively [16]. Their orthogonality relations are (parameters: h, µ > 0) +� ∞ +−∞ +dx φ0(x; λ)2 ˇPn(x; λ) ˇPm(x; λ) = hn(λ)δnm (n, m = 0, 1, . . . , [h]′), +(A.13) +where φ0(x; λ) = ew(x;λ) and [x]′ denotes the greatest integer not equal or exceeding x. The +normalization constants hn(λ) are given by [2, 16] +B : hn(λ) = n! Γ(2h − n + 1) +22h+1(h − n) +, +(A.14) +pJ : hn(λ) = +2πn! 22n−2hΓ(2h − 2n) +(2h − 2n + 1)nΓ(h − n + 1 +2 − iµ)Γ(h − n + 1 +2 + iµ). +(A.15) +A.2 +Polynomials in idQM +We consider the following thirteen polynomials [5]: continuous Hahn (cH), Meixner-Pollaczek +(MP), Wilson (W), continuous dual Hahn (cdH), Askey-Wilson (AW), continuous dual q- +32 + +Hahn (cdqH), Al-Salam-Chihara (ASC), continuous big q-Hermite (cbqHe), continuous q- +Hermite (cqHe), continuous q-Jacobi (cqJ), continuous q-Laguerre (cqL), continuous q-Hahn +(cqH) and q-Meixner-Pollaczek (qMP). The sinusoidal coordinates η(x) and auxiliary func- +tions ϕ(x) are [5] +(i) : η(x) = x, +ϕ(x) = 1, +(ii) : η(x) = x2, +ϕ(x) = 2x, +(iii) : η(x) = cos x, +ϕ(x) = 2 sin x, +(A.16) +(iv) : η(x) = cos(x + φ), +ϕ(x) = 2 sin(x + φ), +and +(i) : cH, MP, +(ii) : W, cdH, +(iii) : AW, cdqH, ASC, cbqHe, cqHe, cqJ, cqL, +(iv) : cqH, qMP. +(A.17) +The constant γ is γ = 1 for non q-polynomials, γ = log q for q-polynomials. The parameters +λ are complex unless mentioned, and satisfy the following; W and AW: {a∗ +1, a∗ +2, a∗ +3, a∗ +4} = +{a1, a2, a3, a4} (as a set), cdH and cdqH: {a∗ +1, a∗ +2, a∗ +3} = {a1, a2, a3} (as a set), ASC and cqH: +{a∗ +1, a∗ +2} = {a1, a2} (as a set). The standard parametrizations [2] are as follows: +cH : (a, b, c, d)standard = (a1, a2, a∗ +1, a∗ +2), +MP : (λ, φ)standard = (a, φ), +W, AW : (a, b, c, d)standard = (a1, a2, a3, a4), +cdH, cdqH : (a, b, c)standard = (a1, a2, a3), +ASC : (a, b)standard = (a1, a2), +cqH : (a, b, c, d, φ)standard = (a1, a2, a1, a2, φ). +(A.18) +Some polynomials are symmetric under the permutations of the following parameters: +W, AW : (a1, a2, a3, a4), +cdH, cdqH : (a1, a2, a3), +cH, ASC, cqH : (a1, a2). +(A.19) +A.2.1 +(i) η(x) = x +cH : λ = (a1, a2), +δ = ( 1 +2, 1 +2), +κ = 1, +b1 +def += a1 + a2 + a∗ +1 + a∗ +2, +En(λ) = n(n + b1 − 1), +fn(λ) = n + b1 − 1, +bn(λ) = n + 1, +V (x; λ) = (a1 + ix)(a2 + ix), +ˇPn(x; λ) = in(a1 + a∗ +1, a1 + a∗ +2)n +n! +3F2 +�−n, n + b1 − 1, a1 + ix +a1 + a∗ +1, a1 + a∗ +2 +��� 1 +� +, +(A.20) +MP : λ = (a, φ), +δ = ( 1 +2, 0), +κ = 1, +a, φ ∈ R, +33 + +En(λ) = 2n sin φ, +fn(λ) = 2 sin φ, +bn(λ) = n + 1, +V (x; λ) = ei( π +2 −φ)(a + ix), +ˇPn(x; λ) = (2a)neinφ +n! +2F1 +�−n, a + ix +2a +��� 1 − e−2iφ� +. +(A.21) +A.2.2 +(ii) η(x) = x2 +W : λ = (a1, a2, a3, a4), +δ = ( 1 +2, 1 +2, 1 +2, 1 +2), +κ = 1, +b1 +def += a1 + a2 + a3 + a4, +En(λ) = n(n + b1 − 1), +fn(λ) = −n(n + b1 − 1), +bn(λ) = −1, +V (x; λ) = (a1 + ix)(a2 + ix)(a3 + ix)(a4 + ix) +2ix(2ix + 1) +, +ˇPn(x; λ) = (a1 + a2, a1 + a3, a1 + a4)n 4F3 +�−n, n + b1 − 1, a1 + ix, a1 − ix +a1 + a2, a1 + a3, a1 + a4 +��� 1 +� +, +(A.22) +cdH : λ = (a1, a2, a3), +δ = ( 1 +2, 1 +2, 1 +2), +κ = 1, +En(λ) = n, +fn(λ) = −n, +bn(λ) = −1, +V (x; λ) = (a1 + ix)(a2 + ix)(a3 + ix) +2ix(2ix + 1) +, +ˇPn(x; λ) = (a1 + a2, a1 + a3)n 3F2 +�−n, a1 + ix, a1 − ix +a1 + a2, a1 + a3 +��� 1 +� +. +(A.23) +A.2.3 +(iii) η(x) = cos x +AW : qλ = (a1, a2, a3, a4), +δ = ( 1 +2, 1 +2, 1 +2, 1 +2), +κ = q−1, +b4 +def += a1a2a3a4, +En(λ) = (q−n − 1)(1 − b4qn−1), +fn(λ) = q +n +2 (q−n − 1)(1 − b4qn−1), +bn(λ) = q− n+1 +2 , +V (x; λ) = (1 − a1eix)(1 − a2eix)(1 − a3eix)(1 − a4eix) +(1 − e2ix)(1 − qe2ix) +, +ˇPn(x; λ) = (a1a2, a1a3, a1a4 ; q)n +an +1 +4φ3 +�q−n, b4qn−1, a1eix, a1e−ix +a1a2, a1a3, a1a4 +��� q ; q +� +, +(A.24) +cdqH : qλ = (a1, a2, a3), +δ = ( 1 +2, 1 +2, 1 +2), +κ = q−1, +En(λ) = q−n − 1, +fn(λ) = q +n +2 (q−n − 1), +bn(λ) = q− n+1 +2 , +V (x; λ) = (1 − a1eix)(1 − a2eix)(1 − a3eix) +(1 − e2ix)(1 − qe2ix) +, +ˇPn(x; λ) = (a1a2, a1a3 ; q)n +an +1 +3φ2 +�q−n, a1eix, a1e−ix +a1a2, a1a3 +��� q ; q +� +, +(A.25) +ASC : qλ = (a1, a2), +δ = ( 1 +2, 1 +2), +κ = q−1, +En(λ) = q−n − 1, +fn(λ) = q +n +2 (q−n − 1), +bn(λ) = q− n+1 +2 , +34 + +V (x; λ) = (1 − a1eix)(1 − a2eix) +(1 − e2ix)(1 − qe2ix) , +ˇPn(x; λ) = (a1a2 ; q)n +an +1 +3φ2 +�q−n, a1eix, a1e−ix +a1a2, 0 +��� q ; q +� +, +(A.26) +cbqHe : qλ = a, +δ = 1 +2, +κ = q−1, +En(λ) = q−n − 1, +fn(λ) = q +n +2 (q−n − 1), +bn(λ) = q− n+1 +2 , +V (x; λ) = +1 − aeix +(1 − e2ix)(1 − qe2ix), +ˇPn(x; λ) = a−n +3φ2 +�q−n, aeix, ae−ix +0, 0 +��� q ; q +� +, +(A.27) +cqHe : λ : none, +δ : none, +κ = q−1, +En(λ) = q−n − 1, +fn(λ) = q +n +2 (q−n − 1), +bn(λ) = q− n+1 +2 , +V (x; λ) = +1 +(1 − e2ix)(1 − qe2ix), +ˇPn(x; λ) = einx +2φ0 +�q−n, 0 +− +��� q ; qne−2ix� +, +(A.28) +cqJ : λ = (α, β), +δ = (1, 1), +κ = q−1, +α, β ∈ R, +En(λ) = (q−n − 1)(1 − qn+α+β+1), +fn(λ) = +q +1 +2(α+ 3 +2)q−n(1 − qn+α+β+1) +(1 + q +1 +2 (α+β+1))(1 + q +1 +2(α+β+2)) +, +bn(λ) = q− 1 +2(α+ 3 +2)qn+1(q−n−1 − 1)(1 + q +1 +2(α+β+1))(1 + q +1 +2 (α+β+2)), +V (x; λ) = (1 − q +1 +2(α+ 1 +2)eix)(1 − q +1 +2(α+ 3 +2)eix)(1 + q +1 +2(β+ 1 +2 )eix)(1 + q +1 +2 (β+ 3 +2)eix) +(1 − e2ix)(1 − qe2ix) +, +ˇPn(x; λ) = (qα+1 ; q)n +(q ; q)n +4φ3 +�q−n, qn+α+β+1, q +1 +2 (α+ 1 +2 )eix, q +1 +2(α+ 1 +2)e−ix +qα+1, −q +1 +2(α+β+1), −q +1 +2 (α+β+2) +��� q ; q +� +, +(A.29) +cqL : λ = α, +δ = 1, +κ = q−1, +α ∈ R, +En(λ) = q−n − 1, +fn(λ) = q +1 +2 (α+ 3 +2)q−n, +bn(λ) = q− 1 +2(α+ 3 +2)qn+1(q−n−1 − 1), +V (x; λ) = (1 − q +1 +2 (α+ 1 +2 )eix)(1 − q +1 +2 (α+ 3 +2 )eix) +(1 − e2ix)(1 − qe2ix) +, +ˇPn(x; λ) = (qα+1 ; q)n +(q ; q)n +3φ2 +�q−n, q +1 +2 (α+ 1 +2 )eix, q +1 +2(α+ 1 +2)e−ix +qα+1, 0 +��� q ; q +� +. +(A.30) +A.2.4 +(iv) η(x; λ) = cos(x + φ) +cqH : qλ = (a1, a2, qφ), +δ = ( 1 +2, 1 +2, 0), +κ = q−1, +b4 +def += a1a2a∗ +1a∗ +2 = a2 +1a2 +2, +φ ∈ R, +En(λ) = (q−n − 1)(1 − b4qn−1), +fn(λ) = q +n +2 (q−n − 1)(1 − b4qn−1), +bn(λ) = q− n+1 +2 , +V (x; λ) = (1 − a1ei(x+2φ))(1 − a2ei(x+2φ))(1 − a1eix)(1 − a2eix) +(1 − e2i(x+φ))(1 − qe2i(x+φ)) +, +35 + +ˇPn(x; λ) = (a1a2e2iφ, a2 +1, a1a2 ; q)n +an +1einφ +4φ3 +�q−n, b4qn−1, a1ei(x+2φ), a1e−ix +a1a2e2iφ, a2 +1, a1a2 +��� q ; q +� +, (A.31) +qMP : qλ = (a, qφ), +δ = ( 1 +2, 0), +κ = q−1, +a, φ ∈ R, +En(λ) = q−n − 1, +fn(λ) = q− n +2 , +bn(λ) = q− n+1 +2 (1 − qn+1), +V (x; λ) = +(1 − aei(x+2φ))(1 − aeix) +(1 − e2i(x+φ))(1 − qe2i(x+φ)), +ˇPn(x; λ) = +1 +(q ; q)n +(a2 ; q)n +aneinφ 3φ2 +�q−n, aei(x+2φ), ae−ix +a2, 0 +��� q ; q +� +. +(A.32) +A.3 +Polynomials in rdQM +The sinusoidal coordinates η(x) and auxiliary functions ϕ(x) are [4] +(i) : η(x) = x, +ϕ(x) = 1, +(ii) : η(x) = x(x + d), +ϕ(x) = 2x + 1 + d +1 + d +, +(iii) : η(x) = 1 − qx, +ϕ(x) = qx, +(A.33) +(iv) : η(x) = q−x − 1, +ϕ(x) = q−x, +(v) : η(x) = (q−x − 1)(1 − dqx), +ϕ(x) = q−x1 − dq2x+1 +1 − dq +. +Note that η(0) = 0 and ϕ(0) = 1. We impose the following normalization condition on the +polynomial (2.1), +ˇPn(0; λ) = Pn(0; λ) = 1. +(A.34) +This normalization condition implies the following two universal expressions. The coefficient +of the highest degree term, cn(λ), is expressed as ((A.14) in [17]), +ˇPn(x; λ) = cn(λ)η(x; λ)n + (lower degree terms), +cn(λ) = (−1)nκ−(n +2) +n +� +j=1 +En(λ) − Ej−1(λ) +η(j; λ)B(0; λ + (j − 1)δ), +(A.35) +and the constants fn(λ) and bn(λ) of the forward and backward shift relations (3.16)–(3.17) +are given by +fn(λ) = En(λ), +bn(λ) = 1. +(A.36) +The parameters λ are real. +36 + +A.3.1 +finite rdQM +We consider the following twelve polynomials [4]: Hahn (H), Krawtchouk (K), Racah (R), +dual Hahn (dH), dual quantum q-Krawtchouk (dqqK) (which is not treated in [2]), q-Hahn +(qH), q-Krawtchouk (qK), quantum q-Krawtchouk (qqK), affine q-Krawtchouk (aqK), q- +Racah (qR), dual q-Hahn (dqH) and dual q-Krawtchouk (dqK). We consider ǫ = ǫ′ = 1 cases +in [4]. Their sinusoidal coordinates (A.33) are +(i) : H, K, +(ii) : R, dH(d = a + b − 1), +(iii) : dqqK, +(iv) : qH, qK, qqK, aqK, +(v) : qR, dqH(d = abq−1), dqK(d = −p). +(A.37) +The standard parametrizations [2] are as follows: +H : (α, β)standard = (a − 1, b − 1), +dH : (γ, δ)standard = (a − 1, b − 1), +qH : (α, β)standard = (aq−1, bq−1), +dqH : (γ, δ)standard = (aq−1, bq−1), +R : (α, β, γ, δ)standard = (a − 1, b + c − d − 1, c − 1, d − c), +(A.38) +qR : (α, β, γ, δ)standard = (aq−1, bcd−1q−1, cq−1, dc−1), +dqK : cstandard = −pqN. +R and qR polynomials are symmetric under the permutations of the following parameters: +R : (b, c) +� +(a, b, c) if a = −N is not imposed +� +, +qR : (b, c) +� +(a, b, c) if a = q−N is not imposed +� +. +(A.39) +A.3.1.1 +(i) η(x) = x +H : λ = (a, b, N), +δ = (1, 1, −1), +κ = 1, +En(λ) = n(n + a + b − 1), +B(x; λ) = (x + a)(N − x), +D(x; λ) = x(b + N − x), +ˇPn(x; λ) = 3F2 +�−n, n + a + b − 1, −x +a, −N +��� 1 +� +, +(A.40) +K : λ = (p, N), +δ = (0, −1), +κ = 1, +En(λ) = n, +B(x; λ) = p(N − x), +D(x; λ) = (1 − p)x, +ˇPn(x; λ) = 2F1 +�−n, −x +−N +��� p−1� +. +(A.41) +A.3.1.2 +(ii) η(x; λ) = x(x + d) +R : We take a = −N and define ˜d = a + b + c − d − 1, +37 + +λ = (a, b, c, d), +δ = (1, 1, 1, 1) +κ = 1, +En(λ) = n(n + ˜d), +B(x; λ) = −(x + a)(x + b)(x + c)(x + d) +(2x + d)(2x + 1 + d) +, +D(x; λ) = −(x + d − a)(x + d − b)(x + d − c)x +(2x − 1 + d)(2x + d) +, +ˇPn(x; λ) = 4F3 +�−n, n + ˜d, −x, x + d +a, b, c +��� 1 +� +, +(A.42) +dH : λ = (a, b, N), +δ = (1, 0, −1), +κ = 1, +En(λ) = n, +B(x; λ) = (x + a)(x + a + b − 1)(N − x) +(2x − 1 + a + b)(2x + a + b) , +D(x; λ) = x(x + b − 1)(x + a + b + N − 1) +(2x − 2 + a + b)(2x − 1 + a + b), +ˇPn(x; λ) = 3F2 +�−n, x + a + b − 1, −x +a, −N +��� 1 +� +. +(A.43) +A.3.1.3 +(iii) η(x) = 1 − qx +dqqK : qλ = (p, qN), +δ = (0, −1), +κ = q−1, +En(λ) = q−n − 1, +B(x; λ) = p−1q−x−N−1(1 − qN−x), +D(x; λ) = (q−x − 1)(1 − p−1q−x), +ˇPn(x; λ) = 2φ1 +�q−n, q−x +q−N +��� q ; pqx+1� +. +(A.44) +A.3.1.4 +(iv) η(x) = q−x − 1 +qH : qλ = (a, b, qN), +δ = (1, 1, −1), +κ = q−1, +En(λ) = (q−n − 1)(1 − abqn−1), +B(x; λ) = (1 − aqx)(qx−N − 1), +D(x; λ) = aq−1(1 − qx)(qx−N − b), +ˇPn(x; λ) = 3φ2 +�q−n, abqn−1, q−x +a, q−N +��� q ; q +� +, +(A.45) +qK : qλ = (p, qN), +δ = (2, −1), +κ = q−1, +En(λ) = (q−n − 1)(1 + pqn), +B(x; λ) = qx−N − 1, +D(x; λ) = p(1 − qx), +ˇPn(x; λ) = 3φ2 +�q−n, q−x, −pqn +q−N, 0 +��� q ; q +� +, +(A.46) +qqK : qλ = (p, qN), +δ = (1, −1), +κ = q, +En(λ) = 1 − qn, +B(x; λ) = p−1qx(qx−N − 1), +D(x; λ) = (1 − qx)(1 − p−1qx−N−1), +ˇPn(x; λ) = 2φ1 +�q−n, q−x +q−N +��� q ; pqn+1� +, +(A.47) +aqK : qλ = (p, qN), +δ = (1, −1), +κ = q−1, +En(λ) = q−n − 1, +B(x; λ) = (qx−N − 1)(1 − pqx+1), +D(x; λ) = pqx−N(1 − qx), +38 + +ˇPn(x; λ) = 3φ2 +�q−n, q−x, 0 +pq, q−N +��� q ; q +� +. +(A.48) +A.3.1.5 +(v) η(x; λ) = (q−x − 1)(1 − dqx) +qR : We take a = q−N and define ˜d = abcd−1q−1, +qλ = (a, b, c, d), +δ = (1, 1, 1, 1), +κ = q−1, +En(λ) = (q−n − 1)(1 − ˜dqn), +B(x; λ) = −(1 − aqx)(1 − bqx)(1 − cqx)(1 − dqx) +(1 − dq2x)(1 − dq2x+1) +, +D(x; λ) = − ˜d (1 − a−1dqx)(1 − b−1dqx)(1 − c−1dqx)(1 − qx) +(1 − dq2x−1)(1 − dq2x) +, +ˇPn(x; λ) = 4φ3 +�q−n, ˜dqn, q−x, dqx +a, b, c +��� q ; q +� +, +(A.49) +dqH : qλ = (a, b, qN), +δ = (1, 0, −1), +κ = q−1, +En(λ) = q−n − 1, +B(x; λ) = (qx−N − 1)(1 − aqx)(1 − abqx−1) +(1 − abq2x−1)(1 − abq2x) +, +D(x; λ) = aqx−N−1(1 − qx)(1 − abqx+N−1)(1 − bqx−1) +(1 − abq2x−2)(1 − abq2x−1) +, +ˇPn(x; λ) = 3φ2 +�q−n, abqx−1, q−x +a, q−N +��� q ; q +� +, +(A.50) +dqK : qλ = (p, qN), +δ = (1, −1), +κ = q−1, +En(λ) = q−n − 1, +B(x; λ) = +(qx−N − 1)(1 + pqx) +(1 + pq2x)(1 + pq2x+1), +D(x; λ) = pq2x−N−1 (1 − qx)(1 + pqx+N) +(1 + pq2x−1)(1 + pq2x), +ˇPn(x; λ) = 3φ2 +�q−n, q−x, −pqx +q−N, 0 +��� q ; q +� +. +(A.51) +A.3.2 +semi-infinite rdQM +We consider the following eight polynomials [4]: Meixner (M), Charlier (C), little q-Jacobi +(lqJ), little q-Laguerre/Wall (lqL), q-Bessel (qB) (=alternative q-Charlier), q-Meixner (qM), +Al-Salam-Carlitz II (ASCII) and q-Charlier (qC). Their sinusoidal coordinates (A.33) are +(i) : M, C, +(iii) : lqJ, lqL, qB, +(iv) : qM, ASCII, qC. +(A.52) +The standard lqJ, lqL, qB and ASCII polynomials [2] do not satisfy the normalization con- +dition (A.34). The standard parametrizations [2] are as follows: +lqJ : (a, b)standard = (aq−1, bq−1), +lqL : astandard = aq−1. +(A.53) +39 + +A.3.2.1 +(i) η(x) = x +M : λ = (β, c), +δ = (1, 0), +κ = 1, +En(λ) = n, +B(x; λ) = +c +1 − c(x + β), +D(x; λ) = +1 +1 − cx, +ˇPn(x; λ) = 2F1 +�−n, −x +β +��� 1 − c−1� +, +(A.54) +C : λ = a, +δ = 0, +κ = 1, +En(λ) = n, +B(x; λ) = a, +D(x; λ) = x, +ˇPn(x; λ) = 2F0 +�−n, −x +− +��� −a−1� +. +(A.55) +A.3.2.2 +(iii) η(x) = 1 − qx +lqJ : qλ = (a, b), +δ = (1, 1), +κ = q−1, +En(λ) = (q−n − 1)(1 − abqn−1), +B(x; λ) = aq−1(q−x − b), +D(x; λ) = q−x − 1, +ˇPn(x; λ) = 3φ1 +�q−n, abqn−1 q−x +b +��� q ; a−1qx+1� += (−a)−nq−(n +2)(a ; q)n +(b ; q)n +2φ1 +�q−n, abqn−1 +a +��� q ; qx+1� +, +(A.56) +lqL : qλ = a, +δ = 1, +κ = q−1, +En(λ) = q−n − 1, +B(x; λ) = aq−x−1, +D(x; λ) = q−x − 1, +ˇPn(x; λ) = 2φ0 +�q−n, q−x +− +��� q ; a−1qx+1� += (a−1q1−n ; q)n 2φ1 +�q−n, 0 +a +��� q ; qx+1� +, (A.57) +qB : qλ = a, +δ = 2, +κ = q−1, +En(λ) = (q−n − 1)(1 + aqn), +B(x; λ) = a, +D(x; λ) = q−x − 1, +ˇPn(x; λ) = (−aqn)−n +2φ0 +�q−n, −aqn +0 +��� q ; qx+1� += qnx +2φ1 +�q−n, q−x +0 +��� q ; −a−1q1−n� +. +(A.58) +A.3.2.3 +(iv) η(x) = q−x − 1 +qM : qλ = (b, c), +δ = (1, −1), +κ = q, +En(λ) = 1 − qn, +B(x; λ) = cqx(1 − bqx+1), +D(x; λ) = (1 − qx)(1 + bcqx), +ˇPn(x; λ) = 2φ1 +�q−n, q−x +bq +��� q ; −c−1qn+1� +, +(A.59) +ASCII : qλ = a, +δ = 0, +κ = q, +En(λ) = 1 − qn, +B(x; λ) = aq2x+1, +D(x; λ) = (1 − qx)(1 − aqx), +40 + +ˇPn(x; λ) = 2φ0 +�q−n, q−x +− +��� q ; a−1qn� +, +(A.60) +qC : qλ = a, +δ = −1, +κ = q, +En(λ) = 1 − qn, +B(x; λ) = aqx, +D(x; λ) = 1 − qx, +ˇPn(x; λ) = 2φ1 +�q−n, q−x +0 +��� q ; −a−1qn+1� +. +(A.61) +A.4 +Polynomials in rdQMJ +We consider the following seven polynomials [10]: big q-Jacobi (bqJ), big q-Laguerre (bqL), +Al-Salam-Carlitz I (ASCI), discrete Hermite I (dqHeI), discrete Hermite II (dqHeII), q- +Laguerre (qL) and Stieltjes-Wigert (SW). (SW is not given in [10] and we comment on this +at the end of this subsection.) +We impose the following normalization condition on the +polynomial Pn(η; λ), +Pn(1; λ) = 1 +: bqJ, bqL, ASCI, dqHeI, +Pn(−i; λ) = (−i)n : dqHeII, +Pn(−1; λ) = 1 +: qL, +(A.62) +Pn(0; λ) = 1 +: SW. +The standard ASCI, dqHeI, dqHeII, qL and SW polynomials [2] do not satisfy this normal- +ization condition (A.62). The standard parametrizations [2] are as follows: +qL : qαstandard = a. +(A.63) +The constants f J +n(λ) and bJ +n(λ) of the forward and backward shift relations (3.27)–(3.28) are +given by +f J +n(λ) = En(λ), +bJ +n(λ) = 1. +(A.64) +The parameters λ are real. +bqJ : qλ = (a, b, c), +δ = (1, 1, 1), +κ = q−1, +En(λ) = (q−n − 1)(1 − abqn+1), +BJ(η; λ) = η−2aq(1 − η)(bη − c), +DJ(η; λ) = η−2(aq − η)(η − cq), +Pn(η; λ) = 3φ2 +�q−n, abqn+1, η +aq, cq +��� q ; q +� +, +(A.65) +bqL : qλ = (a, b), +δ = (1, 1), +κ = q−1, +En(λ) = q−n − 1, +BJ(η; λ) = −η−2abq(1 − η), +DJ(η; λ) = η−2(aq − η)(η − bq), +41 + +Pn(η; λ) = 3φ2 +�q−n, 0, η +aq, bq +��� q ; q +� += (−b)nq +1 +2n(n+1) +(bq ; q)n +2φ1 +�q−n, aqη−1 +aq +��� q ; b−1η +� +, (A.66) +ASCI : qλ = a, +δ = 0, +κ = q−1, +En(λ) = q−n − 1, +BJ(η; λ) = −η−2aq−1, +DJ(η; λ) = η−2(1 − η)(η − a), +Pn(η; λ) = 2φ1 +�q−n, η−1 +0 +��� q ; a−1qη +� +, +(A.67) +dqHeI : qλ : none, +δ : none, +κ = q−1, +En(λ) = q−n − 1, +BJ(η; λ) = η−2q−1, +DJ(η; λ) = η−2(1 − η2), +Pn(η; λ) = 2φ1 +�q−n, η−1 +0 +��� q ; −qη +� +, +(A.68) +dqHeII : qλ : none, +δ : none, +κ = q, +En(λ) = 1 − qn, +BJ(η; λ) = η−2(1 + η2), +DJ(η; λ) = η−2q, +Pn(η; λ) = i−n +2φ0 +�q−n, iη +− +��� q ; −qn� += q(n +2)ηn +2φ1 +�q−n, q1−n +0 +��� q2 ; −q2η−2� +, (A.69) +qL : qλ = a, +δ = 1, +κ = q, +En(λ) = 1 − qn, +BJ(η; λ) = η−1(1 + η), +DJ(η; λ) = η−1a−1, +Pn(η; λ) = 2φ1 +�q−n, −η +0 +��� q ; aqn+1� += (−aqn)nηn +2φ1 +�q−n, a−1q−n +0 +��� q ; −qη−1� += (aq ; q)n 1φ1 +�q−n +aq +��� q ; −aqn+1η +� +, +(A.70) +SW : qλ : none, +δ : none, +κ = q, +En(λ) = 1 − qn, +BJ(η; λ) = 1, +DJ(η; λ) = η−1, +Pn(η; λ) = 1φ1 +�q−n +0 +��� q ; −qn+1η +� +. +(A.71) +The quantum mechanical formulation needs two component formalism with two sinu- +soidal coordinates η(±)(x; λ) (except qL and SW) [10], +bqJ : η(+)(x; λ) = aqx+1, η(−)(x; λ) = cqx+1, +bqL : η(+)(x; λ) = aqx+1, η(−)(x; λ) = bqx+1, +ASCI : η(+)(x; λ) = qx, +η(−)(x; λ) = aqx, +dqHeI : η(+)(x; λ) = qx, +η(−)(x; λ) = −qx, +(A.72) +dqHeII : η(+)(x; λ) = cqx, +η(−)(x; λ) = −cqx, +qL, SW : η(x; λ) = cqx, +where the parameters λ are extended to qλ = c (δ = 1) for dqHeII, qλ = (a, c) (δ = (1, 1)) +42 + +for qL and qλ = c (δ = 2) for SW. The quantum mechanical formulation for SW is not +given in [10], but it is similar to that for qL. There is an infinite sum orthogonality relations +(parameter: c > 0), +SW : +∞ +� +x=−∞ +cxq +1 +2x(x+1)Pn(cqx)Pm(cqx) = δnm q−n(q ; q)n(q, −cq, −c−1 ; q)∞ (n, m ∈ Z≥0), +(A.73) +which is not given in [2]. +References +[1] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, vol. 98 +of Encyclopedia of mathematics and its applications, Cambridge Univ. Press, Cambridge +(2005). +[2] R. Koekoek, P. A. Lesky and R. F. Swarttouw, Hypergeometric orthogonal polynomials +and their q-analogues, Springer-Verlag Berlin-Heidelberg (2010). +[3] S. Odake and R. Sasaki, “Discrete quantum mechanics,” (Topical Review) J. Phys. A44 +(2011) 353001 (47pp), arXiv:1104.0473[math-ph]. Typo in (2.132), c1(η, λ) for H : +−1 +2 ⇒ −η +2. +[4] S. Odake and R. Sasaki, “Orthogonal Polynomials from Hermitian Matrices,” J. Math. +Phys. 49 (2008) 053503 (43pp), arXiv:0712.4106[math.CA]. (For the dual q-Meixner +and dual q-Charlier polynomials, see [10].) +[5] S. Odake and R. Sasaki, “Exactly solvable ‘discrete’ quantum mechanics; shape invari- +ance, Heisenberg solutions, annihilation-creation operators and coherent states,” Prog. +Theor. Phys. 119 (2008) 663-700, arXiv:0802.1075[quant-ph]. +[6] S. Odake and R. Sasaki, “Exactly solvable quantum mechanics and infinite families of +multi-indexed orthogonal polynomials,” Phys. Lett. B702 (2011) 164-170, arXiv:1105. +0508[math-ph]. +[7] S. Odake and R. Sasaki, “Multi-indexed (q-)Racah polynomials,” J. Phys. A 45 (2012) +385201 (21pp), arXiv:1203.5868[math-ph]. +43 + +[8] S. Odake and R. Sasaki, “Multi-indexed Wilson and Askey-Wilson polynomials,” J. +Phys. A46 (2013) 045204 (22pp), arXiv:1207.5584[math-ph]. +[9] S. Odake and R. Sasaki, ““Diophantine” and Factorisation Properties of Finite Orthog- +onal Polynomials in the Askey Scheme,” arXiv:2207.14479[math.CA]. +[10] S. Odake and R. Sasaki, “Orthogonal Polynomials from Hermitian Matrices II,” J. Math. +Phys. 59 (2018) 013504 (42pp), arXiv:1604.00714[math.CA]. +[11] S. Odake and R. Sasaki, “Unified theory of annihilation-creation operators for solvable +(‘discrete’) quantum mechanics,” J. Math. Phys. 47 (2006) 102102 (33pp), arXiv: +quant-ph/0605215. +[12] S. Odake, “New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From +State-Adding Darboux Transformations,” arXiv:2209.12353[math-ph]. +[13] S. Odake, “Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete +Quantum Mechanics with Real Shifts,” Prog. Theor. Exp. Phys 2017(12) (2017) +123A02 (30pp), arXiv:1708.01830[math-ph]. +[14] S. Odake and R. Sasaki, “Multi-indexed Meixner and Little q-Jacobi (Laguerre) Poly- +nomials,” J. Phys. A50 (2017) 165204 (23pp), arXiv:1610.09854[math.CA]. +[15] S. Odake, “Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed +Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types,” Prog. +Theor. Exp. Phy. 2019 (2019) 123A01 (20pp), arXiv:1907.12218[math-ph]. +[16] S. Odake and R. Sasaki, “Extensions of solvable potentials with finitely many discrete +eigenstates,” J. Phys. A46 (2013) 235205 (15pp), arXiv:1301.3980[math-ph]. +[17] S. Odake and R. Sasaki, “Dual Christoffel transformations,” Prog. Theor. Phys. 126 +(2011) 1-34, arXiv:1101.5468[math-ph]. +44 + diff --git a/jtAyT4oBgHgl3EQfx_nV/content/tmp_files/load_file.txt b/jtAyT4oBgHgl3EQfx_nV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..474e8a4f6848860bc761b7fb2827be4770c14fc9 --- /dev/null +++ b/jtAyT4oBgHgl3EQfx_nV/content/tmp_files/load_file.txt @@ -0,0 +1,1826 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf,len=1825 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='00678v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='CA] 2 Jan 2023 DPSU-22-3 Another Type of Forward and Backward Shift Relations for Orthogonal Polynomials in the Askey Scheme Satoru Odake Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan Abstract The forward and backward shift relations are basic properties of the (basic) hyperge- ometric orthogonal polynomials in the Askey scheme (Jacobi, Askey-Wilson, q-Racah, big q-Jacobi etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=') and they are related to the factorization of the differential or differ- ence operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Based on other factorizations, we obtain another type of forward and backward shift relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 1 Introduction The (basic) hypergeometric orthogonal polynomials in the Askey scheme satisfy second or- der differential or difference equations and the forward and backward shift relations are their basic properties [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The orthogonal polynomials in the Askey scheme provide us with exactly solvable quantum mechanical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Conversely, we can use the quantum mechanical formulation as a tool to investigate orthogonal polynomials [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' For example, the forward and backward shift relations are a consequence of the shape invariance [4, 5, 3], and the multi-indexed orthogonal polynomials ([6, 7, 8] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=') are found by using the quantum mechanical formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The Schr¨odinger equation is a second order differential equation for ordinary quantum mechanics (oQM) and a second order difference equation for discrete quantum mechanics (dQM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' There are two types of dQM, dQM with pure imaginary shifts (idQM) and dQM with real shifts (rdQM) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The coordinate x for oQM and idQM is continuous and that for rdQM is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The forward and backward shift relations are related to the factorization of the Hamil- tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Recently another factorization of the Hamiltonian was found in a study of the state-adding Darboux transformations for the finite rdQM systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' It gives another for- ward and backward shift relations for the orthogonal polynomials appearing in the finite rdQM systems (q-Racah etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' ), which were called the forward and backward x-shift rela- tions [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' In this paper, we investigate whether such new factorization and forward and backward shift relations exist for other orthogonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' In addition to the finite rdQM systems (q-Racah etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' ), we examine the oQM systems (Jacobi etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' ), the idQM sys- tems (Askey-Wilson etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' ), the semi-infinite rdQM systems (q-Meixner etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=') and the rdQM systems with the Jackson integral type measure (big q-Jacobi etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We call the last category rdQMJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The quantum mechanical formulation of the rdQMJ systems needs two component formalism [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We consider all the polynomials in chapter 9 and 14 of [2] and the dual quantum q-Krawtchouk polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The orthogonal polynomials in the Askey scheme and their second order differential or difference equations are recalled in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The forward and backward shift relations are reviewed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Section 4 is the main part of this paper and new factorization and new forward and backward shift relations are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Section 5 is for a summary and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' In Appendix A the data for the orthogonal polynomials are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We comment that the oQM systems described by the Bessel and pseudo Jacobi polynomials are the Morse potential and the hyperbolic symmetric top II, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We also comment on an infinite sum orthogonality relations for the Stieltjes-Wigert polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 2 Orthogonal Polynomials in the Askey Scheme In this section we fix the notation and recall the second order differential or difference equations for the orthogonal polynomials in the Askey scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' In our quantum mechanical formulation [3], the orthogonal polynomials in the Askey scheme are expressed as ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) def = Pn � η(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ � : a polynomial of degree n in η(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1) for n ∈ Z≥0 and ˇP−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) def = P−1(η(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) def = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Here x is a coordinate of quantum me- chanical system and η(x) is a sinusoidal coordinate [11], and λ = (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=') are parameters, whose dependence is expressed as f = f(λ) and f(x) = f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The parameter q is 0 < q < 1 and qλ stands for q(λ1,λ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=') = (qλ1, qλ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' ), and we omit writing q-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The data for the orthogonal polynomials treated in this paper are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We remark that the polynomials ˇPn(x) in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1, whose orthogonality holds for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , N, are ill-defined for n > N due to the normalization condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' So we should replace ˇPn(x) (n > N) 2 in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 with the monic version ˇP monic n (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) def = cn(λ)−1 ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) (see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='35)) in Theorem 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 (with the replacements fn(λ) → f monic n (λ) = fn(λ)cn(λ)−1cn−1(λ + δ), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The Schr¨odinger equations of oQM and dQM systems are second order differential and difference equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' By the similarity transformation in terms of the ground state wavefunction, the similarity transformed Hamiltonian � H(λ) is a second order differen- tial or difference operator acting on the eigenpolynomials ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) [3], oQM : � H(λ) def = − d2 dx2 − 2dw(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) dx d dx � = −4c2(η) d2 dη2 − 4c1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) d dη � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2) idQM : � H(λ) def = V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)(eγp − 1) + V ∗(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)(e−γp − 1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3) rdQM : � H(λ) def = B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)(1 − e∂) + D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)(1 − e−∂), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4) where the functions w(x), c1(η), c2(η), V (x), B(x) and D(x) are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' For oQM, the coordinate x is a continuous variable and the Hamiltonian H(λ) is H(λ) = − d2 dx2 + U(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), U(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) def = �dw(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) dx �2 + d2w(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) dx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='5) While the orthogonality relations of ˇPn(x) for B and pJ cases hold only for a finite number of n (see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='13)), we consider all n ∈ Z≥0, because we consider only differential equations (or relations) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' For idQM, the coordinate x is a continuous variable and the momentum p is p = −i d dx, and γ is a real constant (γ = 1 for non q-polynomial, γ = log q for q-polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The operator eαp (α: constant) is a shift operator, eαpf(x) = f(x − iα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The ∗-operation on an analytic function f(x) = � n anxn (an ∈ C) is defined by f ∗(x) = � n a∗ nxn, in which a∗ n is the complex conjugation of an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' For rdQM, the Schr¨odinger equation is a matrix eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The similarity transformed Hamiltonian � H = ( � Hx,y) is a matrix labeled by the coordinate x, which takes discrete values in {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , N} or Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' In this paper, however, we treat x as a continuous variable x ∈ R, because we only deal with difference equations (or relations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The operators e±∂ are shift operators e±∂ = e± d dx, e±∂f(x) = f(x ± 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We consider ˇPn(x) with all n ∈ Z≥0 even for finite systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' For rdQMJ (the rdQM system with Jackson integral type measure such as the big q- Jacobi polynomial), its quantum mechanical formulation needs two component formalism with two sinusoidal coordinates η(±)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Since only difference equations (or relations) are considered in this paper, we use η only (we do not use x) and treat η as a continuous variable η ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The similarity transformed Hamiltonian � HJ(λ) is a second order difference 3 operator acting on the eigenpolynomials Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) [10], rdQMJ : � HJ(λ) def = BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)(1 − qη d dη ) + DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)(1 − q−η d dη ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='6) where the functions BJ(η) and DJ(η) are given in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The operators q±η d dη are q-shift operators, q±η d dη f(η) = f(q±1η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The orthogonal polynomials in the Askey scheme studied in this paper have the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Theorem 1 [1, 2] The polynomials in Appendix A satisfy the second order differential or difference equations for n ∈ Z≥0, oQM, idQM, rdQM : � H(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = En(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='7) rdQMJ : � HJ(λ)Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = En(λ)Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='8) We remark that the constant terms of � H and � HJ are chosen such that E0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' For idQM, the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='7) is invariant under the ∗-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 3 Forward and Backward Shift Relations The similarity transformed Hamiltonians � H(λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4) are factorized as � H(λ) = B(λ)F(λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1) where the forward and backward shift operators, F(λ) and B(λ), are defined by [4, 5], oQM : F(λ) def = cF �dη(x) dx �−1 d dx � = cF d dη � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2) B(λ) def = −c−1 F �dη(x) dx d dx + 4c1 � η(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ �� � = −4c−1 F � c2(η) d dη + c1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3) idQM : F(λ) def = iϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)−1(e γ 2 p − e− γ 2 p), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4) B(λ) def = −i � V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)e γ 2 p − V ∗(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)e− γ 2 p� ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='5) rdQM : F(λ) def = B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)−1(1 − e∂), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='6) B(λ) def = 1 B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) � B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)e−∂� ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='7) and cF and ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Since w(x), B(x), D(x) and V (x) satisfy �dw(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) dx �2 − d2w(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) dx2 = �dw(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) dx �2 + d2w(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) dx2 + E1(λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='8) 4 V (x − iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) = κ ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) ϕ(x − iγ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='9) V (x + iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + V ∗(x − iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = κ � V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) + V ∗(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) � − E1(λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='10) B(x + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) = κ ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) ϕ(x + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) = κ ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) ϕ(x − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='11) B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + D(x + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = κ � B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) + D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) � + E1(λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='12) (κ and δ are given in Appendix A), we obtain F(λ)B(λ) = κB(λ + δ)F(λ + δ) + E1(λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='13) which is the (similarity transformed) shape invariance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The constants fn(λ) and bn(λ) given in Appendix A satisfy En(λ) = fn(λ)bn−1(λ) (n ∈ Z≥0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='14) and the energy eigenvalues En satisfy En+1(λ) = κEn(λ + δ) + E1(λ) (n ∈ Z≥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='15) Corresponding to the factorization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1), the shape invariance combined with the Crum’s theorem give the following relations [4, 5, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 [2] For the polynomials in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3, the forward and back- ward shift relations hold: F(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = fn(λ) ˇPn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) (n ∈ Z≥0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='16) B(λ) ˇPn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) = bn−1(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) (n ∈ Z≥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='17) For idQM, the relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='16)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='17) are invariant under the ∗-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The similarity transformed Hamiltonians � HJ(λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='6) are factorized as � HJ(λ) = BJ(λ)F J(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='18) Here the forward and backward shift operators, F J(λ) and BJ(λ), are defined by [10] rdQMJ : F J(λ) def = A qη(1 − qη d dη ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='19) BJ(λ) def = � BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)q−η d dη �qη A , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='20) 5 where the constant A is given by A = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 −DJ(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) : bqJ, bqL, qL, SW −a : ASCI 1 : dqHeI q : dqHeII .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='21) We can show that BJ(η) and DJ(η) satisfy qBJ(qr−1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = κBJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ), q−1DJ(r−1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = κDJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='22) BJ(r−1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + DJ(qr−1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = κ � BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) + DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) � + E1(λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='23) where r is given by r = \uf8f1 \uf8f2 \uf8f3 q : bqJ, bqL, dqHeII, qL 1 : ASCI, dqHeI q2 : SW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='24) Therefore we obtain F J(λ)BJ(λ) ��� η→r−1η= κBJ(λ + δ)F J(λ + δ) + E1(λ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='25) which is the (similarity transformed) shape invariance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The constants f J n(λ) and bJ n(λ) given in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 satisfy En(λ) = f J n(λ)bJ n−1(λ) (n ∈ Z≥0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='26) and the energy eigenvalues En satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Corresponding to the factorization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='18), we have the following relations [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 [2] For the polynomials in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4, the forward and backward shift re- lations hold: F J(λ)Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = f J n(λ)Pn−1(rη;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) (n ∈ Z≥0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='27) BJ(λ)Pn−1(rη;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + δ) = bJ n−1(λ)Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) (n ∈ Z≥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='28) 4 New Forward and Backward Shift Relations In this section, based on other factorizations of � H and � HJ, we present another type of forward and backward shift relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 Polynomials in oQM systems For the oQM systems described by the polynomials in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 (except He and B), let us define the operators ˜F(λ) and ˜B(λ) as follows: L : (a) : ˜F(λ) def = 1 2x d dx + g − 1 2 � = η d dη + g − 1 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1) ˜B(λ) def = −1 2 1 x d dx + 1 � = − d dη + 1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2) (b) : ˜F(λ) def = −1 2 1 x d dx + 1 � = − d dη + 1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3) ˜B(λ) def = 1 2x d dx + g + 1 2 � = η d dη + g + 1 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4) J : (a) : ˜F(λ) def = 1 2 tan x d dx + g − 1 2 � = −(1 − η) d dη + g − 1 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='5) ˜B(λ) def = −1 2 cot x d dx + h + 1 2 � = (1 + η) d dη + h + 1 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='6) (b) : ˜F(λ) def = −1 2 cot x d dx + h − 1 2 � = (1 + η) d dη + h − 1 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='7) ˜B(λ) def = 1 2 tan x d dx + g + 1 2 � = −(1 − η) d dη + g + 1 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='8) pJ : (a) : ˜F(λ) def = � tanh x + i cosh x � d dx − h − 1 2 − iµ � = (η + i) d dη − h − 1 2 − iµ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='9) ˜B(λ) def = � tanh x − i cosh x � d dx − h + 1 2 + iµ � = (η − i) d dη − h + 1 2 + iµ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='10) (b) : ˜F(λ) def = � tanh x − i cosh x � d dx − h − 1 2 + iµ � = (η − i) d dη − h − 1 2 + iµ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='11) ˜B(λ) def = � tanh x + i cosh x � d dx − h + 1 2 − iµ � = (η + i) d dη − h + 1 2 − iµ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='12) Let us define the constants ˜fn(λ), ˜bn(λ) and ¯δ as follows: L : (a) : ˜fn(λ) = n + g − 1 2, ˜bn(λ) = 1, ¯δ = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='13) (b) : ˜fn(λ) = 1, ˜bn(λ) = n + g + 1 2, ¯δ = −1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='14) J : (a) : ˜fn(λ) = n + g − 1 2, ˜bn(λ) = n + h + 1 2, ¯δ = (1, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='15) (b) : ˜fn(λ) = n + h − 1 2, ˜bn(λ) = n + g + 1 2, ¯δ = (−1, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='16) pJ : (a) : ˜fn(λ) = n − h − 1 2 − iµ, ˜bn(λ) = n − h + 1 2 + iµ, ¯δ = (0, i), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='17) (b) : ˜fn(λ) = n − h − 1 2 + iµ, ˜bn(λ) = n − h + 1 2 − iµ, ¯δ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='18) We remark that the second components of ¯δ for pJ are unphysical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 7 Then we can show that � H(λ) = 4 � ˜B(λ) ˜F(λ) − ˜f0(λ)˜b0(λ) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='19) En(λ) = 4 � ˜fn(λ)˜bn(λ) − ˜f0(λ)˜b0(λ) � (n ∈ Z≥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='20) Corresponding to this factorization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='19), the following relations are obtained by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 For the polynomials in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 (except He and B), the following forward and backward shift relations hold for n ∈ Z≥0, ˜F(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ˜fn(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='21) ˜B(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) = ˜bn(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='22) We think that these identities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='21)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='22) may be known formulas but this interpretation is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 Two formulas with ¯δ and −¯δ are equivalent by interchanging ˜F and ˜B, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='22) for L (b) agree with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='22) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='21) for L (a) with the replacement g → g + 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' For He and B, we do not have new factorization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='19) and new forward and backward shift relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='21)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 Polynomials in idQM systems For the idQM systems described by the polynomials in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2, let us define the operators ˜F(λ) and ˜B(λ) as follows: ˜F(λ) def = V1(x + iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)e γ 2 p + V ∗ 1 (x − iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)e− γ 2 p, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='23) ˜B(λ) def = V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)e γ 2 p + V ∗ 2 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)e− γ 2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='24) The potential functions V1(x) and V2(x) satisfy V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='25) and their explicit forms are given by cH : (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a1 + ix, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a2 + ix, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='26) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a2 + ix, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a1 + ix, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='27) 8 MP : (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a + ix, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ei( π 2 −φ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='28) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ei( π 2 −φ), V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a + ix, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='29) W : Assume {a∗ j, a∗ k} = {aj, ak} (as a set) and set {l, m} = {1, 2, 3, 4}\\{j, k}, Vi(x) = V (j,k) i (x) (i = 1, 2), V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (aj + ix)(ak + ix) 2ix + 1 , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (al + ix)(am + ix) 2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='30) cdH : Assume {a∗ j, a∗ k} = {aj, ak} (as a set) and set {l} = {1, 2, 3}\\{j, k}, Vi(x) = V (j,k) i (x) (i = 1, 2), (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (aj + ix)(ak + ix) 2ix + 1 , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = al + ix 2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='31) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = al + ix 2ix + 1, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (aj + ix)(ak + ix) 2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='32) AW : Assume {a∗ j, a∗ k} = {aj, ak} (as a set) and set {l, m} = {1, 2, 3, 4}\\{j, k}, Vi(x) = V (j,k) i (x) (i = 1, 2), V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − ajeix)(1 − akeix) 1 − qe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − aleix)(1 − ameix) 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='33) cdqH : Assume {a∗ j, a∗ k} = {aj, ak} (as a set) and set {l} = {1, 2, 3}\\{j, k}, Vi(x) = V (j,k) i (x) (i = 1, 2), (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − ajeix)(1 − akeix) 1 − qe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − aleix 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='34) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − aleix 1 − qe2ix, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − ajeix)(1 − akeix) 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='35) ASC : Assume a1, a2 ∈ R for (b) and (c), (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a1eix)(1 − a2eix) 1 − qe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − e2ix, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='36) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − a1eix 1 − qe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − a2eix 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='37) (c) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − a2eix 1 − qe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − a1eix 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='38) (d) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − qe2ix, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a1eix)(1 − a2eix) 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='39) cbqHe : (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − aeix 1 − qe2ix, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − e2ix, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='40) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − qe2ix, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − aeix 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='41) cqHe : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − qe2ix, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − e2ix, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='42) 9 cqJ : (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − q 1 2(α+ 1 2)eix)(1 − q 1 2 (α+ 3 2 )eix) 1 − qe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 + q 1 2(β+ 1 2)eix)(1 + q 1 2 (β+ 3 2 )eix) 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='43) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 + q 1 2(β+ 1 2)eix)(1 + q 1 2 (β+ 3 2)eix) 1 − qe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − q 1 2(α+ 1 2)eix)(1 − q 1 2 (α+ 3 2 )eix) 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='44) cqL : (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − q 1 2(α+ 1 2)eix)(1 − q 1 2 (α+ 3 2 )eix) 1 − qe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − e2ix, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='45) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − qe2ix, V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − q 1 2 (α+ 1 2 )eix)(1 − q 1 2(α+ 3 2)eix) 1 − e2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='46) cqH : (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a1e2iφeix)(1 − a∗ 1eix) 1 − qe2iφe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a2e2iφeix)(1 − a∗ 2eix) 1 − e2iφe2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='47) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a2e2iφeix)(1 − a∗ 2eix) 1 − qe2iφe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a1e2iφeix)(1 − a∗ 1eix) 1 − e2iφe2ix , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='48) qMP : (a) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − ae2iφeix)(1 − aeix) 1 − qe2iφe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − e2iφe2ix, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='49) (b) : V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − qe2iφe2ix , V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − ae2iφeix)(1 − aeix) 1 − e2iφe2ix .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='50) Let us define the constants ˜fn(λ), ˜bn(λ) and ¯δ as follows: cH : (a) : ˜fn(λ) = a1 + a∗ 1 + n − 1, ˜bn(λ) = a2 + a∗ 2 + n, ¯δ = ( 1 2, −1 2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='51) (b) : ˜fn(λ) = a2 + a∗ 2 + n − 1, ˜bn(λ) = a1 + a∗ 1 + n, ¯δ = (−1 2, 1 2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='52) MP : (a) : ˜fn(λ) = 2a + n − 1, ˜bn(λ) = 2 sin φ, ¯δ = ( 1 2, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='53) (b) : ˜fn(λ) = 2 sin φ, ˜bn(λ) = 2a + n, ¯δ = (−1 2, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='54) W : ˜fn(λ) = aj + ak + n − 1, ˜bn(λ) = al + am + n, (¯δ)j = (¯δ)k = 1 2, (¯δ)l = (¯δ)m = −1 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='55) cdH : (a) : ˜fn(λ) = aj + ak + n − 1, ˜bn(λ) = 1, (¯δ)j = (¯δ)k = 1 2, (¯δ)l = −1 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='56) (b) : ˜fn(λ) = 1, ˜bn(λ) = aj + ak + n, (¯δ)l = 1 2, (¯δ)j = (¯δ)k = −1 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='57) AW : ˜fn(λ) = q− n 2 (1 − ajakqn−1), ˜bn(λ) = q− n 2 (1 − alamqn), (¯δ)j = (¯δ)k = 1 2, (¯δ)l = (¯δ)m = −1 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='58) cdqH : (a) : ˜fn(λ) = q− n 2 (1 − ajakqn−1), ˜bn(λ) = q− n 2 , (¯δ)j = (¯δ)k = 1 2, (¯δ)l = −1 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='59) 10 (b) : ˜fn(λ) = q− n 2 , ˜bn(λ) = q− n 2 (1 − ajakqn), (¯δ)l = 1 2, (¯δ)j = (¯δ)k = −1 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='60) ASC : (a) : ˜fn(λ) = q− n 2 (1 − a1a2qn−1), ˜bn(λ) = q− n 2 , ¯δ = ( 1 2, 1 2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='61) (b) : ˜fn(λ) = q− n 2 , ˜bn(λ) = q− n 2 , ¯δ = ( 1 2, −1 2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='62) (c) : ˜fn(λ) = q− n 2 , ˜bn(λ) = q− n 2 , ¯δ = (−1 2, 1 2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='63) (d) : ˜fn(λ) = q− n 2 , ˜bn(λ) = q− n 2 (1 − a1a2qn), ¯δ = (−1 2, −1 2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='64) cbqHe : (a) : ˜fn(λ) = q− n 2 , ˜bn(λ) = q− n 2 , ¯δ = 1 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='65) (b) : ˜fn(λ) = q− n 2 , ˜bn(λ) = q− n 2 , ¯δ = −1 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='66) cqHe : ˜fn(λ) = q− n 2 , ˜bn(λ) = q− n 2 , ¯δ : none, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='67) cqJ : (a) : ˜fn(λ) = 1 − qα+n, ˜bn(λ) = q−n(1 − qβ+n+1), ¯δ = (1, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='68) (b) : ˜fn(λ) = q−n(1 − qβ+n), ˜bn(λ) = 1 − qα+n+1, ¯δ = (−1, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='69) cqL : (a) : ˜fn(λ) = 1 − qα+n, ˜bn(λ) = q−n, ¯δ = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='70) (b) : ˜fn(λ) = q−n, ˜bn(λ) = 1 − qα+n+1, ¯δ = −1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='71) cqH : (a) : ˜fn(λ) = q− n 2 (1 − a1a∗ 1qn−1), ˜bn(λ) = q− n 2 (1 − a2a∗ 2qn), ¯δ = ( 1 2, −1 2, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='72) (b) : ˜fn(λ) = q− n 2 (1 − a2a∗ 2qn−1), ˜bn(λ) = q− n 2 (1 − a1a∗ 1qn), ¯δ = (−1 2, 1 2, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='73) qMP : (a) : ˜fn(λ) = q− n 2 (1 − a2qn−1), ˜bn(λ) = q− n 2 , ¯δ = ( 1 2, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='74) (b) : ˜fn(λ) = q− n 2 , ˜bn(λ) = q− n 2 (1 − a2qn), ¯δ = (−1 2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='75) Then we can show that V1(x) and V2(x) satisfy V1(x + iγ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)V ∗ 2 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + V ∗ 1 (x − iγ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − ˜f0(λ)˜b0(λ) = −V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − V ∗(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='76) and the constants ˜fn and ˜bn satisfy En(λ) = ˜fn(λ)˜bn(λ) − ˜f0(λ)˜b0(λ) (n ∈ Z≥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='77) The relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='76) gives other factorizations of � H(λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3), � H(λ) = ˜B(λ) ˜F(λ) − ˜f0(λ)˜b0(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='78) Corresponding to this factorization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='78), we obtain the following relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 For the polynomials in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2, the following forward and backward shift rela- tions hold for n ∈ Z≥0, ˜F(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ˜fn(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='79) ˜B(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) = ˜bn(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='80) 11 Proof: It is sufficient to show (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='79), because (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='77)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='79) imply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Taking AW (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='33) with (j, k) = (1, 2) as an example, let us prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' It is shown by direct calculation: ˜F(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a1q− 1 2eix)(1 − a2q− 1 2eix) 1 − e2ix (a1a2, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n an 1 × 4φ3 �q−n, a1a2a3a4qn−1, a1q 1 2eix, a1q− 1 2e−ix a1a2, a1a3, a1a4 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � + (1 − a1q− 1 2e−ix)(1 − a2q− 1 2e−ix) 1 − e−2ix (a1a2, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n an 1 × 4φ3 �q−n, a1a2a3a4qn−1, a1q− 1 2eix, a1q 1 2e−ix a1a2, a1a3, a1a4 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � = (a1a2, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n an 1(1 − e2ix) n � k=0 (q−n, a1a2a3a4qn−1, a1q− 1 2eix, a1q− 1 2e−ix ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (a1a2, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k × � (1 − a1eixqk− 1 2)(1 − a2q− 1 2eix) − e2ix(1 − a1e−ixqk− 1 2)(1 − a2q− 1 2e−ix) � = (a1a2, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n an 1(1 − e2ix) n � k=0 (q−n, a1a2a3a4qn−1, a1q− 1 2eix, a1q− 1 2e−ix ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (a1a2, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k × (1 − a1a2qk−1)(1 − e2ix) = (a1a2, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n an 1 n � k=0 (1 − a1a2q−1)(q−n, a1a2a3a4qn−1, a1q− 1 2eix, a1q− 1 2e−ix ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (a1a2q−1, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k = q− n 2 (1 − a1a2qn−1) × (a1a2q−1, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n (a1q− 1 2)n n � k=0 (q−n, a1a2a3a4qn−1, a1q− 1 2eix, a1q− 1 2e−ix ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (a1a2q−1, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k = ˜fn(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The other cases are proved in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 Two formulas with ¯δ and −¯δ are equivalent by interchanging ˜F and ˜B, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='79) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='80) for cH (b) agree with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='80) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='79) for cH (a) with the replacements a1 → a1 + 1 2 and a2 → a2 − 1 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 The relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='79)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='80) are invariant under the ∗-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' In contrast to the x-shift relations studied in [9] (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3), the coordinate x is not shifted, and only the parameters λ are shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We choose the operators ˜F(λ) and ˜B(λ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='23)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='24) to 12 respect this ∗-operation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' See also Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 We can show that V1(x + iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)V2(x − iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = V1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ)V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), V1(x + iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)V ∗ 2 (x − iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + V ∗ 1 (x − iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)V2(x + iγ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − ˜f0(λ)˜b0(λ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='81) = V1(x + iγ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ)V ∗ 2 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) + V ∗ 1 (x − iγ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ)V2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) − ˜f0(λ − ¯δ)˜b0(λ − ¯δ), which imply ˜F(λ) ˜B(λ) − ˜f0(λ)˜b0(λ) = ˜B(λ − ¯δ) ˜F(λ − ¯δ) − ˜f0(λ − ¯δ)˜b0(λ − ¯δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='82) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 Polynomials in rdQM systems For the rdQM systems described by the polynomials in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 (except C and qB), let us define the operators ˜F(λ) and ˜B(λ) as follows: ˜F(λ) def = D1(x + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)e∂, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='83) ˜B(λ) def = B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)e−∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='84) The potential functions B1(x), B2(x), D1(x) and D2(x) satisfy B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='85) and their explicit forms are given by H : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N − x N + 1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (N + 1)(x + a), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x N + 1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (N + 1)(b + N − x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='86) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x + a a − 1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a − 1)(N − x), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x 1 − a, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a)(b + N − x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='87) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a(N − x), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x + a a , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −a(b + N − x), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −x a, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='88) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N(x + a), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N − x N , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N(b + N − x), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x N , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='89) K : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N − x N + 1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (N + 1)p, 13 D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x N + 1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (N + 1)(1 − p), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='90) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = Np, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N − x N , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N(1 − p), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x N , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='91) R : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − (x − N)(x + d) (N + 1)(2x + 1 + d), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (N + 1)(x + b)(x + c) 2x + d , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + d + N)x (N + 1)(2x − 1 + d), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(N + 1)(x + d − b)(x + d − c) 2x + d , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='92) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + b)(x + d) (b − 1)(2x + 1 + d), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(b − 1)(x − N)(x + c) 2x + d , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − (x + d − b)x (b − 1)(2x − 1 + d), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (b − 1)(x + d + N)(x + d − c) 2x + d , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='93) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + c)(x + d) (c − 1)(2x + 1 + d), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(c − 1)(x − N)(x + b) 2x + d , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − (x + d − c)x (c − 1)(2x − 1 + d), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (c − 1)(x + d + N)(x + d − b) 2x + d , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='94) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −c(x − N)(x + b) 2x + 1 + d , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + c)(x + d) c(2x + d) , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = c(x + d + N)(x + d − b) 2x − 1 + d , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(x + d − c)x c(2x + d) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='95) (e) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −b(x − N)(x + c) 2x + 1 + d , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + b)(x + d) b(2x + d) , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = b(x + d + N)(x + d − c) 2x − 1 + d , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(x + d − b)x b(2x + d) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='96) (f) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N(x + b)(x + c) 2x + 1 + d , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(x − N)(x + d) N(2x + d) , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −N(x + d − b)(x + d − c) 2x − 1 + d , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + d + N)x N(2x + d) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='97) dH : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + a + b − 1)(N − x) (N + 1)(2x + a + b) , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (N + 1)(x + a) 2x − 1 + a + b , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x(x + a + b + N − 1) (N + 1)(2x − 2 + a + b), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (N + 1)(x + b − 1) 2x − 1 + a + b , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='98) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + a)(x + a + b − 1) (a − 1)(2x + a + b) , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a − 1)(N − x) 2x − 1 + a + b , 14 D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x(x + b − 1) (1 − a)(2x − 2 + a + b), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a)(x + a + b + N − 1) 2x − 1 + a + b , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='99) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x + a + b − 1 2x + a + b , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + a)(N − x) 2x − 1 + a + b , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x 2x − 2 + a + b, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + b − 1)(x + a + b + N − 1) 2x − 1 + a + b , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='100) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + a)(N − x) 2x + a + b , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x + a + b − 1 2x − 1 + a + b, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + b − 1)(x + a + b + N − 1) 2x − 2 + a + b , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x 2x − 1 + a + b, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='101) (e) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a(N − x) 2x + a + b, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + a)(x + a + b − 1) a(2x − 1 + a + b) , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −a(x + a + b + N − 1) 2x − 2 + a + b , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − x(x + b − 1) a(2x − 1 + a + b), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='102) (f) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N(x + a) 2x + a + b, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + a + b − 1)(N − x) N(2x − 1 + a + b) , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = N(x + b − 1) 2x − 2 + a + b, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x(x + a + b + N − 1) N(2x − 1 + a + b) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='103) dqqK : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−N−1(1 − qN−x) q−N−1 − 1 , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N−1 − 1)p−1q−x, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x − 1 q−N−1 − 1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N−1 − 1)(1 − p−1q−x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='104) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x−1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = p−1q−N(1 − qN−x), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(q−x − 1), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − p−1q−x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='105) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = p−1q−N−1(1 − qN−x), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − p−1q−x), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(q−x − 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='106) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN)p−1q−x−N−1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qN−x 1 − qN , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N − 1)(1 − p−1q−x), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x − 1 q−N − 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='107) qH : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 q−N−1 − 1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N−1 − 1)(1 − aqx), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx 1 − qN+1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN+1)aq−1(qx−N − b), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='108) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − aqx 1 − aq−1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − aq−1)(qx−N − 1), 15 D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aq−1(1 − qx) aq−1 − 1 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (aq−1 − 1)(qx−N − b), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='109) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a)(qx−N − 1), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − aqx 1 − a , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a − 1)q−1(qx−N − b), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a(1 − qx) a − 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='110) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N − 1)(1 − aqx), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 q−N − 1 , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN)aq−1(qx−N − b), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx 1 − qN , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='111) qK : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 q−N−1 − 1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−N−1 − 1, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx 1 − qN+1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN+1)p, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='112) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−N − 1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 q−N − 1 , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN)p, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx 1 − qN , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='113) qqK : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 q−N−1 − 1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N−1 − 1)p−1qx, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx 1 − qN+1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN+1)(1 − p−1qx−N−1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='114) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qqx, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−1p−1(qx−N − 1), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − p−1qx−N−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='115) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = p−1(qx−N − 1), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − p−1qx−N−1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='116) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N − 1)p−1qx, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 q−N − 1 , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN)(1 − p−1qx−N−1), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx 1 − qN , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='117) aqK : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 q−N−1 − 1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N−1 − 1)(1 − pqx+1), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx 1 − qN+1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN+1)pqx−N, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='118) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − pqx+1 1 − p , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − p)(qx−N − 1), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = p(1 − qx) p − 1 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (p − 1)qx−N, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='119) 16 (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − pq)(qx−N − 1), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − pqx+1 1 − pq , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (pq − 1)qx−N−1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = pq(1 − qx) pq − 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='120) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N − 1)(1 − pqx+1), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 q−N − 1 , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N − 1)pqx, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−N(1 − qx) q−N − 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='121) qR : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − (1 − qx−N)(1 − dqx) (q−N−1 − 1)(1 − dq2x+1), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N−1 − 1)(1 − bqx)(1 − cqx) 1 − dq2x , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − dqx+N)(1 − qx) (1 − qN+1)(1 − dq2x−1), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − ˜d (1 − qN+1)(1 − b−1dqx)(1 − c−1dqx) 1 − dq2x , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='122) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − bqx)(1 − dqx) (1 − bq−1)(1 − dq2x+1), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − bq−1)(1 − qx−N)(1 − cqx) 1 − dq2x , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − b−1dqx)(1 − qx) (1 − b−1q)(1 − dq2x−1), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − ˜d (1 − b−1q)(1 − dqx+N)(1 − c−1dqx) 1 − dq2x , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='123) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − cqx)(1 − dqx) (1 − cq−1)(1 − dq2x+1), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − cq−1)(1 − qx−N)(1 − bqx) 1 − dq2x , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − c−1dqx)(1 − qx) (1 − c−1q)(1 − dq2x−1), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − ˜d (1 − c−1q)(1 − dqx+N)(1 − b−1dqx) 1 − dq2x , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='124) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − c)(1 − qx−N)(1 − bqx) 1 − dq2x+1 , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − cqx)(1 − dqx) (1 − c)(1 − dq2x) , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − ˜d (1 − c−1)(1 − dqx+N)(1 − b−1dqx) 1 − dq2x−1 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − c−1dqx)(1 − qx) (1 − c−1)(1 − dq2x) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='125) (e) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − b)(1 − qx−N)(1 − cqx) 1 − dq2x+1 , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − bqx)(1 − dqx) (1 − b)(1 − dq2x) , 17 D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − ˜d (1 − b−1)(1 − dqx+N)(1 − c−1dqx) 1 − dq2x−1 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − b−1dqx)(1 − qx) (1 − b−1)(1 − dq2x) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='126) (f) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N − 1)(1 − bqx)(1 − cqx) 1 − dq2x+1 , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − qx−N)(1 − dqx) (q−N − 1)(1 − dq2x), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − ˜d (1 − qN)(1 − b−1dqx)(1 − c−1dqx) 1 − dq2x−1 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − dqx+N)(1 − qx) (1 − qN)(1 − dq2x) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='127) dqH : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qx−N − 1)(1 − abqx−1) (q−N−1 − 1)(1 − abq2x), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N−1 − 1)(1 − aqx) 1 − abq2x−1 , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qx)(1 − abqx+N−1) (1 − qN+1)(1 − abq2x−2), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN+1)aqx−N−1(1 − bqx−1) 1 − abq2x−1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='128) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − aqx)(1 − abqx−1) (1 − aq−1)(1 − abq2x), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − aq−1)(qx−N − 1) 1 − abq2x−1 , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qx)(1 − bqx−1) (1 − a−1q)(1 − abq2x−2), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a−1q)aqx−N−1(1 − abqx+N−1) 1 − abq2x−1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='129) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − abqx−1 1 − abq2x , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qx−N − 1)(1 − aqx) 1 − abq2x−1 , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = bq−2aqx(1 − qx) 1 − abq2x−2 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−N−1(1 − abqx+N−1)(1 − bqx−1) bq−2(1 − abq2x−1) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='130) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qx−N − 1)(1 − aqx) 1 − abq2x , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − abqx−1 1 − abq2x−1, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−N(1 − abqx+N−1)(1 − bqx−1) b(1 − abq2x−2) , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = baqx−1(1 − qx) 1 − abq2x−1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='131) (e) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a)(qx−N − 1) 1 − abq2x , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − aqx)(1 − abqx−1) (1 − a)(1 − abq2x−1) , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a − 1)qx−N−1(1 − abqx+N−1) 1 − abq2x−2 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a(1 − qx)(1 − bqx−1) (a − 1)(1 − abq2x−1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='132) (f) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−N − 1)(1 − aqx) 1 − abq2x , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qx−N − 1)(1 − abqx−1) (q−N − 1)(1 − abq2x−1), 18 D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN)aqx−N−1(1 − bqx−1) 1 − abq2x−2 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qx)(1 − abqx+N−1) (1 − qN)(1 − abq2x−1) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='133) dqK : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qx−N − 1)(1 + pqx) (q−N−1 − 1)(1 + pq2x+1), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−N−1 − 1 1 + pq2x , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qx)(1 + pqx+N) (1 − qN+1)(1 + pq2x−1), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN+1)pq2x−N−1 1 + pq2x , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='134) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 + pqx 1 + pq2x+1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 1 + pq2x , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −pqx−1(1 − qx) 1 + pq2x−1 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −qx−N(1 + pqx+N) 1 + pq2x , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='135) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1 1 + pq2x+1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 + pqx 1 + pq2x, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −qx−N−1(1 + pqx+N) 1 + pq2x−1 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −pqx(1 − qx) 1 + pq2x , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='136) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−N − 1 1 + pq2x+1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qx−N − 1)(1 + pqx) (q−N − 1)(1 + pq2x), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qN)pq2x−N−1 1 + pq2x−1 , D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qx)(1 + pqx+N) (1 − qN)(1 + pq2x) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='137) M : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x + β β − 1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (β − 1)c 1 − c , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x 1 − β, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − β 1 − c , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='138) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = βc 1 − c, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x + β β , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − β 1 − c, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −x β , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='139) lqJ : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−1(q−x − b) 1 − bq−1 , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − bq−1)a, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x − 1 bq−1 − 1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = bq−1 − 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='140) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − b)aq−1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x − b 1 − b , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = b − 1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x − 1 b − 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='141) lqL : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x−1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(q−x − 1), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='142) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aq−1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(q−x − 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='143) 19 qM : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx+1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = cq−1(1 − bqx+1), D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 + bcqx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='144) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − bqx+1 1 − b , B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − b)cqx, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx 1 − b−1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − b−1)(1 + bcqx), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='145) (c) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − bq)cqx, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − bqx+1 1 − bq , D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − b−1q−1)(1 + bcqx), D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx 1 − b−1q−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='146) (d) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = c(1 − bqx+1), B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 + bcqx, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='147) ASCII : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx+1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aqx, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − aqx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='148) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aqx+1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − aqx, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='149) qC : (a) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx+1, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aq−1, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='150) (b) : B1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a, B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx, D1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1, D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='151) Let us define the constants ˜fn(λ), ˜bn(λ) and ¯δ as follows: H : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (0, 0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='152) (b) : ˜fn(λ) = 1, ˜bn(λ) = −(n + a − 1)(n + b), ¯δ = (1, −1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='153) (c) : ˜fn(λ) = −(n + a)(n + b − 1), ˜bn(λ) = 1, ¯δ = (−1, 1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='154) (d) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (0, 0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='155) K : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='156) (b) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='157) R : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (1, 0, 0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='158) (b) : ˜fn(λ) = 1, ˜bn(λ) = −(n + b − 1)(n + c − d − N), ¯δ = (0, 1, 0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='159) (c) : ˜fn(λ) = 1, ˜bn(λ) = −(n + c − 1)(n + b − d − N), ¯δ = (0, 0, 1, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='160) (d) : ˜fn(λ) = −(n + c)(n + b − d − N − 1), ˜bn(λ) = 1, ¯δ = (0, 0, −1, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='161) 20 (e) : ˜fn(λ) = −(n + b)(n + c − d − N − 1), ˜bn(λ) = 1, ¯δ = (0, −1, 0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='162) (f) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (−1, 0, 0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='163) dH : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (0, 1, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='164) (b) : ˜fn(λ) = 1, ˜bn(λ) = −(n + a − 1), ¯δ = (1, 0, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='165) (c) : ˜fn(λ) = 1, ˜bn(λ) = −(n − b − N + 1), ¯δ = (0, 1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='166) (d) : ˜fn(λ) = −(n − b − N), ˜bn(λ) = 1, ¯δ = (0, −1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='167) (e) : ˜fn(λ) = −(n + a), ˜bn(λ) = 1, ¯δ = (−1, 0, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='168) (f) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (0, −1, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='169) dqqK : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (1, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='170) (b) : ˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − p−1qn−N), ¯δ = (1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='171) (c) : ˜fn(λ) = −q−n(1 − p−1qn−N−1), ˜bn(λ) = 1, ¯δ = (−1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='172) (d) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (−1, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='173) qH : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (0, 0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='174) (b) : ˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − aqn−1)(1 − bqn), ¯δ = (1, −1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='175) (c) : ˜fn(λ) = −q−n(1 − aqn)(1 − bqn−1), ˜bn(λ) = 1, ¯δ = (−1, 1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='176) (d) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (0, 0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='177) qK : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='178) (b) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='179) qqK : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='180) (b) : ˜fn(λ) = 1, ˜bn(λ) = −p−1q−1(1 − pqn+1), ¯δ = (−1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='181) (c) : ˜fn(λ) = −p−1(1 − pqn) ˜bn(λ) = 1, ¯δ = (1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='182) (d) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='183) aqK : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='184) (b) : ˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − pqn), ¯δ = (1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='185) (c) : ˜fn(λ) = −q−n(1 − pqn+1), ˜bn(λ) = 1, ¯δ = (−1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='186) (d) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='187) qR : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (1, 0, 0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='188) (b) : ˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − bqn−1)(1 − cd−1qn−N), ¯δ = (0, 1, 0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='189) 21 (c) : ˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − cqn−1)(1 − bd−1qn−N), ¯δ = (0, 0, 1, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='190) (d) : ˜fn(λ) = −q−n(1 − cqn)(1 − bd−1qn−N−1), ˜bn(λ) = 1, ¯δ = (0, 0, −1, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='191) (e) : ˜fn(λ) = −q−n(1 − bqn)(1 − cd−1qn−N−1), ˜bn(λ) = 1, ¯δ = (0, −1, 0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='192) (f) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (−1, 0, 0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='193) dqH : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (0, 1, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='194) (b) : ˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − aqn−1), ¯δ = (1, 0, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='195) (c) : ˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − b−1qn−N+1), ¯δ = (0, 1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='196) (d) : ˜fn(λ) = −q−n(1 − b−1qn−N), ˜bn(λ) = 1, ¯δ = (0, −1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='197) (e) : ˜fn(λ) = −q−n(1 − aqn), ˜bn(λ) = 1, ¯δ = (−1, 0, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='198) (f) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (0, −1, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='199) dqK : (a) : ˜fn(λ) = 1, ˜bn(λ) = EN+1(λ) − En(λ), ¯δ = (1, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='200) (b) : ˜fn(λ) = 1, ˜bn(λ) = −q−n, ¯δ = (1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='201) (c) : ˜fn(λ) = −q−n, ˜bn(λ) = 1, ¯δ = (−1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='202) (d) : ˜fn(λ) = EN(λ) − En(λ), ˜bn(λ) = 1, ¯δ = (−1, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='203) M : (a) : ˜fn(λ) = 1, ˜bn(λ) = −(n + β − 1), ¯δ = (1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='204) (b) : ˜fn(λ) = −(n + β), ˜bn(λ) = 1, ¯δ = (−1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='205) lqJ : (a) : ˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − aqn)(1 − bqn−1), ¯δ = (−1, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='206) (b) : ˜fn(λ) = −q−n(1 − aqn−1)(1 − bqn), ˜bn(λ) = 1, ¯δ = (1, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='207) lqL : (a) : ˜fn(λ) = 1, ˜bn(λ) = −q−n(1 − aqn), ¯δ = −1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='208) (b) : ˜fn(λ) = −q−n(1 − aqn−1), ˜bn(λ) = 1, ¯δ = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='209) qM : (a) : ˜fn(λ) = 1, ˜bn(λ) = qn + cq−1, ¯δ = (0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='210) (b) : ˜fn(λ) = 1, ˜bn(λ) = −b−1(1 − bqn), ¯δ = (1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='211) (c) : ˜fn(λ) = −b−1q−1(1 − bqn+1), ˜bn(λ) = 1, ¯δ = (−1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='212) (d) : ˜fn(λ) = qn + c, ˜bn(λ) = 1, ¯δ = (0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='213) ASCII : (a) : ˜fn(λ) = 1, ˜bn(λ) = qn, ¯δ = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='214) (b) : ˜fn(λ) = qn, ˜bn(λ) = 1, ¯δ = −1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='215) qC : (a) : ˜fn(λ) = 1, ˜bn(λ) = qn + aq−1, ¯δ = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='216) 22 (b) : ˜fn(λ) = qn + a, ˜bn(λ) = 1, ¯δ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='217) Then we can show that B1(x), B2(x), D1(x) and D2(x) satisfy B1(x − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + D1(x + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − ˜f0(λ)˜b0(λ) = −B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='218) and the constants ˜fn and ˜bn satisfy En(λ) = ˜f0(λ)˜b0(λ) − ˜fn(λ)˜bn(λ) (n ∈ Z≥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='219) The relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='218) gives another factorization of � H(λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4), � H(λ) = − ˜B(λ) ˜F(λ) + ˜f0(λ)˜b0(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='220) Corresponding to this factorization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='220), we obtain the following relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 For the polynomials in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 (except C and qB), the following forward and backward shift relations hold for n ∈ Z≥0, ˜F(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ˜fn(λ) ˇPn(x + s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='221) ˜B(λ) ˇPn(x + s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) = ˜bn(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='222) where s is given by s = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 1 : H (a)(b), K (a), R (a)(b)(c), dH (a)(b)(c), dqqK (a)(b), qH (a)(b), qK (a), qqK (a)(b), aqK (a)(b), qR (a)(b)(c), dqH (a)(b)(c), dqK (a)(b), M (a), lqJ (a), lqL (a), qM (a)(b), ASCII (a), qC (a) 0 : others .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='223) Proof: It is sufficient to show (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='221), because (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='219)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='221) imply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='222).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Taking qR (a) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='122) as an example, let us prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='221).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' It is shown by direct calculation: ˜F(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a−1dqx+1)(1 − qx+1) (1 − a−1q)(1 − dq2x+1) 4φ3 �q−n, abcd−1qn−1, q−x, dqx a, b, c ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � − (1 − aqx)(1 − dqx) (aq−1 − 1)(1 − dq2x+1) 4φ3 �q−n, abcd−1qn−1, q−x−1, dqx+1 a, b, c ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � = 1 (1 − aq−1)(1 − dq2x+1) n � k=0 (q−n, abcd−1qn−1, q−x−1, dqx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (a, b, c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k × � −aq−1(1 − a−1dqx+1)(−qx+1)(1 − q−x+k−1) + (1 − aqx)(1 − dqx+k) � 23 = 1 (1 − aq−1)(1 − dq2x+1) n � k=0 (q−n, abcd−1qn−1, q−x−1, dqx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (a, b, c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (1 − aqk−1)(1 − dq2x+1) = n � k=0 (q−n, abcd−1qn−1, q−x−1, dqx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (aq−1, b, c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k = ˜fn(λ) ˇPn(x + s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The other cases are proved in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 Two formulas with ¯δ and −¯δ are equivalent by interchanging ˜F and ˜B, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='221) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='222) for H (c) agree with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='222) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='221) for H (b) with the replacements a → a + 1 and b → b − 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' For C and qB, we do not have new factorization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='220) and new forward and backward shift relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='221)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='222).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 The relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='221)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='222) for twelve cases ((a) of H, K, R, dH, dqqK, qH, qK, qqK, aqK, qR, dqH, dqK, which have ˜fn = 1, ˜bn(λ) = EN+1(λ) − En(λ), s = 1 and D1(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = B1(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 0) were given in [9] and they were called forward and backward x-shift relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' By considering e−∂ ˜F(λ) and ˜B(λ)e∂, the above results (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='220) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='221)– (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='222) with s = 1 are rewritten as � H(λ) = − � ˜B(λ)e∂�� e−∂ ˜F(λ) � + ˜f0(λ)˜b0(λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='224) � e−∂ ˜F(λ) � ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ˜fn(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='225) � ˜B(λ)e∂� ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) = ˜bn(λ) ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='226) That is, x is not shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' As an identity of polynomial, the x-shift is not essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' However, this x-shift has important implications in the state-adding Darboux transformation for the finite rdQM systems [9, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 AW and qR polynomials are related as [2, 13] eixAW = d 1 2qxqR, (a1, a2, a3, a4) = (ad− 1 2, bd− 1 2, cd− 1 2, d 1 2), ˇP AW n (xAW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λAW) = d− n 2 (a, b, c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n ˇP qR n (xqR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λqR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='227) For the (j, k) = (1, 4) case in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='33), the operators ˜F and ˜B for AW are related to those for qR (a) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='122) as e γ 2 p ˜F AW(λAW) = −(q−N−1 − 1) ˜F qR(λqR), ˜BAW(λAW)e− γ 2 p = (q−N−1 − 1)−1 ˜BqR(λqR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='228) 24 These extra factors e± γ 2 p give the property in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Similarly AW with (j, k) = (2, 4), (3, 4), (1, 2), (1, 3) and (2, 4) cases correspond to qR (b), (c), (d), (e) and (f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 We can show that B1(x − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)B2(x − s + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), D1(x − s + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)D2(x − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), B1(x − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)D2(x − s + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + D1(x − s + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)B2(x − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − ˜f0(λ)˜b0(λ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='229) = B1(x − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ)D2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) + D1(x + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ)B2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) − ˜f0(λ − ¯δ)˜b0(λ − ¯δ), which imply ˜F(λ) ˜B(λ) ��� x→x−s − ˜f0(λ)˜b0(λ) = ˜B(λ − ¯δ) ˜F(λ − ¯δ) − ˜f0(λ − ¯δ)˜b0(λ − ¯δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='230) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 Polynomials in rdQMJ systems For the rdQMJ systems described by the polynomials in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 (except dqHeI, dqHeII and SW), let us define the operators ˜F J(λ) and ˜BJ(λ) as follows: ˜F J(λ) def = DJ 1(qη;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)qη d dη , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='231) ˜BJ(λ) def = BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)q−η d dη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='232) The potential functions BJ 1(η), BJ 2(η), DJ 1(η) and DJ 2(η) satisfy BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='233) and their explicit forms are given by bqJ : (a) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1a(1 − η) 1 − a , BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a)η−1q(bη − c), DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(aq − η) a − 1 , DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a − 1)η−1(η − cq), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='234) (b) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(1 − η) c−1 − 1 , BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (c−1 − 1)η−1aq(bη − c), DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(η − cq) 1 − c , DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − c)η−1(aq − η), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='235) (c) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (c−1q−1 − 1)η−1aq(bη − c), BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(1 − η) c−1q−1 − 1, DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − cq)η−1(aq − η), DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(η − cq) 1 − cq , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='236) (d) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − aq)η−1(bη − c), BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1aq(1 − η) 1 − aq , 25 DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (aq − 1)η−1(η − cq), DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(aq − η) aq − 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='237) bqL : (a) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1a(1 − η) 1 − a , BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − a)η−1bq, DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(aq − η) a − 1 , DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a − 1)η−1(η − bq), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='238) (b) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1b(1 − η) 1 − b , BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − b)η−1aq, DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(η − bq) 1 − b , DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − b)η−1(aq − η), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='239) (c) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (bq − 1)η−1a, BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1bq(1 − η) bq − 1 , DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − bq)η−1(aq − η), DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(η − bq) 1 − bq , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='240) (d) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (aq − 1)η−1b, BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1aq(1 − η) aq − 1 , DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (aq − 1)η−1(η − bq), DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(aq − η) aq − 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='241) ASCI : (a) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1q−1, BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1a, DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1(1 − η), DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1(η − a), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='242) (b) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1aq−1, BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1, DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1(η − a), DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1(1 − η), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='243) qL : (a) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(1 + η), BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1, DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1q, DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −a−1q−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='244) (b) : BJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1, BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(1 + η), DJ 1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −a−1, DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='245) Let us define the constants ˜f J n(λ), ˜bJ n(λ) and ¯δ as follows: bqJ : (a) : ˜f J n(λ) = 1, ˜bJ n(λ) = −q−n(1 − aqn)(1 − bqn+1), ¯δ = (1, −1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='246) (b) : ˜f J n(λ) = 1, ˜bJ n(λ) = −q−n(1 − cqn)(1 − abc−1qn+1), ¯δ = (0, 0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='247) (c) : ˜f J n(λ) = −q−n(1 − cqn+1)(1 − abc−1qn), ˜bJ n(λ) = 1, ¯δ = (0, 0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='248) (d) : ˜f J n(λ) = −q−n(1 − aqn+1)(1 − bqn), ˜bJ n(λ) = 1, ¯δ = (−1, 1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='249) bqL : (a) : ˜f J n(λ) = 1, ˜bJ n(λ) = −q−n(1 − aqn), ¯δ = (1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='250) (b) : ˜f J n(λ) = 1, ˜bJ n(λ) = −q−n(1 − bqn), ¯δ = (0, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='251) 26 (c) : ˜f J n(λ) = −q−n(1 − bqn+1), ˜bJ n(λ) = 1, ¯δ = (0, −1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='252) (d) : ˜f J n(λ) = −q−n(1 − aqn+1), ˜bJ n(λ) = 1, ¯δ = (−1, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='253) ASCI : (a) : ˜f J n(λ) = 1, ˜bJ n(λ) = −q−n, ¯δ = −1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='254) (b) : ˜f J n(λ) = −q−n, ˜bJ n(λ) = 1, ¯δ = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='255) qL : (a) : ˜f J n(λ) = 1, ˜bJ n(λ) = −a−1q−1(1 − aqn+1), ¯δ = −1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='256) (b) : ˜f J n(λ) = −a−1(1 − aqn), ˜bJ n(λ) = 1, ¯δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='257) Then we can show that BJ 1(η), BJ 2(η), DJ 1(η) and DJ 2(η) satisfy BJ 1(q−1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + DJ 1(qη;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − ˜f J 0 (λ)˜bJ 0(λ) = −BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='258) and the constants ˜f J n and ˜bJ n satisfy En(λ) = ˜f J 0 (λ)˜bJ 0(λ) − ˜f J n(λ)˜bJ n(λ) (n ∈ Z≥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='259) The relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='258) give other factorizations of � H(λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='6), � HJ(λ) = − ˜BJ(λ) ˜F J(λ) + ˜f J 0 (λ)˜bJ 0(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='260) Corresponding to this factorization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='260), we obtain the following relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 For the polynomials in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 (except dqHeI, dqHeII and SW), the following forward and backward shift relations hold for n ∈ Z≥0, ˜F J(λ)Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ˜f J n(λ)Pn(r′η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='261) ˜BJ(λ)Pn(r′η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) = ˜bJ n(λ)Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='262) where r′ is given by r′ = � q : bqJ (c)(d), bqL (c)(d), ASCI (a), qL (b) 1 : bqJ (a)(b), bqL (a)(b), ASCI (b), qL (a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='263) Proof: It is sufficient to show (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='261), because (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='259)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='261) imply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='262).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Taking bqJ (a) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='234) as an example, let us prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='261).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' It is shown by direct calculation: ˜F J(λ)Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(a − η) a − 1 3φ2 �q−n, abqn+1, η aq, cq ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � + η−1a(1 − η) 1 − a 3φ2 �q−n, abqn+1, qη aq, cq ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � 27 = η−1 1 − a n � k=0 (q−n, abqn+1, η ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (aq, cq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k � −(a − η) + a(1 − ηqk) � = η−1 1 − a n � k=0 (q−n, abqn+1, η ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (aq, cq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (1 − aqk)η = n � k=0 (q−n, abqn+1, η ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (a, cq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k qk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k = ˜f J n(λ)Pn(r′η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The other cases are proved in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 Two formulas with ¯δ and −¯δ are equivalent by interchanging ˜F J and ˜BJ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='261) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='262) for bqJ (d) agree with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='262) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='261) for bqJ (a) with the replacements a → aq and b → bq−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' For dqHeI, dqHeII and SW, we do not have new factorization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='260) and new forward and backward shift relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='261)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='262).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 As in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2, by considering q−η d dη ˜F J(λ) and ˜BJ(λ)qη d dη , the above results (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='260) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='261)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='262) with r′ = q are rewritten as � HJ(λ) = − � ˜BJ(λ)qη d dη �� q−η d dη ˜F J(λ) � + ˜f J 0 (λ)˜bJ 0(λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='264) � q−η d dη ˜F J(λ) � Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ˜f J n(λ)Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='265) � ˜BJ(λ)qη d dη � Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) = ˜bJ n(λ)Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='266) That is, η is not q-shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' As an identity of polynomial, the q-shift of η is not essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 We can show that BJ 1(r′ −1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)BJ 2(qr′ −1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), DJ 1(qr′ −1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)DJ 2(r′ −1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ), BJ 1(r′ −1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)DJ 2(qr′ −1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) + DJ 1(qr′ −1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)BJ 2(r′ −1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) − ˜f J 0 (λ)˜bJ 0(λ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='267) = BJ 1(q−1η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ)DJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) + DJ 1(qη;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ)BJ 2(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ − ¯δ) − ˜f J 0 (λ − ¯δ)˜bJ 0(λ − ¯δ), which imply ˜F J(λ) ˜BJ(λ) ��� η→r′ −1η − ˜f J 0 (λ)˜bJ 0(λ) = ˜BJ(λ − ¯δ) ˜F J(λ − ¯δ) − ˜f J 0 (λ − ¯δ)˜bJ 0(λ − ¯δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='268) 5 Summary and Comments The orthogonal polynomials in the Askey scheme satisfy second order differential or difference equations (Theorem 1) and we study them by using quantum mechanical formulation (oQM, 28 idQM, rdQM, rdQMJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The forward and backward shift relations are their basic properties (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2), in which the degree n and the parameters λ are shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' They are based on the factorizations of the differential or difference operators � H (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1) and � HJ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Motivated by the recently found forward and backward x-shift relations [9], in which the coordinate x and parameters λ are shifted, we have tried to find new forward and backward relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We have found new factorizations of � H (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='19), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='78), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='220) and � HJ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='260), and based on them, we have obtained another type of forward and backward shift relations (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' In these new forward and backward shift relations except for some cases of rdQM and rdQMJ, only the parameters λ are shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' As an identity of polynomial, the x-shift (or q-shift of η) is not essential (Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The forward and backward shift relations are related to the shape invariance property of quantum mechanical systems [4, 5, 3, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' It is an interesting problem to investigate the quantum mechanical implications of the new forward and backward shift relations obtained in this paper (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Especially the twelve finite rdQM cases in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 are interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' In these cases, the x-shift has important implications related to the state- adding Darboux transformations [9, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We will report this topic elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The case-(1) multi-indexed orthogonal polynomials are constructed for R and qR [7], W and AW [8], M, lqJ and lqL [14], cH and MP [15], and they have shape invariant property, namely, satisfy the forward and backward shift relations like Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' It is an interesting problem to investigate whether these multi-indexed polynomials satisfy new forward and backward shift relations such as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Acknowledgements This work is supported by JSPS KAKENHI Grant Number JP19K03667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' A Data for Orthogonal Polynomials We give the data for the orthogonal polynomials (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1) in the Askey scheme (all the poly- nomials in chapters 9 and 14 of [2] and the dual quantum q-Krawtchouk), which satisfy second order differential or difference equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The parametrization of some polynomials are different from the conventional ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Since we do not consider orthogonality relations in 29 this paper, we do not care about concrete ranges of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The Pochhammer symbol (shifted factorial), the hypergeometric series and their q-versions are defined by (a)n def = n−1 � j=0 (a + j), (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , ar)n def = r� k=1 (ak)n, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1) (a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n def = n−1 � j=0 (1 − aqj), (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , ar ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n def = r� k=1 (ak ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2) rFs �a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , ar b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , bs ��� x � def = ∞ � k=0 (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , ar)k (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , bs)k xk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3) rφs �a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , ar b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , bs ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' z � def = ∞ � k=0 (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , ar ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , bs ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k (−1)(1+s−r)kq(1+s−r)(k 2) zk (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)k , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4) with the conventions �n−1 j=n ∗ = 0 and �n−1 j=n ∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 Polynomials in oQM We consider the following five polynomials [2, 3, 16]: Hermite (He), Laguerre (L), Jacobi (J), Bessel (B) and pseudo Jacobi (pJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The sinusoidal coordinates η(x) are (i) : η(x) = x : He, (ii) : η(x) = x2 : L, (iii) : η(x) = cos 2x : J, (iv) : η(x) = ex : B, (v) : η(x) = sinh x : pJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='5) The functions c1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) and c2(η) are defined by 4c1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) def = d2η(x) dx2 + 2dw(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) dx dη(x) dx , 4c2(η) def = �dη(x) dx �2 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='6) and the potential U(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The parameters λ are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The standard parametrizations [2] are as follows: L : αstandard = g − 1 2, J : (α, β)standard = (g − 1 2, h − 1 2), B : astandard = −2h − 1, pJ : (N, ν)standard = (h − 1 2, −µ), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='7) and h of pJ is a continuous parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 30 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 (i) η(x) = x He : λ : none, δ : none, κ = 1, En(λ) = 2n, fn(λ) = 2n, bn(λ) = 1, c1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −1 2η, c2(η) = 1 4, cF = 1, w(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −1 2x2, U(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x2 − 1, Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (2η)n 2F0 �−n 2, n−1 2 − ��� − 1 η2 � = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' [ n 2 ] � k=0 (−1)k(2η)n−2k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (n − 2k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='8) where [x] denotes the greatest integer not exceeding x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 (ii) η(x) = x2 L : λ = g, δ = 1, κ = 1, En(λ) = 4n, fn(λ) = −2, bn(λ) = −2(n + 1), c1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = g + 1 2 − η, c2(η) = η, cF = 2, w(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −1 2x2 + g log x, U(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x2 + g(g − 1) x2 − 2g − 1, Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (g + 1 2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 1F1 � −n g + 1 2 ��� η � = L (g− 1 2) n (η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='9) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 (iii) η(x) = cos 2x J : λ = (g, h), δ = (1, 1), κ = 1, En(λ) = 4n(n + g + h), fn(λ) = −2(n + g + h), bn(λ) = −2(n + 1), c1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = h − g − (g + h + 1)η, c2(η) = 1 − η2, cF = −4, w(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = g log sin x + h log cos x, U(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = g(g − 1) sin x2 + h(h − 1) cos x2 − (g + h)2, Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (g + 1 2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 2F1 �−n, n + g + h g + 1 2 ��� 1 − η 2 � = P (g− 1 2 ,h− 1 2) n (η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='10) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 (iv) η(x) = ex B : λ = h, δ = −1, κ = 1, En(λ) = n(2h − n), fn(λ) = −1 2n(2h − n), bn(λ) = −2, 4c1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2 + (1 − 2h)η, 4c2(η) = η2, cF = 1, w(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −hx − e−x, U(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = e−2x − (2h + 1)e−x + h2, 31 Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2F0 �−n, n − 2h − ��� −η 2 � = (n − 2h)n �η 2 �n 1F0 � −n 2h + 1 − 2n ��� 2 η � = (−1)nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' �η 2 �n L(2h−2n) n (2η−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='11) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='5 (v) η(x) = sinh x pJ : λ = (h, µ), δ = (−1, 0), κ = 1, En(λ) = n(2h − n), fn(λ) = n, bn(λ) = 2h − n − 1, 4c1(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − 2h)η − 2µ, 4c2(η) = 1 + η2, cF = 1, w(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −h log cosh x − µ tan−1 sinh x, U(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −h(h + 1) + µ2 + µ(2h + 1) sinh x cosh2 x + h2, Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (−2i)n(−h + 1 2 − iµ)n (n − 2h)n 2F1 � −n, n − 2h −h + 1 2 − iµ ��� 1 − iη 2 � = (η + i)n 2F1 �−n, h + 1 2 + iµ − n 2h + 1 − 2n ��� 2 1 − iη � = (−2i)nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (n − 2h)n P (−h− 1 2−iµ,−h− 1 2 +iµ) n (iη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='12) The Bessel and pseudo Jacobi polynomials have not been treated in our previous papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Their oQM systems are the Morse potential (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 of [16] with the replacement x → −x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The parameter µ can be taken as µ = 1 by shifting x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=') and the hyperbolic symmetric top II, respectively [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Their orthogonality relations are (parameters: h, µ > 0) � ∞ −∞ dx φ0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)2 ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) ˇPm(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = hn(λ)δnm (n, m = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' , [h]′), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='13) where φ0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ew(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='λ) and [x]′ denotes the greatest integer not equal or exceeding x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The normalization constants hn(λ) are given by [2, 16] B : hn(λ) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Γ(2h − n + 1) 22h+1(h − n) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='14) pJ : hn(λ) = 2πn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 22n−2hΓ(2h − 2n) (2h − 2n + 1)nΓ(h − n + 1 2 − iµ)Γ(h − n + 1 2 + iµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='15) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 Polynomials in idQM We consider the following thirteen polynomials [5]: continuous Hahn (cH), Meixner-Pollaczek (MP), Wilson (W), continuous dual Hahn (cdH), Askey-Wilson (AW), continuous dual q- 32 Hahn (cdqH), Al-Salam-Chihara (ASC), continuous big q-Hermite (cbqHe), continuous q- Hermite (cqHe), continuous q-Jacobi (cqJ), continuous q-Laguerre (cqL), continuous q-Hahn (cqH) and q-Meixner-Pollaczek (qMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The sinusoidal coordinates η(x) and auxiliary func- tions ϕ(x) are [5] (i) : η(x) = x, ϕ(x) = 1, (ii) : η(x) = x2, ϕ(x) = 2x, (iii) : η(x) = cos x, ϕ(x) = 2 sin x, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='16) (iv) : η(x) = cos(x + φ), ϕ(x) = 2 sin(x + φ), and (i) : cH, MP, (ii) : W, cdH, (iii) : AW, cdqH, ASC, cbqHe, cqHe, cqJ, cqL, (iv) : cqH, qMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='17) The constant γ is γ = 1 for non q-polynomials, γ = log q for q-polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The parameters λ are complex unless mentioned, and satisfy the following;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' W and AW: {a∗ 1, a∗ 2, a∗ 3, a∗ 4} = {a1, a2, a3, a4} (as a set), cdH and cdqH: {a∗ 1, a∗ 2, a∗ 3} = {a1, a2, a3} (as a set), ASC and cqH: {a∗ 1, a∗ 2} = {a1, a2} (as a set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The standard parametrizations [2] are as follows: cH : (a, b, c, d)standard = (a1, a2, a∗ 1, a∗ 2), MP : (λ, φ)standard = (a, φ), W, AW : (a, b, c, d)standard = (a1, a2, a3, a4), cdH, cdqH : (a, b, c)standard = (a1, a2, a3), ASC : (a, b)standard = (a1, a2), cqH : (a, b, c, d, φ)standard = (a1, a2, a1, a2, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='18) Some polynomials are symmetric under the permutations of the following parameters: W, AW : (a1, a2, a3, a4), cdH, cdqH : (a1, a2, a3), cH, ASC, cqH : (a1, a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='19) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 (i) η(x) = x cH : λ = (a1, a2), δ = ( 1 2, 1 2), κ = 1, b1 def = a1 + a2 + a∗ 1 + a∗ 2, En(λ) = n(n + b1 − 1), fn(λ) = n + b1 − 1, bn(λ) = n + 1, V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a1 + ix)(a2 + ix), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = in(a1 + a∗ 1, a1 + a∗ 2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 3F2 �−n, n + b1 − 1, a1 + ix a1 + a∗ 1, a1 + a∗ 2 ��� 1 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='20) MP : λ = (a, φ), δ = ( 1 2, 0), κ = 1, a, φ ∈ R, 33 En(λ) = 2n sin φ, fn(λ) = 2 sin φ, bn(λ) = n + 1, V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = ei( π 2 −φ)(a + ix), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (2a)neinφ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 2F1 �−n, a + ix 2a ��� 1 − e−2iφ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='21) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 (ii) η(x) = x2 W : λ = (a1, a2, a3, a4), δ = ( 1 2, 1 2, 1 2, 1 2), κ = 1, b1 def = a1 + a2 + a3 + a4, En(λ) = n(n + b1 − 1), fn(λ) = −n(n + b1 − 1), bn(λ) = −1, V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a1 + ix)(a2 + ix)(a3 + ix)(a4 + ix) 2ix(2ix + 1) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a1 + a2, a1 + a3, a1 + a4)n 4F3 �−n, n + b1 − 1, a1 + ix, a1 − ix a1 + a2, a1 + a3, a1 + a4 ��� 1 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='22) cdH : λ = (a1, a2, a3), δ = ( 1 2, 1 2, 1 2), κ = 1, En(λ) = n, fn(λ) = −n, bn(λ) = −1, V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a1 + ix)(a2 + ix)(a3 + ix) 2ix(2ix + 1) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a1 + a2, a1 + a3)n 3F2 �−n, a1 + ix, a1 − ix a1 + a2, a1 + a3 ��� 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='23) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 (iii) η(x) = cos x AW : qλ = (a1, a2, a3, a4), δ = ( 1 2, 1 2, 1 2, 1 2), κ = q−1, b4 def = a1a2a3a4, En(λ) = (q−n − 1)(1 − b4qn−1), fn(λ) = q n 2 (q−n − 1)(1 − b4qn−1), bn(λ) = q− n+1 2 , V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a1eix)(1 − a2eix)(1 − a3eix)(1 − a4eix) (1 − e2ix)(1 − qe2ix) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a1a2, a1a3, a1a4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n an 1 4φ3 �q−n, b4qn−1, a1eix, a1e−ix a1a2, a1a3, a1a4 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='24) cdqH : qλ = (a1, a2, a3), δ = ( 1 2, 1 2, 1 2), κ = q−1, En(λ) = q−n − 1, fn(λ) = q n 2 (q−n − 1), bn(λ) = q− n+1 2 , V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a1eix)(1 − a2eix)(1 − a3eix) (1 − e2ix)(1 − qe2ix) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a1a2, a1a3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n an 1 3φ2 �q−n, a1eix, a1e−ix a1a2, a1a3 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='25) ASC : qλ = (a1, a2), δ = ( 1 2, 1 2), κ = q−1, En(λ) = q−n − 1, fn(λ) = q n 2 (q−n − 1), bn(λ) = q− n+1 2 , 34 V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a1eix)(1 − a2eix) (1 − e2ix)(1 − qe2ix) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a1a2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n an 1 3φ2 �q−n, a1eix, a1e−ix a1a2, 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='26) cbqHe : qλ = a, δ = 1 2, κ = q−1, En(λ) = q−n − 1, fn(λ) = q n 2 (q−n − 1), bn(λ) = q− n+1 2 , V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − aeix (1 − e2ix)(1 − qe2ix), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a−n 3φ2 �q−n, aeix, ae−ix 0, 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='27) cqHe : λ : none, δ : none, κ = q−1, En(λ) = q−n − 1, fn(λ) = q n 2 (q−n − 1), bn(λ) = q− n+1 2 , V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 (1 − e2ix)(1 − qe2ix), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = einx 2φ0 �q−n, 0 − ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' qne−2ix� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='28) cqJ : λ = (α, β), δ = (1, 1), κ = q−1, α, β ∈ R, En(λ) = (q−n − 1)(1 − qn+α+β+1), fn(λ) = q 1 2(α+ 3 2)q−n(1 − qn+α+β+1) (1 + q 1 2 (α+β+1))(1 + q 1 2(α+β+2)) , bn(λ) = q− 1 2(α+ 3 2)qn+1(q−n−1 − 1)(1 + q 1 2(α+β+1))(1 + q 1 2 (α+β+2)), V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − q 1 2(α+ 1 2)eix)(1 − q 1 2(α+ 3 2)eix)(1 + q 1 2(β+ 1 2 )eix)(1 + q 1 2 (β+ 3 2)eix) (1 − e2ix)(1 − qe2ix) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qα+1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n 4φ3 �q−n, qn+α+β+1, q 1 2 (α+ 1 2 )eix, q 1 2(α+ 1 2)e−ix qα+1, −q 1 2(α+β+1), −q 1 2 (α+β+2) ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='29) cqL : λ = α, δ = 1, κ = q−1, α ∈ R, En(λ) = q−n − 1, fn(λ) = q 1 2 (α+ 3 2)q−n, bn(λ) = q− 1 2(α+ 3 2)qn+1(q−n−1 − 1), V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − q 1 2 (α+ 1 2 )eix)(1 − q 1 2 (α+ 3 2 )eix) (1 − e2ix)(1 − qe2ix) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qα+1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n 3φ2 �q−n, q 1 2 (α+ 1 2 )eix, q 1 2(α+ 1 2)e−ix qα+1, 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='30) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 (iv) η(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = cos(x + φ) cqH : qλ = (a1, a2, qφ), δ = ( 1 2, 1 2, 0), κ = q−1, b4 def = a1a2a∗ 1a∗ 2 = a2 1a2 2, φ ∈ R, En(λ) = (q−n − 1)(1 − b4qn−1), fn(λ) = q n 2 (q−n − 1)(1 − b4qn−1), bn(λ) = q− n+1 2 , V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − a1ei(x+2φ))(1 − a2ei(x+2φ))(1 − a1eix)(1 − a2eix) (1 − e2i(x+φ))(1 − qe2i(x+φ)) , 35 ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (a1a2e2iφ, a2 1, a1a2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n an 1einφ 4φ3 �q−n, b4qn−1, a1ei(x+2φ), a1e−ix a1a2e2iφ, a2 1, a1a2 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='31) qMP : qλ = (a, qφ), δ = ( 1 2, 0), κ = q−1, a, φ ∈ R, En(λ) = q−n − 1, fn(λ) = q− n 2 , bn(λ) = q− n+1 2 (1 − qn+1), V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − aei(x+2φ))(1 − aeix) (1 − e2i(x+φ))(1 − qe2i(x+φ)), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 (q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n (a2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n aneinφ 3φ2 �q−n, aei(x+2φ), ae−ix a2, 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='32) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 Polynomials in rdQM The sinusoidal coordinates η(x) and auxiliary functions ϕ(x) are [4] (i) : η(x) = x, ϕ(x) = 1, (ii) : η(x) = x(x + d), ϕ(x) = 2x + 1 + d 1 + d , (iii) : η(x) = 1 − qx, ϕ(x) = qx, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='33) (iv) : η(x) = q−x − 1, ϕ(x) = q−x, (v) : η(x) = (q−x − 1)(1 − dqx), ϕ(x) = q−x1 − dq2x+1 1 − dq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Note that η(0) = 0 and ϕ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We impose the following normalization condition on the polynomial (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1), ˇPn(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = Pn(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='34) This normalization condition implies the following two universal expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The coefficient of the highest degree term, cn(λ), is expressed as ((A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='14) in [17]), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = cn(λ)η(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)n + (lower degree terms), cn(λ) = (−1)nκ−(n 2) n � j=1 En(λ) − Ej−1(λ) η(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ)B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ + (j − 1)δ), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='35) and the constants fn(λ) and bn(λ) of the forward and backward shift relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='16)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='17) are given by fn(λ) = En(λ), bn(λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='36) The parameters λ are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 36 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 finite rdQM We consider the following twelve polynomials [4]: Hahn (H), Krawtchouk (K), Racah (R), dual Hahn (dH), dual quantum q-Krawtchouk (dqqK) (which is not treated in [2]), q-Hahn (qH), q-Krawtchouk (qK), quantum q-Krawtchouk (qqK), affine q-Krawtchouk (aqK), q- Racah (qR), dual q-Hahn (dqH) and dual q-Krawtchouk (dqK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' We consider ǫ = ǫ′ = 1 cases in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Their sinusoidal coordinates (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='33) are (i) : H, K, (ii) : R, dH(d = a + b − 1), (iii) : dqqK, (iv) : qH, qK, qqK, aqK, (v) : qR, dqH(d = abq−1), dqK(d = −p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='37) The standard parametrizations [2] are as follows: H : (α, β)standard = (a − 1, b − 1), dH : (γ, δ)standard = (a − 1, b − 1), qH : (α, β)standard = (aq−1, bq−1), dqH : (γ, δ)standard = (aq−1, bq−1), R : (α, β, γ, δ)standard = (a − 1, b + c − d − 1, c − 1, d − c), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='38) qR : (α, β, γ, δ)standard = (aq−1, bcd−1q−1, cq−1, dc−1), dqK : cstandard = −pqN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' R and qR polynomials are symmetric under the permutations of the following parameters: R : (b, c) � (a, b, c) if a = −N is not imposed � , qR : (b, c) � (a, b, c) if a = q−N is not imposed � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='39) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 (i) η(x) = x H : λ = (a, b, N), δ = (1, 1, −1), κ = 1, En(λ) = n(n + a + b − 1), B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + a)(N − x), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x(b + N − x), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3F2 �−n, n + a + b − 1, −x a, −N ��� 1 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='40) K : λ = (p, N), δ = (0, −1), κ = 1, En(λ) = n, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = p(N − x), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − p)x, ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2F1 �−n, −x −N ��� p−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='41) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 (ii) η(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x(x + d) R : We take a = −N and define ˜d = a + b + c − d − 1, 37 λ = (a, b, c, d), δ = (1, 1, 1, 1) κ = 1, En(λ) = n(n + ˜d), B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(x + a)(x + b)(x + c)(x + d) (2x + d)(2x + 1 + d) , D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(x + d − a)(x + d − b)(x + d − c)x (2x − 1 + d)(2x + d) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 4F3 �−n, n + ˜d, −x, x + d a, b, c ��� 1 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='42) dH : λ = (a, b, N), δ = (1, 0, −1), κ = 1, En(λ) = n, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (x + a)(x + a + b − 1)(N − x) (2x − 1 + a + b)(2x + a + b) , D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x(x + b − 1)(x + a + b + N − 1) (2x − 2 + a + b)(2x − 1 + a + b), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3F2 �−n, x + a + b − 1, −x a, −N ��� 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='43) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 (iii) η(x) = 1 − qx dqqK : qλ = (p, qN), δ = (0, −1), κ = q−1, En(λ) = q−n − 1, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = p−1q−x−N−1(1 − qN−x), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−x − 1)(1 − p−1q−x), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2φ1 �q−n, q−x q−N ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' pqx+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='44) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 (iv) η(x) = q−x − 1 qH : qλ = (a, b, qN), δ = (1, 1, −1), κ = q−1, En(λ) = (q−n − 1)(1 − abqn−1), B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − aqx)(qx−N − 1), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aq−1(1 − qx)(qx−N − b), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3φ2 �q−n, abqn−1, q−x a, q−N ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='45) qK : qλ = (p, qN), δ = (2, −1), κ = q−1, En(λ) = (q−n − 1)(1 + pqn), B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx−N − 1, D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = p(1 − qx), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3φ2 �q−n, q−x, −pqn q−N, 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='46) qqK : qλ = (p, qN), δ = (1, −1), κ = q, En(λ) = 1 − qn, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = p−1qx(qx−N − 1), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qx)(1 − p−1qx−N−1), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2φ1 �q−n, q−x q−N ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' pqn+1� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='47) aqK : qλ = (p, qN), δ = (1, −1), κ = q−1, En(λ) = q−n − 1, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qx−N − 1)(1 − pqx+1), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = pqx−N(1 − qx), 38 ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3φ2 �q−n, q−x, 0 pq, q−N ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='48) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='5 (v) η(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (q−x − 1)(1 − dqx) qR : We take a = q−N and define ˜d = abcd−1q−1, qλ = (a, b, c, d), δ = (1, 1, 1, 1), κ = q−1, En(λ) = (q−n − 1)(1 − ˜dqn), B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −(1 − aqx)(1 − bqx)(1 − cqx)(1 − dqx) (1 − dq2x)(1 − dq2x+1) , D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = − ˜d (1 − a−1dqx)(1 − b−1dqx)(1 − c−1dqx)(1 − qx) (1 − dq2x−1)(1 − dq2x) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 4φ3 �q−n, ˜dqn, q−x, dqx a, b, c ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='49) dqH : qλ = (a, b, qN), δ = (1, 0, −1), κ = q−1, En(λ) = q−n − 1, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qx−N − 1)(1 − aqx)(1 − abqx−1) (1 − abq2x−1)(1 − abq2x) , D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aqx−N−1(1 − qx)(1 − abqx+N−1)(1 − bqx−1) (1 − abq2x−2)(1 − abq2x−1) , ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3φ2 �q−n, abqx−1, q−x a, q−N ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='50) dqK : qλ = (p, qN), δ = (1, −1), κ = q−1, En(λ) = q−n − 1, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (qx−N − 1)(1 + pqx) (1 + pq2x)(1 + pq2x+1), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = pq2x−N−1 (1 − qx)(1 + pqx+N) (1 + pq2x−1)(1 + pq2x), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3φ2 �q−n, q−x, −pqx q−N, 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='51) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 semi-infinite rdQM We consider the following eight polynomials [4]: Meixner (M), Charlier (C), little q-Jacobi (lqJ), little q-Laguerre/Wall (lqL), q-Bessel (qB) (=alternative q-Charlier), q-Meixner (qM), Al-Salam-Carlitz II (ASCII) and q-Charlier (qC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' Their sinusoidal coordinates (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='33) are (i) : M, C, (iii) : lqJ, lqL, qB, (iv) : qM, ASCII, qC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='52) The standard lqJ, lqL, qB and ASCII polynomials [2] do not satisfy the normalization con- dition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The standard parametrizations [2] are as follows: lqJ : (a, b)standard = (aq−1, bq−1), lqL : astandard = aq−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='53) 39 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='1 (i) η(x) = x M : λ = (β, c), δ = (1, 0), κ = 1, En(λ) = n, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = c 1 − c(x + β), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 1 − cx, ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2F1 �−n, −x β ��� 1 − c−1� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='54) C : λ = a, δ = 0, κ = 1, En(λ) = n, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a, D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = x, ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2F0 �−n, −x − ��� −a−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='55) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2 (iii) η(x) = 1 − qx lqJ : qλ = (a, b), δ = (1, 1), κ = q−1, En(λ) = (q−n − 1)(1 − abqn−1), B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aq−1(q−x − b), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x − 1, ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3φ1 �q−n, abqn−1 q−x b ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' a−1qx+1� = (−a)−nq−(n 2)(a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n (b ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n 2φ1 �q−n, abqn−1 a ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' qx+1� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='56) lqL : qλ = a, δ = 1, κ = q−1, En(λ) = q−n − 1, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aq−x−1, D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x − 1, ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2φ0 �q−n, q−x − ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' a−1qx+1� = (a−1q1−n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n 2φ1 �q−n, 0 a ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' qx+1� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='57) qB : qλ = a, δ = 2, κ = q−1, En(λ) = (q−n − 1)(1 + aqn), B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = a, D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = q−x − 1, ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (−aqn)−n 2φ0 �q−n, −aqn 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' qx+1� = qnx 2φ1 �q−n, q−x 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' −a−1q1−n� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='58) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='3 (iv) η(x) = q−x − 1 qM : qλ = (b, c), δ = (1, −1), κ = q, En(λ) = 1 − qn, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = cqx(1 − bqx+1), D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qx)(1 + bcqx), ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2φ1 �q−n, q−x bq ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' −c−1qn+1� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='59) ASCII : qλ = a, δ = 0, κ = q, En(λ) = 1 − qn, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aq2x+1, D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (1 − qx)(1 − aqx), 40 ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2φ0 �q−n, q−x − ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' a−1qn� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='60) qC : qλ = a, δ = −1, κ = q, En(λ) = 1 − qn, B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aqx, D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 − qx, ˇPn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2φ1 �q−n, q−x 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' −a−1qn+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='61) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='4 Polynomials in rdQMJ We consider the following seven polynomials [10]: big q-Jacobi (bqJ), big q-Laguerre (bqL), Al-Salam-Carlitz I (ASCI), discrete Hermite I (dqHeI), discrete Hermite II (dqHeII), q- Laguerre (qL) and Stieltjes-Wigert (SW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (SW is not given in [10] and we comment on this at the end of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=') We impose the following normalization condition on the polynomial Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ), Pn(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 : bqJ, bqL, ASCI, dqHeI, Pn(−i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = (−i)n : dqHeII, Pn(−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 : qL, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='62) Pn(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1 : SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The standard ASCI, dqHeI, dqHeII, qL and SW polynomials [2] do not satisfy this normal- ization condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The standard parametrizations [2] are as follows: qL : qαstandard = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='63) The constants f J n(λ) and bJ n(λ) of the forward and backward shift relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='27)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='28) are given by f J n(λ) = En(λ), bJ n(λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='64) The parameters λ are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' bqJ : qλ = (a, b, c), δ = (1, 1, 1), κ = q−1, En(λ) = (q−n − 1)(1 − abqn+1), BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−2aq(1 − η)(bη − c), DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−2(aq − η)(η − cq), Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3φ2 �q−n, abqn+1, η aq, cq ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='65) bqL : qλ = (a, b), δ = (1, 1), κ = q−1, En(λ) = q−n − 1, BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−2abq(1 − η), DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−2(aq − η)(η − bq), 41 Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 3φ2 �q−n, 0, η aq, bq ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q � = (−b)nq 1 2n(n+1) (bq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n 2φ1 �q−n, aqη−1 aq ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' b−1η � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='66) ASCI : qλ = a, δ = 0, κ = q−1, En(λ) = q−n − 1, BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −η−2aq−1, DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−2(1 − η)(η − a), Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2φ1 �q−n, η−1 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' a−1qη � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='67) dqHeI : qλ : none, δ : none, κ = q−1, En(λ) = q−n − 1, BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−2q−1, DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−2(1 − η2), Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2φ1 �q−n, η−1 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' −qη � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='68) dqHeII : qλ : none, δ : none, κ = q, En(λ) = 1 − qn, BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−2(1 + η2), DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−2q, Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = i−n 2φ0 �q−n, iη − ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' −qn� = q(n 2)ηn 2φ1 �q−n, q1−n 0 ��� q2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' −q2η−2� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='69) qL : qλ = a, δ = 1, κ = q, En(λ) = 1 − qn, BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1(1 + η), DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1a−1, Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 2φ1 �q−n, −η 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' aqn+1� = (−aqn)nηn 2φ1 �q−n, a−1q−n 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' −qη−1� = (aq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n 1φ1 �q−n aq ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' −aqn+1η � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='70) SW : qλ : none, δ : none, κ = q, En(λ) = 1 − qn, BJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1, DJ(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = η−1, Pn(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = 1φ1 �q−n 0 ��� q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' −qn+1η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='71) The quantum mechanical formulation needs two component formalism with two sinu- soidal coordinates η(±)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) (except qL and SW) [10], bqJ : η(+)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aqx+1, η(−)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = cqx+1, bqL : η(+)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aqx+1, η(−)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = bqx+1, ASCI : η(+)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx, η(−)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = aqx, dqHeI : η(+)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = qx, η(−)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −qx, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='72) dqHeII : η(+)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = cqx, η(−)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = −cqx, qL, SW : η(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' λ) = cqx, where the parameters λ are extended to qλ = c (δ = 1) for dqHeII, qλ = (a, c) (δ = (1, 1)) 42 for qL and qλ = c (δ = 2) for SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' The quantum mechanical formulation for SW is not given in [10], but it is similar to that for qL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' There is an infinite sum orthogonality relations (parameter: c > 0), SW : ∞ � x=−∞ cxq 1 2x(x+1)Pn(cqx)Pm(cqx) = δnm q−n(q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)n(q, −cq, −c−1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' q)∞ (n, m ∈ Z≥0), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content='73) which is not given in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' E.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} +page_content=' 44' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAyT4oBgHgl3EQfx_nV/content/2301.00678v1.pdf'} diff --git a/k9E3T4oBgHgl3EQfhwor/content/tmp_files/2301.04573v1.pdf.txt b/k9E3T4oBgHgl3EQfhwor/content/tmp_files/2301.04573v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e97b5f2420a8ac667f056c1f0a17e982fb97c012 --- /dev/null +++ b/k9E3T4oBgHgl3EQfhwor/content/tmp_files/2301.04573v1.pdf.txt @@ -0,0 +1,630 @@ +1 +Experimental demonstration of bandwidth enhancement in +photonic time delay reservoir computing +Irene Est´ebanez, Apostolos Argyris, and Ingo Fischer +Abstract—Time delay reservoir computing (TDRC) using semi- +conductor lasers (SLs) has proven to be a promising photonic +analog approach for information processing. One appealing +property is that SLs subject to delayed optical feedback and +external optical injection, allow tuning the response bandwidth by +changing the level of optical injection. Here we use strong optical +injection, thereby expanding the SL’s modulation response up to +tens of GHz. Performing a nonlinear time series prediction task, +we demonstrate experimentally that for appropriate operating +conditions, our TDRC system can operate with sampling times as +small as 11.72 ps, without sacrificing computational performance. +I. INTRODUCTION +Photonic systems that perform analog information process- +ing have been demonstrated in recent years as an interesting +alternative to conventional digital computing. In particular, +the use of photonic devices in brain-inspired computing and +machine-learning schemes has attracted significant attention, +helping to reduce learning costs and power consumption +[1]–[4]. Among the different techniques, reservoir computing +(RC) [5]–[7] has proven to be a powerful method, drasti- +cally simplifying the implementation and training of recurrent +neural networks. The time-delay reservoir computing (TDRC) +approach [1], [8]–[10] represents a very successful minimal +design approach of RC. TDRC uses time-multiplexing imple- +mented via temporal masking to recurrently connect virtual +nodes in a delayed feedback loop. This allows the storage +of past information and the generation of different responses +depending on the previous inputs. While the recurrence is +established by the feedback loop (time delay, τ), the coupling +among virtual nodes can be introduced via a mismatch of delay +τ and masking period Tm, and via the inertia of the transient +response of the real node. For achieving coupling through +inertia, the separation between the virtual nodes θ must be +smaller than the response time of the nonlinear node T, but +not too small, to obtain an acceptable level of signal-to-noise +ratio (SNR) responses. The masked information is introduced +into the reservoir by an external drive laser. For this drive- +response configuration, we show that the reservoir’s response +bandwidth can be increased by the high power of the injected +optical carrier. +In this letter, we experimentally demonstrate that a sig- +nificantly higher processing speed of a photonic TDRC can +be achieved by exploiting the bandwidth enhancement of the +response SL. Specifically, under a strong optical injection of +I. Est´ebanez, A. Argyris, and I. Fischer are with Instituto de F´ısica Inter- +disciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus UIB, Palma de +Mallorca, 07122, Spain (e-mail: irene@ifisc.uib-csic.es; apostolos@ifisc.uib- +csic.es; ingo@ifisc.uib-csic.es). +the input information, we reduce the virtual node separation +to only 11.72 ps - the smallest reported so far - while +preserving the TDRC’s computational performance obtained +for larger separations (e.g. 93.75 ps). This work complements +and confirms our previously published numerical results [11]. +II. EXPERIMENTAL REALIZATION +A. Photonic reservoir +The experimental single-mode fiber-based (SMF) photonic +reservoir is shown in Fig. 1. The input signal to the system +is generated by multiplying each value of the to-be-processed +information sequence with a mask to expand its dimensional- +ity. This mask is a periodically repeated sequence of length +Tm, drawn randomly from a uniform distribution [0, 1]. The +masked input signal is uploaded into an arbitrary waveform +generator (AWG - Keysight M8196A, 92 GSa/s, 32 GHz) +and transformed into an electrical modulation signal with a +sampling rate of 85.33 GSa/s. Each value is assigned to only +one sample, resulting in an analog bandwidth-limited AWG +output signal (Fig. 2). The analog bandwidth limitation causes +some additional correlations between virtual nodes, which can +be beneficial to computing tasks. +The AWG’s output is amplified with a 55 GHz SHF-S807C +broadband RF amplifier (RFA) and modulates the optical +carrier of a drive DFB laser (SL) via a 40 GHz Mach-Zehnder +intensity modulator (MZM - iXblue MX-LN-40). The resulting +optical signal is injected into a response DFB laser via a 50/50 +optical coupler (CPL-1) and an optical circulator (CIR), while +its strength is controlled by an optical attenuator (ATT 1). +The photonic reservoir is implemented using a fiber loop with +a roundtrip delay of τ = 24.5 ns. The setup is realized using +polarization-maintaining (PM) SMF and components, ensuring +robust operation over time. The response laser is emitting at +1545.5 nm and is biased below threshold at 10.6 mA (Ith = +10.8 mA). The drive SL emits at a similar wavelength that +can be tuned via temperature control. The frequency detuning +between the drive and the response laser ∆f = fd − fr can +be changed with a resolution of 0.01 K (∼ 125 MHz) and is +a crucial control parameter since it determines the dynamical +response to the input sequence. A 10 dB optical attenuator +(ATT 2) sets the optical feedback strength, and a 50/50 optical +coupler (CPL-2) closes the feedback loop. The optical output +of the response laser is amplified by a semiconductor optical +amplifier (SOA), filtered by a tunable optical filter (OF), and +detected by a 40 GHz photoreceiver (PD). The converted +electrical signal is obtained via a real-time oscilloscope (Osc +- Keysight UXR0404A, 256 GSa/s, 40 GHz). Finally, the +TDRC’s virtual node responses obtained from recorded time +arXiv:2301.04573v1 [eess.SP] 11 Jan 2023 + +2 +Injection +SL +ISO +MZM +ATT 1 +ATT 2 +CPL-1 +CPL-2 +Response +SL +ISO +SOA +OF +PD +RFA +AWG +CIR +Osc. +Fig. 1. Experimental photonic TDRC. ISO: optical isolator +Fig. 2. Pre-uploaded masking sequence to the AWG (gray) and RF bandwidth- +limited generated masking sequence by the AWG (red). +series - obtained after 256 averages for SNR improvement - +are used to train a linear (LR) classifier and to evaluate the +computing task performance. +B. TDRC and benchmark task +Given the feedback delay τ = 24.5 ns of our photonic +reservoir (Fig. 1), the 85.33 GSa/s sampling rate sets the space +for 2090 virtual nodes in the TDRC. We use 2080 of the +available virtual nodes. Thus, the mask length is asynchronous +to the delay (Tm ̸= τ), increasing the connectivity in the reser- +voir [12]. We investigate two different cases of virtual node +separation. For the smaller separation (θs = 11.72 ps), every +encoded input information is masked with a random value +at the sampling rate of the AWG. For the larger separation +(θl = 93.75 ps), every encoded input information is masked +with a random value that is repeated eight times, setting an +effective sampling rate for processing to 10.67 GSa/s. This +results in a TDRC with 2080/8 = 260 virtual nodes. For a +fair comparison of the TDRC performance, we implement the +same number of virtual nodes (N = 260) also when using θs, +by repeating the encoded masked input sequence eight times +to fill the delay τ. +We evaluate the TDRC’s performance via a benchmark +test that has commonly been used in the reservoir computing +community, the Santa Fe time series prediction [13]. The aim +is to predict the future value of a chaotic time series, y(t+1), +by considering its previous values up to the time t. At the +output layer of the TDRC, we consider the responses from +the first 3000 points of the Santa Fe time series to train the +system via an offline, ridge regression algorithm with ridge +parameter λ = 0.01. We discard the next 500 points to +eliminate prediction bias, and we apply the calculated weights +from the training process to a test set of the next 1000 data +points. We use the normalized mean square error (NMSE) to +quantify the prediction performance: +NMSE = 1 +L +L +� +n=1 +[y(n) − ¯y(n)]2 +(1) +where L is the number of data points used in the test set. y +is the predicted value, ¯y is the expected value, and are both +normalized to zero mean and unit variance. +III. RESULTS +The dynamical response of the TDRC defines the nonlinear +transformation of the input signal, which determines the com- +puting performance. One attribute of the dynamical response is +the TDRC’s operating bandwidth, which has to be sufficient to +generate large enough response signals despite small θ [14], +[15]. To obtain the best computing performance, additional +conditions must also be fulfilled; attributes, such as fading +memory and consistency of the nonlinear input-output trans- +formation, are critical for the final performance [16]. In the +following, we evaluate the effect of the bandwidth-enhanced +operation on the Santa Fe prediction task for several frequency +detuning conditions (∆f). We identify those conditions that +result in bandwidth-enhanced operation, and we show how +we can benefit from faster transient states and smaller θ. +A. Bandwidth enhancement +It is known that in injection-locked SL lasers with strong +optical injection, the response bandwidth can be several times +the free-running relaxation oscillation bandwidth [17]–[23]. +To measure the bandwidth enhancement of our system, we +upload random values chosen from a uniform distribution to +the AWG, and choose 85.33 GSa/s as the output sampling rate. +The optical output is averaged 2048 times before calculating +the corresponding spectra. We define the reservoir’s response +bandwidth as the frequency where a 10 dB power spectral den- +sity reduction from the low-frequency regime (MHz) occurs. +Fig. 3 shows the response bandwidth of the photonic +reservoir versus the frequency detuning between the drive and +the reservoir laser (∆f), for two different levels of average +optical injection power - 0.1 mW (red) and 1 mW (green) +- and a frequency resolution of 1 GHz. If the lasers are +injection locked (unlocked), the points in this graph are color +filled (empty). We consider that the lasers are unlocked if we +can identify an additional spectral peak centered at an RF +frequency equal to ∆f, with a 3dB cutoff. When increasing +the average optical injection power from 0.1 mW to 1 mW, +we observe the following: first, there is an enhancement of +the response bandwidth of the system of at least 5 GHz; +second, the injection locking is observed for a wider region +of the ∆f parameter space. For both injection conditions, +we observe a local dip in the response bandwidth, but for +different ∆f. The dip emerges from a dynamical bistability + +6 +:n +(arb. l +4 +Amplitude +2 +0 +Original input +AWG output +-2 +0 +100 +200 +300 +Time (ps)SSS1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +2 +4 +8 +10 +12 -20 +Intensity (dB) +40 +-60 +975.0 +975.5 +976.0 +976.5 +977.0 +977.5 +Wavelength (nm)0.0004 +0.0003 +0.0002 +voltage (V) +0.0001 +0.0000 +-0.0001 +-0.0002 +2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +time(s) +1e-83 +-40 +-30 +-20 +-10 +0 +Frequency detuning (GHz) +15 +20 +25 +30 +35 +40 +45 +SL response +bandwidth (GHz) +0.1 mW +1 mW +Fig. 3. +Response bandwidth of the photonic TDRC. The empty symbols +indicate that the calculation of the response bandwidth of the reservoir does +not originate only from the bandwidth enhancement effect. +region of the reservoir’s response and is associated with the +boundary between locking and unlocking. The dip is located +at ∆f = −12 GHz for the case of 0.1 mW, and at ∆f = −25 +GHz for the case of 1 mW injection. For lower ∆f than for the +location of the dip, partial unlocking of the drive laser starts to +appear, resulting in an RF frequency close to ∆f. This signal +originates from the beating of injected light and response laser +emission in the unlocked regime. Thus, our calculation of the +response bandwidth of the reservoir does not originate only +from the bandwidth enhancement effect (empty symbols in +Fig. 3). +B. TDRC performance +Bandwidth enhancement is favorable for computing at faster +rates. The final computing TDRC performance depends, how- +ever, also on other attributes that contribute to the nonlinear +transformation between the input and output response. Here, +we explore the impact of injection strength and frequency +detuning frequency. In Fig. 4, we show the computing per- +formance for the Santa Fe one-step-ahead prediction task, for +θl = 93.75 ps and θs = 11.72 ps, and for two levels of optical +injection (0.1 mW and 1 mW), versus ∆f. For θl = 93.75 ps +(Fig. 4 a), when a moderate injection is considered (0.1 mW), +we identify two regions (∆f ∼ 0 GHz and ∆f ∼ −12 GHz) +with the lowest values of the NMSE (0.076). When a strong +injection is considered (1 mW), the lowest NMSE is 0.054 +and achieved for ∆f = −31 GHz. The smallest NMSE is +found for a frequency detuning slightly lower than the one of +the bistability region, independently of the injection level. For +these frequency detuning conditions, the lasers are partially +locked, and the output response exhibits high consistency. For +more negative ∆f, once the effect of partial locking fades out +and the lasers become completely unlocked, the SNR of the +response signal decreases, and the NMSE of the computing +task increases drastically. +For θs = 11.72 ps, we again observe a reduction of the +NMSE for the strong injection case, as shown in Fig. 4 b. Here, +the NMSE reaches its minimum value again at slightly lower +∆f than the detuning where we observe bistable reservoir +operation. The minima of the prediction error are obtained +for ∆f = −12 GHz, for the case of moderate injection, and +∆f = −31 GHz, for the case of strong injection, with NMSE +(a) +(b) +Fig. 4. NMSE in the Santa Fe one-step-ahead prediction task, for (a) θl = +93.75 ps, and (b) θs = 11.72 ps, and for two levels of average optical injection +power: 0.1 mW (red) and 1 mW (green). +of 0.078 and 0.046, respectively. As for the larger virtual +node separation, the prediction error increases in the unlocked +regime. +C. TDRC dynamics +To gain a better understanding of the computing perfor- +mance for strong injection and θs, we study the persistence +plots of the reservoir’s temporal response (Fig. 5 a-c) and +the corresponding spectra (Fig. 5 d-f), for different ∆f. The +spectra are obtained with a 44 GHz bandwidth electrical +spectrum analyzer (Keysight, EXA N9010B). For ∆f = 0 +GHz, the response laser is injection-locked to the drive laser. +The obtained RF spectrum (Fig. 5 d) consists of different +frequency peaks that result from the characteristic time scales +of the encoded masked information. The persistence plot (Fig. +5 a) shows the system’s response to a masked input of a 2 +ns duration. When setting ∆f ¡ 0 GHz, and in the presence +of injection-locked operation, the system’s response to the +masked input also includes higher frequency components. For +∆f = −20 GHz, this is reflected in the RF spectrum (Fig. +5 e), where frequencies above 30 GHz appear, due to the +bandwidth enhancement of the response laser. The persistence +plot in this case (Fig. 5 b) shows that the dynamical behavior +of the response laser is affected by the underlying bistability +that is associated with the locking-unlocking transition. The +bistability results in time intervals with bimodal distributions +of the intensity, as, e.g., visible for the time intervals [0 ns, 0.5 +ns] and [1.0 ns, 1.2 ns]. For larger frequency detunings, such +as ∆f = −30 GHz (Fig. 5 c), the response laser is no longer +completely locked to the drive laser. In the corresponding +spectral distribution of Fig. 5 f, we observe the appearance +of accumulated, high-power, spectral components centered at + +-1mw +0.2 +-0.1 mw +NMSE +0.15 +0.1 +0.05 +-40 +-30 +-20 +-10 +0 +Freguency detuning (GHz)1mw +0.2 +-0.1 mw +NMSE +0.15 +0.1 +0.05 +-40 +-30 +-20 +-10 +0 +Freguency detuning (GHz)4 +(a) +(b) +(c) +(f) +(e) +(d) +f = - 20 GHz +f = 0 GHz +f = - 31 GHz +Fig. 5. +Dynamical response of the photonic reservoir after photodetection, for different ∆f conditions. a-c: Temporal persistence plots of the reservoir’s +response to the masked input. d-f: The corresponding power spectra. PSD: Power spectral density. +an RF frequency close to ∆f. This is also reflected in the +time series of Fig. 5 c, with the existence of much faster +signal changes. This operating region, which combines high +consistency and bandwidth-enhanced operation, is the one that +exhibits the lowest prediction errors. Even more negative ∆f +lead to a completely unlocked operation of the response laser, +with reduced dependence on the input signal, lower SNR +responses, and higher NMSE. +IV. CONCLUSIONS +We +demonstrated +experimentally +the +advantage +of +bandwidth-enhanced operation of a photonic TDRC, speeding +up the computation without sacrificing performance. The +induced fast nonlinear transient responses through bandwidth +enhancement allowed our TDRC system to operate with +a virtual node time separation of 11.72 ps - the smallest +reported experimentally to our knowledge – while achieving +a very low prediction error (NMSE = 0.046) for the nonlinear +Santa Fe time series prediction task. Therefore, even when +being restricted to short delays, reasonable numbers of +virtual nodes can still be implemented. This opens further +perspectives for the application of integrated semiconductor +laser-based TDRC implementations. +ACKNOLEDGEMENTS +This work has been partially supported by the Mar´ıa +de +Maeztu +project +CEX2021-001164-M +funded +by +the +MCIN/AEI/10.13039/501100011033. +The +work +of +I. +Est´ebanez has been supported by MICINN, AEI, FEDER and +the University of the Balearic Islands through a predoctoral +fellowship (MDM-2017-0711-18-2). +REFERENCES +[1] D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel +photonic information processing at gigabyte per second data rates using +transient states,” Nature Communications, vol. 4, no. 1, p. 1364, 2013. +[2] E. Strubell, A. Ganesh, and A. McCallum, “Energy and policy con- +siderations for deep learning in nlp,” arXiv preprint arXiv:1906.02243, +2019. +[3] A. Lugnan, A. Katumba, F. Laporte, M. 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Simpson, “Modulation bandwidth, +noise, and stability of a semiconductor laser subject to strong injection +locking,” IEEE Photonics Technology Letters, vol. 9, no. 10, pp. 1325– +1327, 1997. +[21] E.-K. +Lee, +H.-S. +Pang, +J.-D. +Park, +and +H. +Lee, +“Bistability +and chaos in an injection-locked semiconductor laser,” Phys. Rev. +A, vol. 47, pp. 736–739, Jan 1993. [Online]. Available: https: +//link.aps.org/doi/10.1103/PhysRevA.47.736 +[22] A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical +chaotic signal generated by a semiconductor laser with optical feedback,” +IEEE Photonics Technology Letters, vol. 20, no. 19, pp. 1633–1635, +2008. +[23] A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift +and bandwidth enhancement in semiconductor lasers with strong light +injection,” IEEE Journal of Quantum Electronics, vol. 39, no. 10, pp. +1196–1204, 2003. + diff --git a/k9E3T4oBgHgl3EQfhwor/content/tmp_files/load_file.txt b/k9E3T4oBgHgl3EQfhwor/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0606b9e989c46e42ca930f73500f09bcca02716 --- /dev/null +++ b/k9E3T4oBgHgl3EQfhwor/content/tmp_files/load_file.txt @@ -0,0 +1,506 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf,len=505 +page_content='1 Experimental demonstration of bandwidth enhancement in photonic time delay reservoir computing Irene Est´ebanez, Apostolos Argyris, and Ingo Fischer Abstract—Time delay reservoir computing (TDRC) using semi- conductor lasers (SLs) has proven to be a promising photonic analog approach for information processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' One appealing property is that SLs subject to delayed optical feedback and external optical injection, allow tuning the response bandwidth by changing the level of optical injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Here we use strong optical injection, thereby expanding the SL’s modulation response up to tens of GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Performing a nonlinear time series prediction task, we demonstrate experimentally that for appropriate operating conditions, our TDRC system can operate with sampling times as small as 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='72 ps, without sacrificing computational performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' INTRODUCTION Photonic systems that perform analog information process- ing have been demonstrated in recent years as an interesting alternative to conventional digital computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' In particular, the use of photonic devices in brain-inspired computing and machine-learning schemes has attracted significant attention, helping to reduce learning costs and power consumption [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Among the different techniques, reservoir computing (RC) [5]–[7] has proven to be a powerful method, drasti- cally simplifying the implementation and training of recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The time-delay reservoir computing (TDRC) approach [1], [8]–[10] represents a very successful minimal design approach of RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' TDRC uses time-multiplexing imple- mented via temporal masking to recurrently connect virtual nodes in a delayed feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' This allows the storage of past information and the generation of different responses depending on the previous inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' While the recurrence is established by the feedback loop (time delay, τ), the coupling among virtual nodes can be introduced via a mismatch of delay τ and masking period Tm, and via the inertia of the transient response of the real node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For achieving coupling through inertia, the separation between the virtual nodes θ must be smaller than the response time of the nonlinear node T, but not too small, to obtain an acceptable level of signal-to-noise ratio (SNR) responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The masked information is introduced into the reservoir by an external drive laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For this drive- response configuration, we show that the reservoir’s response bandwidth can be increased by the high power of the injected optical carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' In this letter, we experimentally demonstrate that a sig- nificantly higher processing speed of a photonic TDRC can be achieved by exploiting the bandwidth enhancement of the response SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Specifically, under a strong optical injection of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Est´ebanez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Argyris, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Fischer are with Instituto de F´ısica Inter- disciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus UIB, Palma de Mallorca, 07122, Spain (e-mail: irene@ifisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='uib-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='es;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' apostolos@ifisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='uib- csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='es;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' ingo@ifisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='uib-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' the input information, we reduce the virtual node separation to only 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='72 ps - the smallest reported so far - while preserving the TDRC’s computational performance obtained for larger separations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='75 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' This work complements and confirms our previously published numerical results [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' EXPERIMENTAL REALIZATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Photonic reservoir The experimental single-mode fiber-based (SMF) photonic reservoir is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The input signal to the system is generated by multiplying each value of the to-be-processed information sequence with a mask to expand its dimensional- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' This mask is a periodically repeated sequence of length Tm, drawn randomly from a uniform distribution [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The masked input signal is uploaded into an arbitrary waveform generator (AWG - Keysight M8196A, 92 GSa/s, 32 GHz) and transformed into an electrical modulation signal with a sampling rate of 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='33 GSa/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Each value is assigned to only one sample, resulting in an analog bandwidth-limited AWG output signal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The analog bandwidth limitation causes some additional correlations between virtual nodes, which can be beneficial to computing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The AWG’s output is amplified with a 55 GHz SHF-S807C broadband RF amplifier (RFA) and modulates the optical carrier of a drive DFB laser (SL) via a 40 GHz Mach-Zehnder intensity modulator (MZM - iXblue MX-LN-40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The resulting optical signal is injected into a response DFB laser via a 50/50 optical coupler (CPL-1) and an optical circulator (CIR), while its strength is controlled by an optical attenuator (ATT 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The photonic reservoir is implemented using a fiber loop with a roundtrip delay of τ = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The setup is realized using polarization-maintaining (PM) SMF and components, ensuring robust operation over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The response laser is emitting at 1545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 nm and is biased below threshold at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='6 mA (Ith = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='8 mA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The drive SL emits at a similar wavelength that can be tuned via temperature control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The frequency detuning between the drive and the response laser ∆f = fd − fr can be changed with a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='01 K (∼ 125 MHz) and is a crucial control parameter since it determines the dynamical response to the input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' A 10 dB optical attenuator (ATT 2) sets the optical feedback strength, and a 50/50 optical coupler (CPL-2) closes the feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The optical output of the response laser is amplified by a semiconductor optical amplifier (SOA), filtered by a tunable optical filter (OF), and detected by a 40 GHz photoreceiver (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The converted electrical signal is obtained via a real-time oscilloscope (Osc Keysight UXR0404A, 256 GSa/s, 40 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Finally, the TDRC’s virtual node responses obtained from recorded time arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='04573v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='SP] 11 Jan 2023 2 Injection SL ISO MZM ATT 1 ATT 2 CPL-1 CPL-2 Response SL ISO SOA OF PD RFA AWG CIR Osc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Experimental photonic TDRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' ISO: optical isolator Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Pre-uploaded masking sequence to the AWG (gray) and RF bandwidth- limited generated masking sequence by the AWG (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' series - obtained after 256 averages for SNR improvement - are used to train a linear (LR) classifier and to evaluate the computing task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' TDRC and benchmark task Given the feedback delay τ = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 ns of our photonic reservoir (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 1), the 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='33 GSa/s sampling rate sets the space for 2090 virtual nodes in the TDRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' We use 2080 of the available virtual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Thus, the mask length is asynchronous to the delay (Tm ̸= τ), increasing the connectivity in the reser- voir [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' We investigate two different cases of virtual node separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For the smaller separation (θs = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='72 ps), every encoded input information is masked with a random value at the sampling rate of the AWG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For the larger separation (θl = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='75 ps), every encoded input information is masked with a random value that is repeated eight times, setting an effective sampling rate for processing to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='67 GSa/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' This results in a TDRC with 2080/8 = 260 virtual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For a fair comparison of the TDRC performance, we implement the same number of virtual nodes (N = 260) also when using θs, by repeating the encoded masked input sequence eight times to fill the delay τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' We evaluate the TDRC’s performance via a benchmark test that has commonly been used in the reservoir computing community, the Santa Fe time series prediction [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The aim is to predict the future value of a chaotic time series, y(t+1), by considering its previous values up to the time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' At the output layer of the TDRC, we consider the responses from the first 3000 points of the Santa Fe time series to train the system via an offline, ridge regression algorithm with ridge parameter λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' We discard the next 500 points to eliminate prediction bias, and we apply the calculated weights from the training process to a test set of the next 1000 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' We use the normalized mean square error (NMSE) to quantify the prediction performance: NMSE = 1 L L � n=1 [y(n) − ¯y(n)]2 (1) where L is the number of data points used in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' y is the predicted value, ¯y is the expected value, and are both normalized to zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' RESULTS The dynamical response of the TDRC defines the nonlinear transformation of the input signal, which determines the com- puting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' One attribute of the dynamical response is the TDRC’s operating bandwidth, which has to be sufficient to generate large enough response signals despite small θ [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' To obtain the best computing performance, additional conditions must also be fulfilled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' attributes, such as fading memory and consistency of the nonlinear input-output trans- formation, are critical for the final performance [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' In the following, we evaluate the effect of the bandwidth-enhanced operation on the Santa Fe prediction task for several frequency detuning conditions (∆f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' We identify those conditions that result in bandwidth-enhanced operation, and we show how we can benefit from faster transient states and smaller θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Bandwidth enhancement It is known that in injection-locked SL lasers with strong optical injection, the response bandwidth can be several times the free-running relaxation oscillation bandwidth [17]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' To measure the bandwidth enhancement of our system, we upload random values chosen from a uniform distribution to the AWG, and choose 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='33 GSa/s as the output sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The optical output is averaged 2048 times before calculating the corresponding spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' We define the reservoir’s response bandwidth as the frequency where a 10 dB power spectral den- sity reduction from the low-frequency regime (MHz) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 3 shows the response bandwidth of the photonic reservoir versus the frequency detuning between the drive and the reservoir laser (∆f), for two different levels of average optical injection power - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 mW (red) and 1 mW (green) and a frequency resolution of 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' If the lasers are injection locked (unlocked), the points in this graph are color filled (empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' We consider that the lasers are unlocked if we can identify an additional spectral peak centered at an RF frequency equal to ∆f, with a 3dB cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' When increasing the average optical injection power from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 mW to 1 mW, we observe the following: first, there is an enhancement of the response bandwidth of the system of at least 5 GHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' second, the injection locking is observed for a wider region of the ∆f parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For both injection conditions, we observe a local dip in the response bandwidth, but for different ∆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The dip emerges from a dynamical bistability 6 :n (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' l 4 Amplitude 2 0 Original input AWG output 2 0 100 200 300 Time (ps)SSS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 0 2 4 8 10 12 -20 Intensity (dB) 40 60 975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 Wavelength (nm)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0002 voltage (V) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 time(s) 1e-83 40 30 20 10 0 Frequency detuning (GHz) 15 20 25 30 35 40 45 SL response bandwidth (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 mW 1 mW Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Response bandwidth of the photonic TDRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The empty symbols indicate that the calculation of the response bandwidth of the reservoir does not originate only from the bandwidth enhancement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' region of the reservoir’s response and is associated with the boundary between locking and unlocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The dip is located at ∆f = −12 GHz for the case of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 mW, and at ∆f = −25 GHz for the case of 1 mW injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For lower ∆f than for the location of the dip, partial unlocking of the drive laser starts to appear, resulting in an RF frequency close to ∆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' This signal originates from the beating of injected light and response laser emission in the unlocked regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Thus, our calculation of the response bandwidth of the reservoir does not originate only from the bandwidth enhancement effect (empty symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' TDRC performance Bandwidth enhancement is favorable for computing at faster rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The final computing TDRC performance depends, how- ever, also on other attributes that contribute to the nonlinear transformation between the input and output response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Here, we explore the impact of injection strength and frequency detuning frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 4, we show the computing per- formance for the Santa Fe one-step-ahead prediction task, for θl = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='75 ps and θs = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='72 ps, and for two levels of optical injection (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 mW and 1 mW), versus ∆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For θl = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='75 ps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 4 a), when a moderate injection is considered (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 mW), we identify two regions (∆f ∼ 0 GHz and ∆f ∼ −12 GHz) with the lowest values of the NMSE (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='076).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' When a strong injection is considered (1 mW), the lowest NMSE is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='054 and achieved for ∆f = −31 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The smallest NMSE is found for a frequency detuning slightly lower than the one of the bistability region, independently of the injection level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For these frequency detuning conditions, the lasers are partially locked, and the output response exhibits high consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For more negative ∆f, once the effect of partial locking fades out and the lasers become completely unlocked, the SNR of the response signal decreases, and the NMSE of the computing task increases drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For θs = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='72 ps, we again observe a reduction of the NMSE for the strong injection case, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 4 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Here, the NMSE reaches its minimum value again at slightly lower ∆f than the detuning where we observe bistable reservoir operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The minima of the prediction error are obtained for ∆f = −12 GHz, for the case of moderate injection, and ∆f = −31 GHz, for the case of strong injection, with NMSE (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' NMSE in the Santa Fe one-step-ahead prediction task, for (a) θl = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='75 ps, and (b) θs = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='72 ps, and for two levels of average optical injection power: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 mW (red) and 1 mW (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='078 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='046, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' As for the larger virtual node separation, the prediction error increases in the unlocked regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' TDRC dynamics To gain a better understanding of the computing perfor- mance for strong injection and θs, we study the persistence plots of the reservoir’s temporal response (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5 a-c) and the corresponding spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5 d-f), for different ∆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The spectra are obtained with a 44 GHz bandwidth electrical spectrum analyzer (Keysight, EXA N9010B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For ∆f = 0 GHz, the response laser is injection-locked to the drive laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The obtained RF spectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5 d) consists of different frequency peaks that result from the characteristic time scales of the encoded masked information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The persistence plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5 a) shows the system’s response to a masked input of a 2 ns duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' When setting ∆f ¡ 0 GHz, and in the presence of injection-locked operation, the system’s response to the masked input also includes higher frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For ∆f = −20 GHz, this is reflected in the RF spectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5 e), where frequencies above 30 GHz appear, due to the bandwidth enhancement of the response laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The persistence plot in this case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5 b) shows that the dynamical behavior of the response laser is affected by the underlying bistability that is associated with the locking-unlocking transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The bistability results in time intervals with bimodal distributions of the intensity, as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=', visible for the time intervals [0 ns, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 ns] and [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='0 ns, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='2 ns].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' For larger frequency detunings, such as ∆f = −30 GHz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5 c), the response laser is no longer completely locked to the drive laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' In the corresponding spectral distribution of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5 f, we observe the appearance of accumulated, high-power, spectral components centered at 1mw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 mw NMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='05 40 30 20 10 0 Freguency detuning (GHz)1mw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 mw NMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='05 40 30 20 10 0 Freguency detuning (GHz)4 (a) (b) (c) (f) (e) (d) f = - 20 GHz f = 0 GHz f = - 31 GHz Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Dynamical response of the photonic reservoir after photodetection, for different ∆f conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' a-c: Temporal persistence plots of the reservoir’s response to the masked input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' d-f: The corresponding power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' PSD: Power spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' an RF frequency close to ∆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' This is also reflected in the time series of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 5 c, with the existence of much faster signal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' This operating region, which combines high consistency and bandwidth-enhanced operation, is the one that exhibits the lowest prediction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Even more negative ∆f lead to a completely unlocked operation of the response laser, with reduced dependence on the input signal, lower SNR responses, and higher NMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' CONCLUSIONS We demonstrated experimentally the advantage of bandwidth-enhanced operation of a photonic TDRC, speeding up the computation without sacrificing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The induced fast nonlinear transient responses through bandwidth enhancement allowed our TDRC system to operate with a virtual node time separation of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='72 ps - the smallest reported experimentally to our knowledge – while achieving a very low prediction error (NMSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='046) for the nonlinear Santa Fe time series prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Therefore, even when being restricted to short delays, reasonable numbers of virtual nodes can still be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' This opens further perspectives for the application of integrated semiconductor laser-based TDRC implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' ACKNOLEDGEMENTS This work has been partially supported by the Mar´ıa de Maeztu project CEX2021-001164-M funded by the MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' The work of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' Est´ebanez has been supported by MICINN, AEI, FEDER and the University of the Balearic Islands through a predoctoral fellowship (MDM-2017-0711-18-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' REFERENCES [1] D.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='8 Time (ns)-60 PSD (dBm) 70 80 90 0 10 20 30 40 Freguency (GHz)-60 PSD (dBm) 70 80 90 0 10 20 30 40 Frequency (GHz)-60 PSD (dBm) 70 80 90 0 10 20 30 40 Frequency (GHz)Freguency detuning △ w =30 65 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='17 (mv 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='33 Voltage( 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content='67 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} +page_content=' 1196–1204, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQfhwor/content/2301.04573v1.pdf'} diff --git a/kb_50/content/tmp_files/kb_50.pdf.txt b/kb_50/content/tmp_files/kb_50.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..85fad03adaab2cbbf8f6b2a85100004e8110d445 --- /dev/null +++ b/kb_50/content/tmp_files/kb_50.pdf.txt @@ -0,0 +1,1769 @@ +RESEARCH +Open Access +A metagenomics roadmap to the +uncultured genome diversity in hypersaline +soda lake sediments +Charlotte D. Vavourakis1 +, Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y. Sorokin3,4 +and Gerard Muyzer1* +Abstract +Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity. Despite the +high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but +the microbiome of soda lake sediments received much less attention of microbiologists. Here, we performed metagenomic +sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, +extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages. +Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and +a salt content between 70 and 400 g L−1. The recovered 16S rRNA gene sequences were mostly from Bacteria, even in +the salt-saturated lakes. Most OTUs were assigned to uncultured families. We reconstructed 871 metagenome-assembled +genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla +Radiation (CPR). Five new species of CPR were among the most dominant community members. Novel dominant +lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen +cycling. Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla +never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the +Actinobacteria. +Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important +advances. First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR +and several hundred other novel prokaryote lineages. The soda lake CPR is a functionally diverse group, but the most +abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation. Second, +we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those +encompassing known homo-acetogens, sulfate-reducers, and methanogens. Since only few environmental metagenomics +studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant +not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine +and freshwater sediments. +Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl +pathway +* Correspondence: G.Muijzer@uva.nl +†Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this +work. +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands +Full list of author information is available at the end of the article +© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 +International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and +reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to +the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver +(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. +Vavourakis et al. Microbiome (2018) 6:168 +https://doi.org/10.1186/s40168-018-0548-7 + +MicrobiomeBackground +Soda lakes are evaporative, athallasic salt lakes with low cal- +cium and magnesium concentrations and a high-alkaline +pH up to 11 buffered by dissolved (bi-) carbonate ions [1]. +They are constrained to arid regions across the globe, +mainly the tropical East African Rift Valley [2], the Libyan +Desert [3], the deserts in California and Nevada [4], and the +dry steppe belt of Central Asia that spans to southern Si- +beria, north-eastern Mongolia, and Inner Mongolia in +China [1]. On top of the extreme salinity and alkaline pH, +the Eurasian soda lakes experience extreme seasonal +temperature differences, causing highly unstable water re- +gimes and fluctuating salinities [5]. Yet, soda lakes harbor +diverse communities of haloalkaliphilic microbes, mostly +prokaryotes that are well adapted to survive and grow in +these extreme environments and consist of similar func- +tional groups in soda lakes around the world [1, 2, 6]. The +relative abundance of different groups is typically governed +by the salinity of the brine [1, 7, 8], and microbial-mediated +nutrient +cycles +become +partially +hampered +only +at +salt-saturating conditions [1]. +So far, all characterized prokaryotic lineages cultured +from soda lakes comprise over 70 different species within +more than 30 genera [1, 6, 9, 10]. From these, only a lim- +ited number of genomes have been sequenced today, +mostly from chemolithoautotrophic sulfur-oxidizing bac- +teria belonging to the genus Thioalkalivibrio (class Gam- +maproteobacteria) [1, 11, 12]. It is well established that +metagenomics enables the recovery of genomes and the +identification of novel genetic diversity where culturing ef- +forts fail [13, 14]. In recent years, next-generation sequen- +cing has recovered a massive number of genomes from +previously unknown groups of prokaryotes [15, 16], +including a strikingly large and diverse group called +“Candidate Phyla Radiation” (CPR), only distantly related +to other cultured bacterial lineages [17]. Previously, we +conducted a metagenomics study on soda lakes and re- +constructed novel genomes from uncultured Bacteroidetes +and “Candidatus Nanohaloarchaeaota” living in hypersa- +line Siberian soda brines [7]. Here, we turned our atten- +tion to the far more complex prokaryotic communities +living in the sediments of the hypersaline soda lakes from +the same region. We give a broad overview of all the +taxonomic groups sequenced and focus on the metabolic +diversity found in the reconstructed genomes of the most +abundant, uncultured organisms. +Results +Overall prokaryote community structure +The salinities from the studied soda lakes ranged from +moderately hypersaline (between 70 and 110 g L−1) to +salt-saturated (400 g L−1 salt). The soluble carbonate al- +kalinity was in the molar range, and the pH in all lakes +was around ten (see Additional file 1: Table S1). To give +an overview of the overall prokaryotic community com- +position in each of the samples, we looked at the taxo- +nomic classification of 16S rRNA genes recovered both +by amplicon sequencing and direct metagenomics se- +quencing (Fig. 1, see also Additional file 2: Figure S1; +Additional file 3). The prokaryotic communities of all +five sediment samples were highly diverse and consisted +mostly of uncultured taxonomic groups. Bacteria were +more abundant than Archaea, regardless of the salinity +of the overlaying brine [7] (Fig. 1). Euryarchaeota were +the second and third largest group in the sediments of +the two salt-saturated lakes comprising ~ 10 and ~ 20% +of the 16S rRNA genes in the metagenomes. Most +Euryarchaeota-related OTUs detected by amplicon se- +quencing belonged either to the uncultured Thermoplas- +mata group KTK 4A (SILVA classification) or the genera +Halohasta and Halorubrum (class Halobacteria). In ac- +cordance with cultivation-dependent studies [6], most +OTUs assigned to methanogens were from the class +Methanomicrobia, +especially +the +lithotrophic +genus +Methanocalculus (up to ~ 3%) and the methylotrophic +genus Methanosalsum (Additional file 3). +The varying ratio of the three dominant bacterial groups, +Firmicutes, Bacteroidetes (including the newly proposed +phyla +Rhodothermaeota +and +Balneolaeota +[18]), +and +Gammaproteobacteria, showed no clear trend in relation to +the salinity in the lakes, but when Firmicutes were domin- +ant, Bacteroidetes were less abundant and vice versa. Most +Firmicutes belonged to the order Clostridales. Uncultured +members from the family Syntrophomonadaceae had a +relative abundance of more than 5% in all five metagen- +omes and comprised in two lakes even ~ 11–20% of the +recovered amplicon sequences. The second most abundant +Firmicutes order was Halanaerobiales, particularly the +genus Halanaerobium (family Halanaerobiaceae) and un- +cultured members of the Halobacteroidaceae. The majority +of Bacteroidetes-related OTUs could not be assigned down +to the genus level. The uncultured ML635J-40 aquatic +group (order Bacteroidales) comprised at least 5% of all five +prokaryotic communities. This group has been previously +found to be abundant in Mono Lake [4] (a soda lake) and +in an anoxic bioreactor degrading cyanobacterial biomass +under haloalkaline conditions [19]. Two other highly abun- +dant (up to ~ 8%) uncultured groups from the class Balneo- +lia (proposed new phylum Balneolaeota [18]) were also +detected in other soda lakes before [3, 4]. Within the Gam- +maproteobacteria, the genus Thioalkalivibrio was abundant +(~ 3% of the total community), but also uncultured +members of HOC36 were prevailing at moderate salinities. +Members of the Deltaproteobacteria, Alphaproteobacteria, +and Chloroflexi comprised up to ~ 10% of the detected 16S +rRNA gene in some of the metagenomes. The GIF9 family +of the class Dehalococcoidia was among the top three most +abundant OTUs in two lakes. The extremely salt-tolerant +Vavourakis et al. Microbiome (2018) 6:168 +Page 2 of 18 + +and alkaliphilic genera Desulfonatronobacter (order Desulfo- +bacterales) and Desulfonatronospira (order Desulfovibrio- +nales) +were +the +dominant +Deltaproteobacteria. +Highly +abundant OTUs, within the Actinobacteria belonged to the +class Nitriliruptoria and within the Alphaproteobacteria to +the family Rhodobacteraceae and the genus Roseibaca. The +important nitrifying genus Nitrobacter (Alphaproteobacteria) +was present in only one of the lakes with moderate salinity +(Additional file 3). +Some bacterial top-level taxa appeared less dominant +(< 5%) from the 16S rRNA genes recovered from the +metagenomes but were represented mainly by a single +highly abundant OTU in the amplicon sequences, in- +cluding the haloalkaliphilic genus Truepera within the +phylum Deinococcus-Thermus, the genus Spirochaeata +within the phylum Spirochaetes, the family BSN166 +within the phylum Ignavibacteriae, the BD2–11 terres- +trial group within the Gemmatimonadetes, and the +WCHB1–41 +order +within +the +Verrucomicrobia. +All +OTUs +within +the +Thermotogae +and +Lentisphaerae +belonged to uncultured genera from the family Kosmoto- +gaceae and Oligosphaeraceae, respectively. All Tenericu- +tes-related OTUs belonged to the class Mollicutes, and +especially the order NB1-n was dominant. In contrast, +the phylum Planctomycetes was relatively diverse, with +at least 11 different genus-level OTUs spread over four +class-level groups. +High-throughput genome recovery +We obtained 717 medium-quality (≥ 50% complete, +< 10% contamination) and 154 near-complete (≥ 90% +complete, < 5% contamination) metagenome-assembled +genomes (MAGs) across three major prokaryote groups: +Archaea, Bacteria, and CPR (see Additional file 4 and +Additional file 2: Figure S2). Figures 2 and 3 show the +top-level phylogeny of all MAGs based on 16 ribosomal +proteins. The reference database used contains a repre- +sentative for each major prokaryote lineage [17]. We +a +b +Fig. 1 Abundant prokaryotic groups in five hypersaline soda lake sediments. a Relative abundance of the top-level taxa (those with ≥ 1% abundance +in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets. b Relative abundance of the 16S rRNA OTUs (those with sum +of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level. Three of the assessed soda +lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) +Vavourakis et al. Microbiome (2018) 6:168 +Page 3 of 18 + +colored the different phyla from which we obtained a +MAG +in +alternate +blue +and +orange +colors, +and +highlighted the MAGs obtained here in a darker shade. +Many MAGs belonged to uncultured groups commonly +detected in soda lake 16S rRNA gene surveys, over 100 +MAGs still belonged to candidate prokaryote phyla and +divisions that to our knowledge were never detected be- +fore in soda lakes, including CPR. Although only few +MAGs had near-complete 16S rRNA genes, in most +cases we were able to link available taxonomic gene an- +notations and ribosomal protein phylogeny to the SILVA +taxonomy of the OTUs assigned to the amplicon se- +quences, while cross-checking the abundance profiles of +both MAGs (Additional file 5) and OTUs. +The soda lake CPR recovered from the metagenomes was +restricted to a few distinct phyla within the Parcubacteria +group, mostly affiliating with “Candidatus Nealsonbacteria” +and “Ca. Zambryskibacteria” [15] (Fig. 2). The first group of +MAGs encompassed four different branches in our riboso- +mal protein tree, suggesting a high-phylogenetic diversity, +with 33 putative new species sampled here (ANI and con- +DNA matrices given in Additional file 6). The “Ca. Zambrys- +kibacteria-”related MAGs consisted of at least five new +species. Few MAGs were recovered from CPR groups also +detected by amplicon sequencing (see Additional file 2: +Figure S1), namely the “Ca. Dojkabacteria” (former WS6), +“Ca. Saccharibacteria” (former TM7), CPR2, and “Ca. +Katanobacteria” (former WWE3). +Fig. 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins. The archaeal tree is unrooted. The CPR tree is rooted +to the Wirthbacteria. Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of +this study are highlighted by darker shades (labeled as “MAG present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this +study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show +confidence percentage of the bootstraps analysis (100×) +Vavourakis et al. Microbiome (2018) 6:168 +Page 4 of 18 + +Most archaeal MAGs belonged to the phylum Euryarch- +aeota and the abundant classes Halobacteria, Methanomi- +crobia, and Thermoplasmata (related to OTU KTK 4A) +within. In addition, three Thermoplasmata-related MAGs +that encoded for the key enzyme for methanogenesis +(methyl-coenzyme M reductase, mcr) affiliated with refer- +ence genomes from Methanomassilicoccales, the seventh +order of methanogens have been recovered [20, 21]. +Another MCR-encoding MAG was closely related to the +latest +discovered +group +of +poly-extremophilic, +methyl-reducing methanogens from hypersaline lakes +from the class Methanonatronarchaeia [9] (related to +OTU ST-12K10A). We recovered also one MAG from the +class Methanobacteria and a high-quality MAG from the +WCHA1–57 +group +(“Candidatus +Methanofastidiosa” +[22]) in the candidate division WSA2 (Arc I). Several +MAGs were recovered from the DPANN archaeal +groups “Ca. Diapherotrites,” “Ca. Aenigmarchaeota,” +(see Additional file 2: Figure S3) and “Ca. Woesearch- +aeota” (former Deep Sea Hydrothermal Vent Group 6, +DHVEG-6). Although we did not reconstruct any +reasonable-sized MAGs from the TACK superphylum, +we found several 16S rRNA genes on the assembled +contigs that affiliated to the Thaumarchaeota (see +Additional file 1: Table S2). +Nearly every known bacterial phylum had an extremo- +philic lineage sampled from our hypersaline soda lake +sediments (Fig. 3). In most cases, the soda lake lineages +clearly formed separate branches appearing as sister +groups to known reference lineages. The highest genome +recovery was from the same top-level taxonomic groups +that were also abundant in our 16S rRNA gene analysis. +From the Verrucomicrobia, most MAGs belonged to the +order WCHB1-41 (16S rRNA gene identity 92–100%). +However, in our ribosomal protein tree, they branched +within the phylum Lentisphaerae. Sixteen Tenericutes +MAGs from at least 12 different species (Additional file 6) +were closely related to the NB1-n group of Mollicutes. +Based on the recovered genome size and encoded meta- +bolic potential, these organisms are free-living anaerobic +fermenters of simple sugars, similar to what has recently +been +proposed +for +“Candidatus +Izimaplasma” +[23]. +Fig. 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins. Alternate orange and blue colors show phyla/ +classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG +present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not +present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) +Vavourakis et al. Microbiome (2018) 6:168 +Page 5 of 18 + +Several MAGs belonged to the candidate phyla “Ca. +Omnitrophica,” “Ca. Atribacteria,” and “Ca. Acetother- +mia” (former OP1), which were moderately abundant +also in some sediment (see Additional file 2: Figure S1). +For the latter phylum, we suspect that four MAGs were +more closely related to ca. div. WS1 and “Ca. Lindow- +bacteria” for which only few reference genomes are +currently available in NCBI (see Additional file 2: +Figure S4). Due to a high-sequencing coverage, we also +managed to reconstruct several MAGs from rare Bacteria +(< 100 amplicon sequences detected, see Additional file 2: +Figure S1), including the phyla “Ca. Hydrogenedentes,” +“Ca. Cloacimonetes,” ca. div. BRC1, Elusimicrobia, Caldi- +serica, and “Ca. Latescibacteria.” The MAGs from the +latter phylum were more closely related to the recently +proposed phylum “Ca. Handelsmanbacteria” [15]. Two +additional MAGs with 16S rRNA gene fragments with +94–95% identity to the class MD2898-B26 (Nitrospinae) +were more likely members of ca. div. KSB3 (proposed +“Ca. Moduliflexus” [24], see Additional file 2: Figure S5). +Draft genomes of haloalkaliphilic CPR +Strikingly, members of the CPR related to “Ca. Nealson- +bacteria” and “Ca. Vogelbacteria” were among the top +5% of abundant organisms in the surface sediments of +the soda lakes, especially those with moderate salinity +(Fig. 4). Like most members of the CPR, the MAGs of +the four most abundant “Ca. Nealsonbacteria” seem to +be anaerobic fermenters [25]. They lacked a complete +TCA cycle and most complexes from the oxidative elec- +tron transfer chain, except for the subunit F of a +NADH-quinone oxidoreductase (complex I, nuoF, nuoG, +nuoA) and coxB genes (complex II). All CPR MAGs had +a near-complete glycolysis pathway (Embden-Meyerhof- +Parnas) encoded, but pentose phosphate pathways were +severely truncated. The commonly encoded F- and +V-type ATPase can establish a membrane potential for +symporter-antiporters by utilizing the ATP formed by +substrate-level phosphorylation during fermentation. All +CPR have V-type ATPases that can translocate Na+ in +addition to H+ (see Additional file 2: Figure S6), while +only two members of the “Ca. Falkowbacteria” had puta- +tive Na+-coupled F-type ATPases (see Additional file 2: +Figure S7). The coupling of ATP hydrolysis to sodium +translocation is advantageous to maintain pH homeosta- +sis in alkaline environments. Interestingly, with only two +exceptions [26, 27], all CPR genomes recovered from +other environments with neutral pH were reported to +encode only F-type ATPases [28–32]. One low-abundant +MAG affiliated to “Ca. Peregrinibacteria” contained both +the +large +subunit +of +a +RuBisCO +(type +II/III, +see +Additional file 2: Figure S8) and a putative phosphoribu- +lokinase (PRK, K00855) encoded in the same contig. +This is remarkable because PRK homologs were not +previously identified among CPR, and RuBisCo form II/ +III was inferred to function in a nucleoside salvage path- +way [33]. One “Ca. Saccharibacteria” MAG encoded for +a putative channelrhodopsin (see Additional file 2: +Figure S9). This is the first rhodopsin found among the +CPR and suggests that this enigmatic group of organ- +isms may have acquired evolutionary adaptations to a +life in sunlit surface environments. +A previous study showed that most CPR has coccoid +cell morphotypes with a monoderm cell envelope resem- +bling those from Gram-positives and Archaea but with a +distinct S-layer [34]. Thick peptidoglycans coated with +acidic surface polymers such as teichoic acids help pro- +tect the cells of Gram-positives against reactive hydroxyl +ions in highly alkaline environments [35] (Fig. 5a). All +soda lake CPR had indeed the capability for peptidogly- +can biosynthesis, but we found proteins typical for +Gram-negatives for the biosynthesis of lipopolysaccha- +rides (see Additional file 1: Table S3), homologous to the +inner membrane proteins of type II secretion systems +and +to +several +proteins +associated +to +the +outer +membrane and peptidoglycan, including OmpA. +It remains to be determined whether the soda lake +CPR also lacks an outer membrane and perhaps anchor +lipopolysaccharides, S-layer proteins, and lipoproteins to +the inner cell membrane or peptidoglycan. We also +found gene encoding cardiolipin and squalene synthases. +Increased levels of cardiolipin and the presence of squa- +lene make the cytoplasmic membrane less leaky for +protons [36]. In addition, cation/proton exchangers are +known to play a crucial role for pH homeostasis in alka- +liphilic prokaryotes as they help acidify the cytoplasm +during the extrusion of cations [35]. Putative Na+/H+ +exchangers of the Nha-type and multi-subunit Mnh-type +were found only within a few soda lake CPR. Secondary +active transport of K+ might be mediated in most soda +lake CPR by KefB (COG0475)/kch Kef-type, glutathione- +dependent K+ transport systems, with or without H+ +antiport (67,68). +Various soda lake CPR had an acidic proteome, with +pI curves resembling those found in extremely halophilic +Bacteria. Intracellular proteins enriched in acidic amino +acids might be an adaptation to a “salt-in” strategy, i.e., +maintaining high intracellular potassium (K+) concentra- +tions to keep osmotic balance [7, 37] (Fig. 5b; see +Additional file 2: Figure S10). Such a strategy is energet- +ically favorable over de novo synthesis or import of +osmolytes such as ectoine and glycine betaine. We did +not find genes for the synthesis of organic osmolytes and +homologs of ABC-type transporters for primary active +uptake of proline/glycine betaine which were encoded +only in one MAG (Fig. 5a). For the “Ca. Nealsonbac- +teria” and “Ca. Vogelbacteria,” the salt-in strategy might +be a unique feature for the soda lake species explaining +Vavourakis et al. Microbiome (2018) 6:168 +Page 6 of 18 + +their high abundance in the hypersaline soda lake sedi- +ments, as we did not found an acidic proteome pre- +dicted from genomes obtained from other non-saline +environments (See Additional file 2: Figure S11). The +uptake of K+ ions remains enigmatic for most soda lake +CPR. Low-affinity Trk-type K+ uptake transporters (gen- +erally with symport of H+) (67,68) were encoded only by +a limited number of MAGs. We found three MAGs +Fig. 4 Relative abundance and metabolic potential of the dominant species. Abundance values, expressed as reads per kilobase of MAG per gigabase +of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, +Additional file 6). Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets +(cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom. The metabolic potential derived from functional marker +genes (Additional file 7) is depicted by the colored symbols. CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, +fix. = fixation, red. = reduction, ox. = oxidation, dis. = disproportionation +Vavourakis et al. Microbiome (2018) 6:168 +Page 7 of 18 + +a +b +Fig. 5 (See legend on next page.) +Vavourakis et al. Microbiome (2018) 6:168 +Page 8 of 18 + +encoding for Kdp-type sensor kinases (kdpD) but no +corresponding genes for the response regulator (kdpE) +or for Kdp-ATPases that function as the inducible, high- +affinity K+ transporters in other Bacteria (67,68). Finally, +mechanosensitive ion channels (mscS, mscL) and ABC- +type multidrug transport systems (AcrAB, ccmA, EmrA, +MdlB, NorM) and sodium efflux permeases (NatB) were +encoded in almost every MAG. The first might rapidly +restore the turgor pressure under fluctuating salinity +conditions by releasing cytoplasmic ions [38]. +Novel abundant groups involved in sulfur, nitrogen, and +carbon cycles +A new species of Thioalkalivibrio (family Ectothiorhodospir- +aceae) was by far the most abundant in the sediments of +the two salt-saturated lakes (Fig. 4). In the sediment of +Bitter-1, also a purple sulfur bacterium from the same fam- +ily was highly abundant. It was closely related to Halorho- +dospira, a genus also frequently cultured from hypersaline +soda lakes [1]. None of the abundant Ectothiorhodospira- +ceae spp. had already a species-representative genome +sequenced (Additional file 6). The potential of the Thioalk- +alivibrio spp. for chemolithotrophic sulfur oxidation was +evident (Additional file 7; see Additional file 8: Information +S1). Interestingly, the encoded nitrogen metabolisms were +quite versatile. While Thioalkalivibrio sp. 1 had the poten- +tial for nitrate reduction to nitrite, Thioalkalivibrio sp. 2 +might perform dissimilatory nitrite reduction to ammonia +(DNRA; see Additional file 2: Figure S12). +Two +deltaproteobacterial +lineages +of +dissimilatory +sulfate-reducing bacteria (SRB) were highly abundant in +the soda lake sediment of Bitter-1. One MAG from the +family Desulfobacteraceae (order Desulfobacterales) is +the first genome from the genus Desulfonatronobacter. It +encodes the genes for complete sulfate reduction to sul- +fide using various electron donors, as well as for the +complete oxidation of volatile fatty acids and alcohols, a +unique +feature +for +the +genus +Desulfonatronobacter +among haloalkaliphilic SRB [10] (see Additional file 8: +Information S2). Fumarate and nitrite (DNRA, NrfAH) +could be used as alternative electron acceptors. The sec- +ond dominant lineage was a new species from the genus +Desulfonatronospira (family Desulfohalobiaceae, order +Desulfovibrionales). Like other members of this genus, it +had the potential to reduce or disproportionate partially +reduced sulfur compounds. In addition, it could also use +nitrite as an alternative electron acceptor (NrfAH) [6]. +A novel lineage of gammaproteobacterial SOB was +highly abundant in the sediments of the moderately hy- +persaline Cock Soda Lake. It appeared as a sister group of +the family Xanthomonadaceae in the ribosomal protein +tree. This heterotrophic organism could conserve energy +through aerobic respiration. It might detoxify sulfide by +oxidizing it to elemental sulfur (sqr) with subsequent re- +duction or disproportionation of the polysulfides (psrA) +chemically formed from the sulfur. It also encoded the po- +tential for DNRA (nrfA and napC). Genes likely involved +in sulfide detoxification (sqr and psrA) were found also in +several other abundant MAGs of heterotrophs, including +one new abundant species from the family of Nitrilirup- +toraceae (class Nitriliruptoria, phylum Actinobacteria). +We found a wide variety of carbohydrate-active enzymes +in these MAGs, such as cellulases (GH1 family) in +addition to genes for glycolysis and TCA cycle and a +chlorophyll/bacteriochlorophyll a/b synthase (bchG gene). +The latter was also found in other Actinobacteria from the +genus Rubrobacter [39]. No evidence was found for +nitrile-degrading potential. +A second novel, uncultured lineage of Gammaproteo- +bacteria that was highly abundant at moderate salinities +branched in our ribosomal protein tree as a sister group +to the family Halothiobacillaceae. The MAGs encoded +for a versatile metabolism typical for purple non-sulfur +bacteria. The MAGs contained puf genes, bch genes, +genes for carotenoid biosynthesis (not shown), and a +Calvin cycle for photoautotrophic growth. Alternatively, +energy may be conserved through aerobic respiration, +while acetate and proprionate could be taken up via an +acetate permease (actP) and further used for acetyl-CoA +biosynthesis and carbon assimilation. Since the sqr gene +was present, but no dsr or sox genes, the organism +might oxidize sulfide only to elemental sulfur. One bin +contained also nifDKH genes suggesting putative diazo- +trophy, as well as a coenzyme F420 hydrogenase suggest- +ing photoproduction of hydrogen [40]. +The abundant Euryarchaeota organism showed a clear +preference for higher salinities. We obtained one highly +abundant MAG from the class Thermoplasmata that +encoded a full-length 16S rRNA gene only distantly re- +lated (91,2% identity, e value 0) to that of a member of +the KTK 4A group found in a hypersaline endoevaporitic +microbial mat [8]. The abundant soda lake organism is +likely a new genus and species. All KTK 4A-related +MAGs found here encoded for similar heterotrophic, +fermentative +metabolisms, +with +the +potential +for +(See figure on previous page.) +Fig. 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla. a Membrane transporters, +channels, and lipids. Peptidoglycan is depicted in gray and S-layer proteins in cyan. b Predicted isoelectric points (bin width 0.2) for the coding +sequences of MAGs. A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also +Additional file 2: Figure S11) +Vavourakis et al. Microbiome (2018) 6:168 +Page 9 of 18 + +anaerobic formate and CO oxidation. The KTK 4A +might be also primary degraders since they encoded pu- +tative cellulases (CAZY-families GH1, GH5) and chiti- +nases (GH18). Interestingly, half of the MAGs encoded a +putative +chlorophyll/bacteriochlorophyll +a/b +synthase +(bchG), which is highly unusual for Archaea. Although +little can be inferred from the presence of only one +marker gene, a functional bchG was previously also +found in Crenarchaeota [41]. The remaining two highly +abundant Euryarchaeota-related MAGs belonged to a +new species of Halorubrum (Additional file 6). +Key genes of the Wood-Ljungdahl pathway found in +novel phylogenetic groups +More than 50 MAGs were related to the family Syntro- +phomonadaceae (class Clostridia, phylum Firmicutes) +based on ribosomal protein phylogeny. All 16S rRNA +gene sequences found in the MAGS had 86–95% iden- +tity to sequences obtained from uncultured organisms +related to the genus Dethiobacter. While an isolated +strain of Dethiobacter alkaliphilus is a facultative auto- +troph +that +respires +thiosulfate, +elemental +sulfur +or +polysulfides with hydrogen as an electron donor [42] or +disproportionates +sulfur +[43], +other +haloalkaliphilic +members +of +the +Syntrophomonadaceae +are +reverse +acetogens, oxidizing acetate in syntrophy with a hydro- +genotrophic partner [44]. Two populations (different +species, Additional file 6) were especially abundant in +Cock Soda Lake (Fig. 4). They encoded for a full +CODH/ACS complex, the key enzyme for the reductive +acetyl-CoA or Wood-Ljungdahl pathway (WL) and a +complete +Eastern +branch +for +CO2 +conversion +to +5-methyl-tetrahydrofolate (Additional file 9) [45, 46]. +Acetogens use the WL to reduce CO2 to acetyl-CoA, +which can be fixed into the cell or used to conserve en- +ergy via acetogenesis. Syntrophic acetate oxidizers, some +sulfate reducing bacteria and aceticlastic methanogens +run the WL in reverse. Syntrophomonadaceae sp. 2 +encoded for a putative thiosulfate/polysulfide reductase +as well as a phosphotransacetylase (pta) and an acetate +kinase (ack) for the ATP-dependent conversion of acet- +ate to acetyl-CoA. Although alternative pathways for the +latter interconversion can exist, this second species has +the complete potential for (reversed) acetogenesis. +Highly remarkable was the presence of a bacterial-type +CODH/ACS +complex +and +a +near-complete +eastern +branch of the WL in a highly abundant species in Cock +Soda Lake from the family Coriobacteriaceae (phylum +Actinobacteria). This prompted us to scan all 871 MAGs +for the presence of acsB encoding for the beta-subunit +of the oxido-reductase module of CODH/ACS. We con- +firmed an encoded +(near)-complete +WL in several +additional organisms belonging to phylogenetic groups +not +previously +associated +with +this +pathway +[46] +(Additional file 9). We removed the Coriobacteriaceae +acsB genes from the final dataset to construct a phylo- +genetic tree since they were < 500 aa (Fig. 6) but found +seven MAGs from the OPB41 class within the Actino- +bacteria (16S rRNA gene fragment identity 94–96%). +The eastern branch of WL can function independently +in folate-dependent C1 metabolism [45], but the pres- +ence of the Western-branch in a phylum that comprises +mostly aerobic isolates is very surprising. The WL in +combination with the potential for acetate to acetyl-CoA +interconversion (pta/ack) and a glycolysis pathway were +also present in the soda lake MAGs from the phyla “Ca. +Handelsmanbacteria,” “Ca. Atribacteria” (latter branched +within the “Ca. Acetothermia”), and the class LD1-PA32 +(Chlamydiae), suggesting all these uncultured organisms +might be heterotrophic acetogens. However, it should be +noted that a PFOR typically connecting glycolysis to the +WL was only encoded in the LD1-PA32 MAGs. More- +over, from the genetic make-up alone, it cannot be +excluded that acetate is activated, and the WL run in +reverse for syntrophic acetate oxidation. Finally, the +novel acsB genes from soda lake Halanaerobiaceae, +Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) +and from Brocadiaceae and Planctomycetaceae (Plancto- +mycetes) disrupt the previously proposed dichotomy +between Terrabacteria and Gracilicutes bacterial groups +unifying 16S rRNA and acsB gene phylogenies [46] and +suggest a far more complex evolutionary history of the +WL pathway than previously anticipated. +Discussion +Extensive +classical +microbiology +efforts +have +been +already undertaken to explore the unique extremophilic +microbial communities inhabiting soda lakes. These un- +covered the presence of most of the functional groups +participating in carbon, nitrogen, sulfur, and minor +element cycling at haloalkaline conditions. The main re- +sults of this work are summarized in several recent re- +views [1, 6, 47, 48]. Since most microbes, including +those living in soda lakes, still evade all cultivation ef- +forts, a very effective way to discover new microbes and +assess their physiology based on their genetic repertoire +is either through single cell genomics or by directly se- +quenced environmental DNA. This exploratory metage- +nomics +study +performed +on +soda +lake +sediments +effectively overcame the existing cultivation bottleneck. +First, we expanded the known diversity of CPR consider- +ably with the first genomes of poly-extremophiles sam- +pled from soda lake sediments. Although the presence of +16S rRNA genes from CPR in marine sediments and hy- +persaline microbial mats was previously shown [34], +until now, CPR MAGs were mainly obtained from deep, +subsurface environments [15, 26, 29, 32, 49–52], and hu- +man microbiota [30]. Despite being highly abundant +Vavourakis et al. Microbiome (2018) 6:168 +Page 10 of 18 + +100 % +90-100 % +70-90 % +50-70 % +some MAGs +all MAGs +Bootstraps +Genes present +Glycolysis (EMP) +PFOR +WL-Eastern branch +H4MPT +TH4 +WL-Western branch +CODH/ACS +Acetogenesis/ +acetate activation +(pta/ack) +0.4 +PVC group (Chlamydiae LD1-PA32) +Syntrophorhabdus aromaticivorans +PVC group bacterium CSSed11_184 +Aerophobetes bacterium SCGC_AAA255-F10 +Ca. Acetothermia +Ca. Handelsmanbacteria +Planctomycetaceae +Anaerolineae +Firmicutes +Brocadiaceae +Planctomycetes +Methanomassiliicoccales +Halobacteroidaceae +Natranaerobiaceae +Methanomicrobiales +Desulfonatronospira +Firmicutes +Dehalococcoidia +Armatimonadetes bacterium CSP1-3 +Deltaproteobacteria +Thermodesulfobacteria +Desulfobulbaceae +Halanaerobiaceae +Nitrospirae +Actinobacteria (OPB41) +Fig. 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs. Only sequences ≥ 500 aa +were included. Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence +of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see +also Additional file 9: Dataset S6). Additional lineages found in this study are marked in blue. The three was rooted according to [46]. +Circles at the nodes show confidence percentage of the bootstraps analysis (100×). EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin +oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = +tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase. PVC group bacterium CSSed11_184 is likely a member +of the WCHB1-41 class within the Verrucomicrobia +Vavourakis et al. Microbiome (2018) 6:168 +Page 11 of 18 + +here, CPR went unnoticed in previous amplicon sequen- +cing studies. This might be due to the fact that many +CPR representatives have random inserts of various +length in their 16S rRNA genes or due to primer mis- +matches [29, 34]. This illustrates also that direct metage- +nomics should not only be preferred over amplicon +sequencing to infer functional potential, but the former +is far more effective for the discovery of novel organ- +isms. Second, we obtained many more genomes from +“traditional” bacterial phyla such as the Planctomycetes +and Chloroflexi, as well as candidate phyla, for which no +soda lake isolates, hence no genomes were previously +obtained. Third, even within the sulfur cycle, the most +active and frequently studied element cycle in soda lakes +[1], we found considerable metabolic novelty. Finally, we +found the Wood-Ljungdahl pathway in several novel +phyla, not closely related to any known acetogens, +methanogens, or sulfate-reducing bacteria [46]. The lat- +ter shows that our sequencing recovery effort has also +significantly contributed to the discovery of metabolic +novelty within various prokaryote phylogenetic groups. +Salinity is often considered to be the major factor +shaping prokaryote community composition in diverse +habitats [53, 54]. Extreme halophilic Euryarchaeota +seem to be always the dominant group in salt-saturated +hypersaline brines, both those with neutral or alkaline +pH [1, 7, 37]. Here, we found that although these +haloarchaea are still relatively more abundant in the sed- +iments exposed to brines with salt-saturating conditions, +the clear majority of microbes in all investigated hyper- +saline soda lake sediments are Bacteria. It could be +hypothesized that the sediment is a hide-out for the +extreme alkalinity and salinity governing the water +column, and that sediment stratification, especially in +the anoxic part, offers plenty of opportunities for niche +diversification. On the other hand, it should no longer +be a surprise that soda lakes are such productive and +biodiverse +systems. +First, +it +has +been +previously +elaborated that soda lake organisms are exposed to +approximately half the osmotic pressure in sodium +carbonate-dominated +brines +compared +to +sodium +chloride-dominated brines with the same Na+ molarity +[47]. Second, nitrogen limitation in the community can +be overcome when many members contribute to the +fixation of atmospheric N2, and various forms of organic +nitrogen are efficiently recycled. The soda lakes exam- +ined in this study were also eutrophic, and sulfur com- +pounds were abundant. Sulfide is also far less toxic at +high pH as it mostly occurs in the form of bisulfide +(HS−). Besides the evident high metabolic and taxo- +nomic diversity of dissimilatory sulfur-cycling bacteria, a +diverse heterotrophic community can be sustained com- +prising both generalist and very specialized carbon de- +graders. Less eutrophic soda lakes might not suffer from +carbon +limitation +either, +due +to +a +presence +of +high-bicarbonate concentrations. These effectively elim- +inate the inorganic carbon limitation for primary pro- +ducers who are highly active in soda lakes, especially +Cyanobacteria [55, 56]. Third, light that penetrates the +surface of the sediment seems to stimulate oxygenic and +anoxygenic phototrophic growth. Moreover, various het- +erotrophs, such as the rhodopsin-containing haloarchaea +and Bacteroidetes, have the option to tap into this un- +limited energy source for example to help sustain the +costly maintenance of osmotic balance. Unexpectedly, +we even found the first rhodopsin encoded by a member +of the CPR. Fourth, tight syntrophic relations, as pro- +posed for CPR members and Syntrophomonadaceae +spp., might be the solution to successful growth in an +energetically challenging environment. +Since our metagenomes are snapshots in time and space, +the failure to reconstruct specific MAGs gives no conclu- +sive evidence for the absence of certain microbial-mediated +element transformation in hypersaline soda lake sediments. +Additionally, technical limitations of the assembly and bin- +ning of highly micro-diverse genome populations might +hamper genome recovery [57]. More importantly, the +abundance of a specific microbe is not necessarily corre- +lated to the importance of its performance in an ecosystem. +Many metabolic capacities are redundant, and often key +transformations are reserved for a few rare organisms that +might proliferate for a short time-span when specific condi- +tions allow for it. For example, although no MAGs were re- +covered from chemolithoautotrophic nitrifiers [58], we did +detect a Nitrobacter-related OTU by amplicon sequencing +and assembled 16S rRNA genes from Thaumarchaeota, +suggesting bacterial and archaeal nitrifiers are present in +the surface sediments of soda lakes at very low abundance. +Finally, the method of DNA isolation might impact the +community composition apparent in the final metagenome +sequenced. Environmental samples contain complex mix- +tures of different organisms, and it is impossible to find a +protocol where the DNA from every single organism is ex- +tracted as efficiently without compromising the final quality +of the extracted DNA. However, since we find all the im- +portant taxonomic and functional groups known from pre- +vious cultivation-dependent studies back in either our +amplicon sequencing datasets or our directly sequenced +metagenomes, we are confident that the community com- +position and the MAGs presented here are representative +for the microbiomes of the soda lake sediments in the +Kulunda Steppe. +Conclusion +Years of intensive microbiological research on soda lakes +seem to have paid off, since many of the described gen- +era we could detect here have a cultured representative +from soda lakes. However, as many of the abundant +Vavourakis et al. Microbiome (2018) 6:168 +Page 12 of 18 + +lineages and groups found in soda lake sediments are +still uncultured, metagenomics proved to be a helpful +tool to gain primary insights in the potential physiology +and ecology of these poly-extremophilic prokaryotes. +We reconstructed the first genomes for many of such +organisms and proposed new functional roles for the +most abundant ones. Future studies should provide +more in depth analyses of these genomes, especially +from the less abundant organisms that might perform +key ecological processes, such as methanogens and nitri- +fiers. In addition, they should focus on gaining physio- +logical culture-based evidence or proof for in situ +activity for the abundant organisms described here. The +key metabolic insights provided by this metagenomics +study can lead to the design of new cultivation strategies. +In general, sediment communities are far more complex +than those found in the corresponding water column +[53, 59] and are therefore often considered too complex +for efficient metagenomic analysis. Many of the novel +lineages found here may therefore have related neutro- +philic lineages in marine and freshwater sediments that +await discovery. We demonstrate here that, by providing +sufficient sequencing depth, the “state of the art metage- +nomics toolbox” can effectively be used on sediments as +well. +Methods +Site description and sample collection +The top 10 cm sediments from four hypersaline, eutrophic +soda lakes located in the Kulunda Steppe (south-western +Siberia, Altai, Russia) were sampled in July of 2010 and +2011. General features and exact location of the sampled +soda lakes are summarized in Additional file 1: Table S1; a +map of the area was published previously [5]. Cock Soda +Lake (a stand-alone lake, sampled both in 2010 and 2011) +and Tanatar-3 (Tanatar system) were moderately hypersa- +line (~ 100 g L−1) with sandy sediment, while Tanatar-1 +and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) +with sulfide-rich sapropel sediments underlined by rock +trona deposits [7, 60]. Especially, Bitter-1 harbors a very +active microbial community, probably due to its high- +organic and -mineral content. Surface sediments were col- +lected by a plastic corer into sterile glass containers and +transported to the laboratory in a cooler. +DNA isolation, 16S rRNA amplicon, and metagenomic +sequencing +The colloidal fraction of each sediment sample (~ 10% +of 50 g) was separated from the course sandy fraction by +several short (30–60 s) low-speed (1–2,000 rpm in +50 mL Falcon tubes) centrifugation steps and washed +with 1–2 M NaCl solution. The pelleted colloidal sedi- +ment fraction was first subjected to 3 cycles of freezing +in liquid nitrogen/thawing, then re-suspended in 0.1 M +Tris (pH 8)/10 mM EDTA, and then subjected to harsh +bead beating treatment. Next, the samples were incu- +bated with lysozyme (15 mg/mL) for 2 h at 37 °C +followed by a SDS (10% w/v) and proteinase K (10 μg/ +mL) treatment for 30 min. at 45 °C. High molecular +weight DNA was isolated using phenol/chloroform ex- +traction, quality-checked, and sequenced as described +previously [7]. Direct high-throughput sequencing of the +DNA was performed on an Illumina HiSeq 2000 plat- +form to generate 150 b paired-end reads. Amplification +of the V4-V6 region of prokaryote 16S rRNA genes +using barcoded 926F-1392R primers, amplicon purifica- +tion, quantification, and Roche (454)-sequencing was +performed together in a batch with brine samples from +the same sampling campaigns. Barcodes and adapter se- +quences were removed from de-multiplexed amplicon +sequence reads and analyzed with the automated NGS +analysis pipeline of the SILVA rRNA gene database pro- +ject [61] (SILVAngs 1.3, database release version 128) +using default parameters. The OTUs (97% identity) +assigned down to the genus level were only considered +when they had a relative abundance ≥ 0.1% in at least +one of the five datasets. +Processing metagenomics reads, assembly, binning, and +post-binning +Metagenomic raw reads were quality trimmed using +Sickle [62] (version 1.33), and only reads ≥ 21 b were +retained. The prokaryotic community structure at taxo- +nomic top levels was extrapolated from ten million ran- +domly sampled singletons from each dataset. Candidate +16S rRNA fragments > 90 b were identified [63] and +compared against the SILVA SSU database 128 (blastn, +min. length 90, min. identity 80%, e value 1e-5). To ver- +ify that the microbial community composition was in- +deed +mostly +prokaryotic, +we +did +a +more +general +screening of the metagenomics reads that identified also +candidate 18S rRNA fragments > 90 b (see Additional +file 1: Tables S4-S5). The complete trimmed read sets +were assembled into contigs ≥ 1 kb with MEGAHIT [64] +(v1.0.3–6-gc3983f9) using paired-end mode, k min = 21, +k max = 131, k step = 10. Genes were predicted using +Prodigal [65] (v.2.6.2) and RNAs with rna_hmm3 [66] +and tRNAscan-SE [67]. Assembled 16S rRNA sequences +were compared to a manually curated version from the +SILVA SSU database (e value ≥ 1e-5). Predicted protein +sequences +were +annotated +against +KEGG +with +GhostKOALA (genus_prokaryotes + family_eukaryotes ++ viruses) [68]. Marker genes for central metabolic +pathways and key environmental element transforma- +tions were identified based on K number assignments +[15, 69–71]. +Contigs ≥ 2.5 kb were binned with METABAT [72] +(superspecific mode) based on differential coverage +Vavourakis et al. Microbiome (2018) 6:168 +Page 13 of 18 + +values obtained by mapping all five trimmed readsets to +all five contig sets with Bowtie2 [73]. The bins were sub- +jected to post-binning (an overview of the workflow is +given in Additional file 2: Figure S13). Bins were +assessed with lineage-specific single copy genes using +CheckM [74] and further processed with the metage- +nomics workflow in Anvi’o [75] (v2.3.2). Since Candidate +Phyla Radiant (CPR) is not included in the CheckM ref- +erence trees and are likely to have low-genome com- +pleteness, we used an existing training file of 797 CPR +genomes to identify putative CPR bins [76]. Bins with +CheckM-completeness ≥ 50% (884 out of 1778) and an +additional four CPR bins were further processed. Coding +sequences +were +annotated +for +taxonomy +against +NCBI-nr (July, 2017) with USEARCH [77] (5.2.32) to +verify that most hits in each bin were to prokaryotic ref- +erences. Phage or viral contigs were manually removed. +Genome +contamination (redundancy) +was estimated +based on marker sets of universal single copy genes +identified for Bacteria [30] and Archaea [78] as imple- +mented in Anvi’o. Genome coverage was obtained by +mapping trimmed reads with BBMap [79] v36.x (kfilter +31, subfilter 15, maxindel 80). Bins with ≥ 5% redun- +dancy were further refined with Anvi’o using circle phy- +lograms +(guide +trees +tnf-cov: +euclidian +ward) +and +scanned again for CPR. Post-binning resulted in a total +of 2499 metagenome-assembled genomes (MAGs), of +which 871 were either medium-quality genome drafts +(CheckM estimated completeness ≥ 50% and contamin- +ation ≤ 10% [80], Additional file 4) or lower quality draft +genomes from CPR. +Phylogeny of the MAGs was assessed based on 16 +single-copy ribosomal proteins and representative refer- +ence genomes of major prokaryote lineages across the +tree of life [17]. Individual ribosomal proteins in our +MAGs were identified by K number assignments. Only +ribosomal proteins ≥ 80 aa were considered. Initial +maximum-likelihood (ML) trees were constructed to de- +termine which organisms belonged to the Archaea, Bac- +teria, or CPR with FastTree 2 [81] (WAG + CAT). Final +separate trees for the three distant evolutionary groups +were constructed in the same manner. Each ribosomal +protein set was aligned separately with MAFFT [82] +(v7.055b, − auto) and concatenated only if a MAG +encoded at least 8 out of 16 proteins. For all trees, a +100× posterior bootstraps +analysis +was +performed. +Phylogenetic trees were visualized together with gen- +ome statistics and abundance information using iTOL +[83]. We cross-checked the taxonomic assignments +based on the phylogeny of the ribosomal protein cas- +sette +with +the +top +hit +contig annotations +against +NCBI-nr and with the reference lineage obtained with +CheckM. Lastly, we manually corrected the MAGs for +misplaced 16S rRNA genes. The final trees presented +in the manuscript were redrawn using FigTree v1.4.3 +[84]. +Detailed genome analyses +CPR +MAGs +were +re-annotated +more +thoroughly: +genes were predicted with Prokka [85], and functional +predictions were performed by running InterProScan +5 locally on the supplied COG, CDD, TIGRFAMs, +HAMAP, Pfam, and SMART databases [86]. BLAST +Koala was used for KEGG pathway predictions [68]. +To find putative carbohydrate-active enzymes in all +final MAGs, we used the web-resource dbCAN [87] +to annotate all predicted proteins ≥ 80 aa against +CAZy [88]. +To identify the top ten abundant MAGs from each re- +spective dataset, ten million randomly sampled single- +tons were mapped onto each MAG with a cut-off of 95% +identity in minimum of 50 bases. Coverage values were +additionally normalized for genome size and expressed +as reads per kilobase of sequence per gigabase of +mapped reads (RPKG) [89]. A positive score (from 871 +to 1) was assigned to each MAG according to the rank- +ing of the summed RPKG of MAGs in the high-salinity +datasets (B1Sed10 and T1Sed) and a negative score ac- +cording to the ranking of the summed RPKGs in the +moderate salinity datasets (CSSed10, CSSed11, T3Se +d10). Both scores were summed to get a “salinity prefer- +ence score” with MAGs recruiting preferably from high +salinity datasets on the positive end, moderate salinity +datasets in the negative end, and those without prefer- +ence in the middle. +We determined species delineation for the most +abundant MAGs and their closest reference genomes +(NCBI-nr) by Average Nucleotide Identity (ANI) and +conserved DNA-matrices, as follows [90]: ANI ≥ 95%, +conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA +< 69% = might be same species, ANI < 95%, condDNA +< 69% = different species. Single gene trees based on +maximum +likelihood +were +constructed +with +un- +trimmed alignments (MAFFT, L-INS-i model) and +FastTree 2 (WAG + CAT, increased accuracy, -spr4 +-mlacc 2 -slownni) using 100× bootstraps. References +were pulled from eggNOG (v4.5.1) [91] and supple- +mented +with +sequences +from +NCBI-nr +or +refined +according to [7, 33, 46, 92–94]. The curated MAGs +were +scanned +for +the +presence +of +rhodopsin +sequences with the hmmsearch software [95] and a +profile +hidden +Markov +model +(HMM) +of +the +bacteriorhodopsin-like protein family (Pfam accession +number +PF01036). +The +identified +sequences +with +significant similarity were aligned together with a +curated database composed of a collection of type-1 +rhodopsins, using MAFFT (L-INS-i accuracy model) +[82]. This protein alignment was further utilized to +Vavourakis et al. Microbiome (2018) 6:168 +Page 14 of 18 + +construct a maximum likelihood tree with 100× boot- +strap with FastTree 2 [81]. All other genes were +identified using the KEGG annotation. +Additional files +Additional file 1: Table S1. General features of the four sampled soda +lakes at time of sampling. Table S2. SILVA classification of the 16S rRNA +gene sequences found in all ≥1 kb contigs of five soda sediment +metagenomic datasets. Table S3. Enzymes involved in lipopolysaccharide +biosynthesis found among different members of the CPR. Table S4. +Sub-kingdom classification of candidate SSU rRNA gene fragments +found in subsamples of 10 million random forward reads from the +five soda sediment metagenomes. Table S5. Top-level taxonomic +classification of the 18S rRNA gene fragments found in subsamples +of 10 million random forward reads from the five soda sediment +metagenomes. Table S6. Description of the metagenomic datasets, +NCBI Sequence Read Archive (SRA) accession numbers and general +statistics of the assembled contigs. (PDF 740 kb) +Additional file 2: Figure S1. Taxonomic fingerprints determined by 16S +rRNA gene amplicon sequencing. Figure S2. Genome statistics of the +871 MAGs. Figure S3. Phylogeny of MAGs belonging to “Candidatus +Aenigmarchaeota” and “Ca. Nanohaloarchaeota”. Figure S4. Phylogeny of +MAGs related to “Candidatus Acetothermia”, candidate division WS1 and +“Candidatus Lindowbacteria”. Figure S5. Phylogeny of MAGs related to +candidate division KSB3 and “Candidatus Schekmanbacteria”. Figure S6. +Multiple sequence alignment of the V-type ATPase subunits K. Figure S7. +Multiple sequence alignment of the F-type ATPase subunits c. Figure S8. +Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- +like proteins. Figure S9. Maximum likelihood tree of the putative +rhodopsins. Figure S10. Predicted isoelectric points (pI) profiles of all +MAGs from CPR members. Figure S11. Predicted isoelectric points +profiles for members of the “Ca. Nealsonbacteria” and “Ca. Vogelbacteria”. +Figure S12. Multiple sequence alignment of the dissimilatory +cytochrome c nitrite reductases (nrfA/TvNiR, K03385). Figure S13. +Overview of the post-binning workflow used for genome recovery. +(PDF 6548 kb) +Additional file 3: Dataset S1. Relative abundance of the OTUs assigned +to the genus-level within the Archaea, Bacteria and organelles from +Eukaryota detected by 16S rRNA gene amplicon sequencing. The OTUs +with less than 0.1% abundance accross all five datasets are not shown. +The names of highly abundant genera (≥1% in at least one of the data- +sets) are shown in bold. (XLSX 24 kb) +Additional file 4: Dataset S2. Organism names, statistics and general +description incl. Completeness and contamination estimates, phylogeny +and DDBJ/EMBL/Genbank accession numbers of the metagenome +assembled genomes (MAGs) described in this paper. All submitted +versions described in this paper are version XXXX01000000. Size = +recovered genome size, Completeness (Compl1), contamination (Cont), +strain heterogenity (Str het) and Taxon CheckM were inferred from +lineage-specific marker sets and a reference tree build with CheckM [74]. +Additional completeness (compl2) and redundancy (red) estimates were +inferred based on the presence of universal single copy genes for Bacteria +and Archaea [75]. Decision and confidence intervals from the Candidate +Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the +besthit in SILVA when 16S rRNA genes were present. Phylum/class 16 +ribosomal proteins is the taxonomy derived from our ribosomal protein +trees (see main text: Figs. 2 and 3). OTU gives the inferred link of a +population genome with our 16S rRNA gene amplicon dataset +(Additional file 3). (XLSX 253 kb) +Additional file 5: Dataset S3. Estimated abundance and derived +salinity preference from each MAG in each metagenomic dataset +expressed as Reads per Kilobase of MAG per Gigabase of mapped reads +(RPKG) and “salinity preference score” (see Methods section), basis for +Fig. 4. (XLSX 143 kb) +Additional file 6: Dataset S4. Average Nucleotide Identity (ANI) and +conserved DNA (condna) matrices to determine species delineation +between the most abundant MAGs shown in Fig. 4, closely related +(less abundant) MAGs and NCBI reference genomes. Decision matrix +shows: 1 = same species, − 1 = might be same species, 0 = different +species (see Methods section). (XLSX 1161 kb) +Additional file 7: Dataset S5. Sheet 1 Presence and absence of marker +genes and putative carbohydrate-active enzymes in the MAGs to infer putative +roles in C, N and S element cycles based on K-number assignments and CAZy +annotations. Sheet 2 Summary basis for Fig. 4. (XLSX 41 kb) +Additional file 8: Information S1. More detailed description of the +main metabolisms encoded by Thioalkalivibrio-related MAGs. +Information S2 More detailed description of the main metabolisms +encoded by Deltaproteobacterial-related MAGs. (PDF 219 kb) +Additional file 9: Dataset 6. Sheet 1 shows the MAGs positive for the +marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS). The +basis for Fig. 6, namely presence and absence of key genes involved in +the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis +and pyruvate to CO2 conversion is shown for each MAG. Sheet 2 shows +the MAGs positive for the marker gene cdhC (K00193) encoding for the +beta subunit of an acetyl-CoA decarboxylase synthase complex. While +acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- +type (methanogens) enzymes with the same function, we found few +discrepancies between marker gene and genome phylogeny within the +Methanomassiliicoccaceae and Chloroflexi. (XLSX 52 kb) +Acknowledgments +We thank Dr. Nikolai Chernych for his technical assistance during the +isolation and purification of metagenomics DNA. We also thank the +Department of Energy Joint Genome Institute for sequencing the +metagenomes. +Funding +CDV and GM were supported by the ERC Advanced Grant PARASOL (no. 322551). +A-SA and RG were supported by the research grant 17-04828S from the Grant +Agency of the Czech Republic. MM was supported by the Czech Academy of +Sciences (Postdoc program PPPLZ application number L200961651). DYS was +supported by the SIAM/Gravitation Program (Dutch Ministry of Education and +Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- +00121). Sequencing was performed by the U.S. Department of Energy Joint +Genome Institute, a DOE Office of Science User Facility, as part of the Community +Sequencing Program (contract no. DE-AC02- 05CH11231). +Availability of data and materials +The raw sequence reads of the five metagenomes have been deposited to +the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession +numbers and read and contig statistics). The final 871 MAGs described in this +paper have been deposited as Whole Genome Shotgun projects at DDBJ/ +EMBL/GenBank, and accession numbers are listed in Additional file 4 +(BioProject ID PRJNA434545). All versions described in this paper are version +XXXX01000000. The cleaned and dereplicated amplicon sequence datasets +are available in FigShare (https://figshare.com/s/7684627445e3621aba24). +Maximum likelihood trees based on the concatenated alignment of 16 +ribosomal proteins, basis for Figs. 2 and 3, in newick format (.tre file) and +complementary datasets (used to plot completeness, contamination, +genome recovery size, G + C mol% and RPKG in iTOL), as well as K number +assignments for the predicted proteins of all MAGs (KEGG-orthologues, +Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions +of this article are also available in FigShare (https://figshare.com/s/ +7684627445e3621aba24). +Authors’ contributions +GM and DYS initiated this study and were responsible for the fieldwork, +sample preparation, and sequencing effort. CDV conceptualized the research +goals under supervision of DYS and GM, and performed the bioinformatics +analysis under close guidance of A-SA and RG. CDV is the primary author of +this manuscript. MM, RG, and CDV prepared the main figures. All authors +read and approved the final manuscript. +Ethics approval and consent to participate +Not applicable. +Vavourakis et al. Microbiome (2018) 6:168 +Page 15 of 18 + +Consent for publication +Not applicable. +Competing interests +The authors declare that they have no competing interests. +Publisher’s Note +Springer Nature remains neutral with regard to jurisdictional claims in +published maps and institutional affiliations. +Author details +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands. 2Department of Aquatic Microbial Ecology, Institute of +Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, +Czech Republic. 3Winogradsky Institute of Microbiology, Research Centre of +Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld. 2, +Moscow, Russian Federation117312. 4Environmental Biotechnology, +Department of Biotechnology, Delft University of Technology, Van der +Maasweg 9, 2629, HZ, Delft, the Netherlands. +Received: 23 June 2018 Accepted: 3 September 2018 +References +1. +Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G. +Microbial diversity and biogeochemical cycling in soda lakes. 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Microbiome (2018) 6:168 +Page 18 of 18 + diff --git a/kb_50/content/tmp_files/load_file.txt b/kb_50/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..602fef5594d1967a71f9049bdf3c05f01bd3c598 --- /dev/null +++ b/kb_50/content/tmp_files/load_file.txt @@ -0,0 +1,1051 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf,len=1050 +page_content='RESEARCH Open Access A metagenomics roadmap to the uncultured genome diversity in hypersaline soda lake sediments Charlotte D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Vavourakis1 , Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sorokin3,4 and Gerard Muyzer1* Abstract Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Despite the high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but the microbiome of soda lake sediments received much less attention of microbiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Here, we performed metagenomic sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and a salt content between 70 and 400 g L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The recovered 16S rRNA gene sequences were mostly from Bacteria, even in the salt-saturated lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Most OTUs were assigned to uncultured families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We reconstructed 871 metagenome-assembled genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla Radiation (CPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Five new species of CPR were among the most dominant community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Novel dominant lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen cycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the Actinobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR and several hundred other novel prokaryote lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The soda lake CPR is a functionally diverse group, but the most abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Second, we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those encompassing known homo-acetogens, sulfate-reducers, and methanogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Since only few environmental metagenomics studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine and freshwater sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl pathway Correspondence: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='Muijzer@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='nl †Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands Full list of author information is available at the end of the article © The Author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='0 International License (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The Creative Commons Public Domain Dedication waiver (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='org/publicdomain/zero/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='0/) applies to the data made available in this article, unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='1186/s40168-018-0548-7 MicrobiomeBackground Soda lakes are evaporative, athallasic salt lakes with low cal- cium and magnesium concentrations and a high-alkaline pH up to 11 buffered by dissolved (bi-) carbonate ions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' They are constrained to arid regions across the globe, mainly the tropical East African Rift Valley , the Libyan Desert , the deserts in California and Nevada , and the dry steppe belt of Central Asia that spans to southern Si- beria, north-eastern Mongolia, and Inner Mongolia in China .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' On top of the extreme salinity and alkaline pH, the Eurasian soda lakes experience extreme seasonal temperature differences, causing highly unstable water re- gimes and fluctuating salinities .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Yet, soda lakes harbor diverse communities of haloalkaliphilic microbes, mostly prokaryotes that are well adapted to survive and grow in these extreme environments and consist of similar func- tional groups in soda lakes around the world .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The relative abundance of different groups is typically governed by the salinity of the brine , and microbial-mediated nutrient cycles become partially hampered only at salt-saturating conditions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' So far, all characterized prokaryotic lineages cultured from soda lakes comprise over 70 different species within more than 30 genera .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' From these, only a lim- ited number of genomes have been sequenced today, mostly from chemolithoautotrophic sulfur-oxidizing bac- teria belonging to the genus Thioalkalivibrio (class Gam- maproteobacteria) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' It is well established that metagenomics enables the recovery of genomes and the identification of novel genetic diversity where culturing ef- forts fail .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In recent years, next-generation sequen- cing has recovered a massive number of genomes from previously unknown groups of prokaryotes , including a strikingly large and diverse group called “Candidate Phyla Radiation” (CPR), only distantly related to other cultured bacterial lineages .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Previously, we conducted a metagenomics study on soda lakes and re- constructed novel genomes from uncultured Bacteroidetes and “Candidatus Nanohaloarchaeaota” living in hypersa- line Siberian soda brines .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Here, we turned our atten- tion to the far more complex prokaryotic communities living in the sediments of the hypersaline soda lakes from the same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We give a broad overview of all the taxonomic groups sequenced and focus on the metabolic diversity found in the reconstructed genomes of the most abundant, uncultured organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Results Overall prokaryote community structure The salinities from the studied soda lakes ranged from moderately hypersaline (between 70 and 110 g L−1) to salt-saturated (400 g L−1 salt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The soluble carbonate al- kalinity was in the molar range, and the pH in all lakes was around ten (see Additional file 1: Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' To give an overview of the overall prokaryotic community com- position in each of the samples, we looked at the taxo- nomic classification of 16S rRNA genes recovered both by amplicon sequencing and direct metagenomics se- quencing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 1, see also Additional file 2: Figure S1; Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The prokaryotic communities of all five sediment samples were highly diverse and consisted mostly of uncultured taxonomic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Bacteria were more abundant than Archaea, regardless of the salinity of the overlaying brine (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Euryarchaeota were the second and third largest group in the sediments of the two salt-saturated lakes comprising ~ 10 and ~ 20% of the 16S rRNA genes in the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Most Euryarchaeota-related OTUs detected by amplicon se- quencing belonged either to the uncultured Thermoplas- mata group KTK 4A (SILVA classification) or the genera Halohasta and Halorubrum (class Halobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In ac- cordance with cultivation-dependent studies , most OTUs assigned to methanogens were from the class Methanomicrobia, especially the lithotrophic genus Methanocalculus (up to ~ 3%) and the methylotrophic genus Methanosalsum (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The varying ratio of the three dominant bacterial groups, Firmicutes, Bacteroidetes (including the newly proposed phyla Rhodothermaeota and Balneolaeota ), and Gammaproteobacteria, showed no clear trend in relation to the salinity in the lakes, but when Firmicutes were domin- ant, Bacteroidetes were less abundant and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Most Firmicutes belonged to the order Clostridales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Uncultured members from the family Syntrophomonadaceae had a relative abundance of more than 5% in all five metagen- omes and comprised in two lakes even ~ 11–20% of the recovered amplicon sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The second most abundant Firmicutes order was Halanaerobiales, particularly the genus Halanaerobium (family Halanaerobiaceae) and un- cultured members of the Halobacteroidaceae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The majority of Bacteroidetes-related OTUs could not be assigned down to the genus level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The uncultured ML635J-40 aquatic group (order Bacteroidales) comprised at least 5% of all five prokaryotic communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' This group has been previously found to be abundant in Mono Lake (a soda lake) and in an anoxic bioreactor degrading cyanobacterial biomass under haloalkaline conditions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Two other highly abun- dant (up to ~ 8%) uncultured groups from the class Balneo- lia (proposed new phylum Balneolaeota ) were also detected in other soda lakes before .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Within the Gam- maproteobacteria, the genus Thioalkalivibrio was abundant (~ 3% of the total community), but also uncultured members of HOC36 were prevailing at moderate salinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Members of the Deltaproteobacteria, Alphaproteobacteria, and Chloroflexi comprised up to ~ 10% of the detected 16S rRNA gene in some of the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The GIF9 family of the class Dehalococcoidia was among the top three most abundant OTUs in two lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The extremely salt-tolerant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 2 of 18 and alkaliphilic genera Desulfonatronobacter (order Desulfo- bacterales) and Desulfonatronospira (order Desulfovibrio- nales) were the dominant Deltaproteobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Highly abundant OTUs, within the Actinobacteria belonged to the class Nitriliruptoria and within the Alphaproteobacteria to the family Rhodobacteraceae and the genus Roseibaca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The important nitrifying genus Nitrobacter (Alphaproteobacteria) was present in only one of the lakes with moderate salinity (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Some bacterial top-level taxa appeared less dominant (< 5%) from the 16S rRNA genes recovered from the metagenomes but were represented mainly by a single highly abundant OTU in the amplicon sequences, in- cluding the haloalkaliphilic genus Truepera within the phylum Deinococcus-Thermus, the genus Spirochaeata within the phylum Spirochaetes, the family BSN166 within the phylum Ignavibacteriae, the BD2–11 terres- trial group within the Gemmatimonadetes, and the WCHB1–41 order within the Verrucomicrobia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All OTUs within the Thermotogae and Lentisphaerae belonged to uncultured genera from the family Kosmoto- gaceae and Oligosphaeraceae, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All Tenericu- tes-related OTUs belonged to the class Mollicutes, and especially the order NB1-n was dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In contrast, the phylum Planctomycetes was relatively diverse, with at least 11 different genus-level OTUs spread over four class-level groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' High-throughput genome recovery We obtained 717 medium-quality (≥ 50% complete, < 10% contamination) and 154 near-complete (≥ 90% complete, < 5% contamination) metagenome-assembled genomes (MAGs) across three major prokaryote groups: Archaea, Bacteria, and CPR (see Additional file 4 and Additional file 2: Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figures 2 and 3 show the top-level phylogeny of all MAGs based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The reference database used contains a repre- sentative for each major prokaryote lineage .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 1 Abundant prokaryotic groups in five hypersaline soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' a Relative abundance of the top-level taxa (those with ≥ 1% abundance in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' b Relative abundance of the 16S rRNA OTUs (those with sum of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Three of the assessed soda lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 3 of 18 colored the different phyla from which we obtained a MAG in alternate blue and orange colors, and highlighted the MAGs obtained here in a darker shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Many MAGs belonged to uncultured groups commonly detected in soda lake 16S rRNA gene surveys, over 100 MAGs still belonged to candidate prokaryote phyla and divisions that to our knowledge were never detected be- fore in soda lakes, including CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Although only few MAGs had near-complete 16S rRNA genes, in most cases we were able to link available taxonomic gene an- notations and ribosomal protein phylogeny to the SILVA taxonomy of the OTUs assigned to the amplicon se- quences, while cross-checking the abundance profiles of both MAGs (Additional file 5) and OTUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The soda lake CPR recovered from the metagenomes was restricted to a few distinct phyla within the Parcubacteria group, mostly affiliating with “Candidatus Nealsonbacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Zambryskibacteria” (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The first group of MAGs encompassed four different branches in our riboso- mal protein tree, suggesting a high-phylogenetic diversity, with 33 putative new species sampled here (ANI and con- DNA matrices given in Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Zambrys- kibacteria-”related MAGs consisted of at least five new species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Few MAGs were recovered from CPR groups also detected by amplicon sequencing (see Additional file 2: Figure S1), namely the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Dojkabacteria” (former WS6), “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Saccharibacteria” (former TM7), CPR2, and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Katanobacteria” (former WWE3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The archaeal tree is unrooted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The CPR tree is rooted to the Wirthbacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 4 of 18 Most archaeal MAGs belonged to the phylum Euryarch- aeota and the abundant classes Halobacteria, Methanomi- crobia, and Thermoplasmata (related to OTU KTK 4A) within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In addition, three Thermoplasmata-related MAGs that encoded for the key enzyme for methanogenesis (methyl-coenzyme M reductase, mcr) affiliated with refer- ence genomes from Methanomassilicoccales, the seventh order of methanogens have been recovered .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Another MCR-encoding MAG was closely related to the latest discovered group of poly-extremophilic, methyl-reducing methanogens from hypersaline lakes from the class Methanonatronarchaeia (related to OTU ST-12K10A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We recovered also one MAG from the class Methanobacteria and a high-quality MAG from the WCHA1–57 group (“Candidatus Methanofastidiosa” ) in the candidate division WSA2 (Arc I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Several MAGs were recovered from the DPANN archaeal groups “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Diapherotrites,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Aenigmarchaeota,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (see Additional file 2: Figure S3) and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Woesearch- aeota” (former Deep Sea Hydrothermal Vent Group 6, DHVEG-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Although we did not reconstruct any reasonable-sized MAGs from the TACK superphylum, we found several 16S rRNA genes on the assembled contigs that affiliated to the Thaumarchaeota (see Additional file 1: Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Nearly every known bacterial phylum had an extremo- philic lineage sampled from our hypersaline soda lake sediments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In most cases, the soda lake lineages clearly formed separate branches appearing as sister groups to known reference lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The highest genome recovery was from the same top-level taxonomic groups that were also abundant in our 16S rRNA gene analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' From the Verrucomicrobia, most MAGs belonged to the order WCHB1-41 (16S rRNA gene identity 92–100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' However, in our ribosomal protein tree, they branched within the phylum Lentisphaerae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sixteen Tenericutes MAGs from at least 12 different species (Additional file 6) were closely related to the NB1-n group of Mollicutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Based on the recovered genome size and encoded meta- bolic potential, these organisms are free-living anaerobic fermenters of simple sugars, similar to what has recently been proposed for “Candidatus Izimaplasma” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Alternate orange and blue colors show phyla/ classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 5 of 18 Several MAGs belonged to the candidate phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Omnitrophica,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Atribacteria,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Acetother- mia” (former OP1), which were moderately abundant also in some sediment (see Additional file 2: Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' For the latter phylum, we suspect that four MAGs were more closely related to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' WS1 and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Lindow- bacteria” for which only few reference genomes are currently available in NCBI (see Additional file 2: Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Due to a high-sequencing coverage, we also managed to reconstruct several MAGs from rare Bacteria (< 100 amplicon sequences detected, see Additional file 2: Figure S1), including the phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Hydrogenedentes,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Cloacimonetes,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' BRC1, Elusimicrobia, Caldi- serica, and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Latescibacteria.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The MAGs from the latter phylum were more closely related to the recently proposed phylum “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Handelsmanbacteria” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Two additional MAGs with 16S rRNA gene fragments with 94–95% identity to the class MD2898-B26 (Nitrospinae) were more likely members of ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' KSB3 (proposed “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Moduliflexus” , see Additional file 2: Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Draft genomes of haloalkaliphilic CPR Strikingly, members of the CPR related to “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Nealson- bacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Vogelbacteria” were among the top 5% of abundant organisms in the surface sediments of the soda lakes, especially those with moderate salinity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Like most members of the CPR, the MAGs of the four most abundant “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Nealsonbacteria” seem to be anaerobic fermenters .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' They lacked a complete TCA cycle and most complexes from the oxidative elec- tron transfer chain, except for the subunit F of a NADH-quinone oxidoreductase (complex I, nuoF, nuoG, nuoA) and coxB genes (complex II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All CPR MAGs had a near-complete glycolysis pathway (Embden-Meyerhof- Parnas) encoded, but pentose phosphate pathways were severely truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The commonly encoded F- and V-type ATPase can establish a membrane potential for symporter-antiporters by utilizing the ATP formed by substrate-level phosphorylation during fermentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All CPR have V-type ATPases that can translocate Na+ in addition to H+ (see Additional file 2: Figure S6), while only two members of the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Falkowbacteria” had puta- tive Na+-coupled F-type ATPases (see Additional file 2: Figure S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The coupling of ATP hydrolysis to sodium translocation is advantageous to maintain pH homeosta- sis in alkaline environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Interestingly, with only two exceptions , all CPR genomes recovered from other environments with neutral pH were reported to encode only F-type ATPases [28–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' One low-abundant MAG affiliated to “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Peregrinibacteria” contained both the large subunit of a RuBisCO (type II/III, see Additional file 2: Figure S8) and a putative phosphoribu- lokinase (PRK, K00855) encoded in the same contig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' This is remarkable because PRK homologs were not previously identified among CPR, and RuBisCo form II/ III was inferred to function in a nucleoside salvage path- way .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' One “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Saccharibacteria” MAG encoded for a putative channelrhodopsin (see Additional file 2: Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' This is the first rhodopsin found among the CPR and suggests that this enigmatic group of organ- isms may have acquired evolutionary adaptations to a life in sunlit surface environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' A previous study showed that most CPR has coccoid cell morphotypes with a monoderm cell envelope resem- bling those from Gram-positives and Archaea but with a distinct S-layer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Thick peptidoglycans coated with acidic surface polymers such as teichoic acids help pro- tect the cells of Gram-positives against reactive hydroxyl ions in highly alkaline environments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All soda lake CPR had indeed the capability for peptidogly- can biosynthesis, but we found proteins typical for Gram-negatives for the biosynthesis of lipopolysaccha- rides (see Additional file 1: Table S3), homologous to the inner membrane proteins of type II secretion systems and to several proteins associated to the outer membrane and peptidoglycan, including OmpA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' It remains to be determined whether the soda lake CPR also lacks an outer membrane and perhaps anchor lipopolysaccharides, S-layer proteins, and lipoproteins to the inner cell membrane or peptidoglycan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We also found gene encoding cardiolipin and squalene synthases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Increased levels of cardiolipin and the presence of squa- lene make the cytoplasmic membrane less leaky for protons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In addition, cation/proton exchangers are known to play a crucial role for pH homeostasis in alka- liphilic prokaryotes as they help acidify the cytoplasm during the extrusion of cations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Putative Na+/H+ exchangers of the Nha-type and multi-subunit Mnh-type were found only within a few soda lake CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Secondary active transport of K+ might be mediated in most soda lake CPR by KefB (COG0475)/kch Kef-type, glutathione- dependent K+ transport systems, with or without H+ antiport (67,68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Various soda lake CPR had an acidic proteome, with pI curves resembling those found in extremely halophilic Bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Intracellular proteins enriched in acidic amino acids might be an adaptation to a “salt-in” strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='e., maintaining high intracellular potassium (K+) concentra- tions to keep osmotic balance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 5b; see Additional file 2: Figure S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Such a strategy is energet- ically favorable over de novo synthesis or import of osmolytes such as ectoine and glycine betaine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We did not find genes for the synthesis of organic osmolytes and homologs of ABC-type transporters for primary active uptake of proline/glycine betaine which were encoded only in one MAG (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' For the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Nealsonbac- teria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Vogelbacteria,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' the salt-in strategy might be a unique feature for the soda lake species explaining Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 6 of 18 their high abundance in the hypersaline soda lake sedi- ments, as we did not found an acidic proteome pre- dicted from genomes obtained from other non-saline environments (See Additional file 2: Figure S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The uptake of K+ ions remains enigmatic for most soda lake CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Low-affinity Trk-type K+ uptake transporters (gen- erally with symport of H+) (67,68) were encoded only by a limited number of MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We found three MAGs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 4 Relative abundance and metabolic potential of the dominant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Abundance values, expressed as reads per kilobase of MAG per gigabase of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets (cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The metabolic potential derived from functional marker genes (Additional file 7) is depicted by the colored symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' = fixation, red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' = reduction, ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' = oxidation, dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' = disproportionation Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 7 of 18 a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 5 (See legend on next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=') Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 8 of 18 encoding for Kdp-type sensor kinases (kdpD) but no corresponding genes for the response regulator (kdpE) or for Kdp-ATPases that function as the inducible, high- affinity K+ transporters in other Bacteria (67,68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Finally, mechanosensitive ion channels (mscS, mscL) and ABC- type multidrug transport systems (AcrAB, ccmA, EmrA, MdlB, NorM) and sodium efflux permeases (NatB) were encoded in almost every MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The first might rapidly restore the turgor pressure under fluctuating salinity conditions by releasing cytoplasmic ions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Novel abundant groups involved in sulfur, nitrogen, and carbon cycles A new species of Thioalkalivibrio (family Ectothiorhodospir- aceae) was by far the most abundant in the sediments of the two salt-saturated lakes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In the sediment of Bitter-1, also a purple sulfur bacterium from the same fam- ily was highly abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' It was closely related to Halorho- dospira, a genus also frequently cultured from hypersaline soda lakes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' None of the abundant Ectothiorhodospira- ceae spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' had already a species-representative genome sequenced (Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The potential of the Thioalk- alivibrio spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' for chemolithotrophic sulfur oxidation was evident (Additional file 7; see Additional file 8: Information S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Interestingly, the encoded nitrogen metabolisms were quite versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' While Thioalkalivibrio sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 1 had the poten- tial for nitrate reduction to nitrite, Thioalkalivibrio sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2 might perform dissimilatory nitrite reduction to ammonia (DNRA; see Additional file 2: Figure S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Two deltaproteobacterial lineages of dissimilatory sulfate-reducing bacteria (SRB) were highly abundant in the soda lake sediment of Bitter-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' One MAG from the family Desulfobacteraceae (order Desulfobacterales) is the first genome from the genus Desulfonatronobacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' It encodes the genes for complete sulfate reduction to sul- fide using various electron donors, as well as for the complete oxidation of volatile fatty acids and alcohols, a unique feature for the genus Desulfonatronobacter among haloalkaliphilic SRB (see Additional file 8: Information S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Fumarate and nitrite (DNRA, NrfAH) could be used as alternative electron acceptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The sec- ond dominant lineage was a new species from the genus Desulfonatronospira (family Desulfohalobiaceae, order Desulfovibrionales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Like other members of this genus, it had the potential to reduce or disproportionate partially reduced sulfur compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In addition, it could also use nitrite as an alternative electron acceptor (NrfAH) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' A novel lineage of gammaproteobacterial SOB was highly abundant in the sediments of the moderately hy- persaline Cock Soda Lake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' It appeared as a sister group of the family Xanthomonadaceae in the ribosomal protein tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' This heterotrophic organism could conserve energy through aerobic respiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' It might detoxify sulfide by oxidizing it to elemental sulfur (sqr) with subsequent re- duction or disproportionation of the polysulfides (psrA) chemically formed from the sulfur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' It also encoded the po- tential for DNRA (nrfA and napC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Genes likely involved in sulfide detoxification (sqr and psrA) were found also in several other abundant MAGs of heterotrophs, including one new abundant species from the family of Nitrilirup- toraceae (class Nitriliruptoria, phylum Actinobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We found a wide variety of carbohydrate-active enzymes in these MAGs, such as cellulases (GH1 family) in addition to genes for glycolysis and TCA cycle and a chlorophyll/bacteriochlorophyll a/b synthase (bchG gene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The latter was also found in other Actinobacteria from the genus Rubrobacter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' No evidence was found for nitrile-degrading potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' A second novel, uncultured lineage of Gammaproteo- bacteria that was highly abundant at moderate salinities branched in our ribosomal protein tree as a sister group to the family Halothiobacillaceae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The MAGs encoded for a versatile metabolism typical for purple non-sulfur bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The MAGs contained puf genes, bch genes, genes for carotenoid biosynthesis (not shown), and a Calvin cycle for photoautotrophic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Alternatively, energy may be conserved through aerobic respiration, while acetate and proprionate could be taken up via an acetate permease (actP) and further used for acetyl-CoA biosynthesis and carbon assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Since the sqr gene was present, but no dsr or sox genes, the organism might oxidize sulfide only to elemental sulfur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' One bin contained also nifDKH genes suggesting putative diazo- trophy, as well as a coenzyme F420 hydrogenase suggest- ing photoproduction of hydrogen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The abundant Euryarchaeota organism showed a clear preference for higher salinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We obtained one highly abundant MAG from the class Thermoplasmata that encoded a full-length 16S rRNA gene only distantly re- lated (91,2% identity, e value 0) to that of a member of the KTK 4A group found in a hypersaline endoevaporitic microbial mat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The abundant soda lake organism is likely a new genus and species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All KTK 4A-related MAGs found here encoded for similar heterotrophic, fermentative metabolisms, with the potential for (See figure on previous page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' a Membrane transporters, channels, and lipids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Peptidoglycan is depicted in gray and S-layer proteins in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' b Predicted isoelectric points (bin width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='2) for the coding sequences of MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also Additional file 2: Figure S11) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 9 of 18 anaerobic formate and CO oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The KTK 4A might be also primary degraders since they encoded pu- tative cellulases (CAZY-families GH1, GH5) and chiti- nases (GH18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Interestingly, half of the MAGs encoded a putative chlorophyll/bacteriochlorophyll a/b synthase (bchG), which is highly unusual for Archaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Although little can be inferred from the presence of only one marker gene, a functional bchG was previously also found in Crenarchaeota .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The remaining two highly abundant Euryarchaeota-related MAGs belonged to a new species of Halorubrum (Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Key genes of the Wood-Ljungdahl pathway found in novel phylogenetic groups More than 50 MAGs were related to the family Syntro- phomonadaceae (class Clostridia, phylum Firmicutes) based on ribosomal protein phylogeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All 16S rRNA gene sequences found in the MAGS had 86–95% iden- tity to sequences obtained from uncultured organisms related to the genus Dethiobacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' While an isolated strain of Dethiobacter alkaliphilus is a facultative auto- troph that respires thiosulfate, elemental sulfur or polysulfides with hydrogen as an electron donor or disproportionates sulfur , other haloalkaliphilic members of the Syntrophomonadaceae are reverse acetogens, oxidizing acetate in syntrophy with a hydro- genotrophic partner .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Two populations (different species, Additional file 6) were especially abundant in Cock Soda Lake (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' They encoded for a full CODH/ACS complex, the key enzyme for the reductive acetyl-CoA or Wood-Ljungdahl pathway (WL) and a complete Eastern branch for CO2 conversion to 5-methyl-tetrahydrofolate (Additional file 9) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Acetogens use the WL to reduce CO2 to acetyl-CoA, which can be fixed into the cell or used to conserve en- ergy via acetogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Syntrophic acetate oxidizers, some sulfate reducing bacteria and aceticlastic methanogens run the WL in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Syntrophomonadaceae sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2 encoded for a putative thiosulfate/polysulfide reductase as well as a phosphotransacetylase (pta) and an acetate kinase (ack) for the ATP-dependent conversion of acet- ate to acetyl-CoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Although alternative pathways for the latter interconversion can exist, this second species has the complete potential for (reversed) acetogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Highly remarkable was the presence of a bacterial-type CODH/ACS complex and a near-complete eastern branch of the WL in a highly abundant species in Cock Soda Lake from the family Coriobacteriaceae (phylum Actinobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' This prompted us to scan all 871 MAGs for the presence of acsB encoding for the beta-subunit of the oxido-reductase module of CODH/ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We con- firmed an encoded (near)-complete WL in several additional organisms belonging to phylogenetic groups not previously associated with this pathway (Additional file 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We removed the Coriobacteriaceae acsB genes from the final dataset to construct a phylo- genetic tree since they were < 500 aa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 6) but found seven MAGs from the OPB41 class within the Actino- bacteria (16S rRNA gene fragment identity 94–96%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The eastern branch of WL can function independently in folate-dependent C1 metabolism , but the pres- ence of the Western-branch in a phylum that comprises mostly aerobic isolates is very surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The WL in combination with the potential for acetate to acetyl-CoA interconversion (pta/ack) and a glycolysis pathway were also present in the soda lake MAGs from the phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Handelsmanbacteria,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Atribacteria” (latter branched within the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Acetothermia”), and the class LD1-PA32 (Chlamydiae), suggesting all these uncultured organisms might be heterotrophic acetogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' However, it should be noted that a PFOR typically connecting glycolysis to the WL was only encoded in the LD1-PA32 MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' More- over, from the genetic make-up alone, it cannot be excluded that acetate is activated, and the WL run in reverse for syntrophic acetate oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Finally, the novel acsB genes from soda lake Halanaerobiaceae, Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) and from Brocadiaceae and Planctomycetaceae (Plancto- mycetes) disrupt the previously proposed dichotomy between Terrabacteria and Gracilicutes bacterial groups unifying 16S rRNA and acsB gene phylogenies and suggest a far more complex evolutionary history of the WL pathway than previously anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Discussion Extensive classical microbiology efforts have been already undertaken to explore the unique extremophilic microbial communities inhabiting soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' These un- covered the presence of most of the functional groups participating in carbon, nitrogen, sulfur, and minor element cycling at haloalkaline conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The main re- sults of this work are summarized in several recent re- views .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Since most microbes, including those living in soda lakes, still evade all cultivation ef- forts, a very effective way to discover new microbes and assess their physiology based on their genetic repertoire is either through single cell genomics or by directly se- quenced environmental DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' This exploratory metage- nomics study performed on soda lake sediments effectively overcame the existing cultivation bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' First, we expanded the known diversity of CPR consider- ably with the first genomes of poly-extremophiles sam- pled from soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Although the presence of 16S rRNA genes from CPR in marine sediments and hy- persaline microbial mats was previously shown , until now, CPR MAGs were mainly obtained from deep, subsurface environments [15, 26, 29, 32, 49–52], and hu- man microbiota .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Despite being highly abundant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 10 of 18 100 % 90-100 % 70-90 % 50-70 % some MAGs all MAGs Bootstraps Genes present Glycolysis (EMP) PFOR WL-Eastern branch H4MPT TH4 WL-Western branch CODH/ACS Acetogenesis/ acetate activation (pta/ack) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='4 PVC group (Chlamydiae LD1-PA32) Syntrophorhabdus aromaticivorans PVC group bacterium CSSed11_184 Aerophobetes bacterium SCGC_AAA255-F10 Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Acetothermia Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Handelsmanbacteria Planctomycetaceae Anaerolineae Firmicutes Brocadiaceae Planctomycetes Methanomassiliicoccales Halobacteroidaceae Natranaerobiaceae Methanomicrobiales Desulfonatronospira Firmicutes Dehalococcoidia Armatimonadetes bacterium CSP1-3 Deltaproteobacteria Thermodesulfobacteria Desulfobulbaceae Halanaerobiaceae Nitrospirae Actinobacteria (OPB41) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Only sequences ≥ 500 aa were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see also Additional file 9: Dataset S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Additional lineages found in this study are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The three was rooted according to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Circles at the nodes show confidence percentage of the bootstraps analysis (100×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' PVC group bacterium CSSed11_184 is likely a member of the WCHB1-41 class within the Verrucomicrobia Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 11 of 18 here, CPR went unnoticed in previous amplicon sequen- cing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' This might be due to the fact that many CPR representatives have random inserts of various length in their 16S rRNA genes or due to primer mis- matches .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' This illustrates also that direct metage- nomics should not only be preferred over amplicon sequencing to infer functional potential, but the former is far more effective for the discovery of novel organ- isms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Second, we obtained many more genomes from “traditional” bacterial phyla such as the Planctomycetes and Chloroflexi, as well as candidate phyla, for which no soda lake isolates, hence no genomes were previously obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Third, even within the sulfur cycle, the most active and frequently studied element cycle in soda lakes , we found considerable metabolic novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Finally, we found the Wood-Ljungdahl pathway in several novel phyla, not closely related to any known acetogens, methanogens, or sulfate-reducing bacteria .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The lat- ter shows that our sequencing recovery effort has also significantly contributed to the discovery of metabolic novelty within various prokaryote phylogenetic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Salinity is often considered to be the major factor shaping prokaryote community composition in diverse habitats .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Extreme halophilic Euryarchaeota seem to be always the dominant group in salt-saturated hypersaline brines, both those with neutral or alkaline pH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Here, we found that although these haloarchaea are still relatively more abundant in the sed- iments exposed to brines with salt-saturating conditions, the clear majority of microbes in all investigated hyper- saline soda lake sediments are Bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' It could be hypothesized that the sediment is a hide-out for the extreme alkalinity and salinity governing the water column, and that sediment stratification, especially in the anoxic part, offers plenty of opportunities for niche diversification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' On the other hand, it should no longer be a surprise that soda lakes are such productive and biodiverse systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' First, it has been previously elaborated that soda lake organisms are exposed to approximately half the osmotic pressure in sodium carbonate-dominated brines compared to sodium chloride-dominated brines with the same Na+ molarity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Second, nitrogen limitation in the community can be overcome when many members contribute to the fixation of atmospheric N2, and various forms of organic nitrogen are efficiently recycled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The soda lakes exam- ined in this study were also eutrophic, and sulfur com- pounds were abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sulfide is also far less toxic at high pH as it mostly occurs in the form of bisulfide (HS−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Besides the evident high metabolic and taxo- nomic diversity of dissimilatory sulfur-cycling bacteria, a diverse heterotrophic community can be sustained com- prising both generalist and very specialized carbon de- graders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Less eutrophic soda lakes might not suffer from carbon limitation either, due to a presence of high-bicarbonate concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' These effectively elim- inate the inorganic carbon limitation for primary pro- ducers who are highly active in soda lakes, especially Cyanobacteria .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Third, light that penetrates the surface of the sediment seems to stimulate oxygenic and anoxygenic phototrophic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Moreover, various het- erotrophs, such as the rhodopsin-containing haloarchaea and Bacteroidetes, have the option to tap into this un- limited energy source for example to help sustain the costly maintenance of osmotic balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Unexpectedly, we even found the first rhodopsin encoded by a member of the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Fourth, tight syntrophic relations, as pro- posed for CPR members and Syntrophomonadaceae spp., might be the solution to successful growth in an energetically challenging environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Since our metagenomes are snapshots in time and space, the failure to reconstruct specific MAGs gives no conclu- sive evidence for the absence of certain microbial-mediated element transformation in hypersaline soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Additionally, technical limitations of the assembly and bin- ning of highly micro-diverse genome populations might hamper genome recovery .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' More importantly, the abundance of a specific microbe is not necessarily corre- lated to the importance of its performance in an ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Many metabolic capacities are redundant, and often key transformations are reserved for a few rare organisms that might proliferate for a short time-span when specific condi- tions allow for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' For example, although no MAGs were re- covered from chemolithoautotrophic nitrifiers , we did detect a Nitrobacter-related OTU by amplicon sequencing and assembled 16S rRNA genes from Thaumarchaeota, suggesting bacterial and archaeal nitrifiers are present in the surface sediments of soda lakes at very low abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Finally, the method of DNA isolation might impact the community composition apparent in the final metagenome sequenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Environmental samples contain complex mix- tures of different organisms, and it is impossible to find a protocol where the DNA from every single organism is ex- tracted as efficiently without compromising the final quality of the extracted DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' However, since we find all the im- portant taxonomic and functional groups known from pre- vious cultivation-dependent studies back in either our amplicon sequencing datasets or our directly sequenced metagenomes, we are confident that the community com- position and the MAGs presented here are representative for the microbiomes of the soda lake sediments in the Kulunda Steppe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Conclusion Years of intensive microbiological research on soda lakes seem to have paid off, since many of the described gen- era we could detect here have a cultured representative from soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' However, as many of the abundant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 12 of 18 lineages and groups found in soda lake sediments are still uncultured, metagenomics proved to be a helpful tool to gain primary insights in the potential physiology and ecology of these poly-extremophilic prokaryotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We reconstructed the first genomes for many of such organisms and proposed new functional roles for the most abundant ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Future studies should provide more in depth analyses of these genomes, especially from the less abundant organisms that might perform key ecological processes, such as methanogens and nitri- fiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In addition, they should focus on gaining physio- logical culture-based evidence or proof for in situ activity for the abundant organisms described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The key metabolic insights provided by this metagenomics study can lead to the design of new cultivation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' In general, sediment communities are far more complex than those found in the corresponding water column and are therefore often considered too complex for efficient metagenomic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Many of the novel lineages found here may therefore have related neutro- philic lineages in marine and freshwater sediments that await discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We demonstrate here that, by providing sufficient sequencing depth, the “state of the art metage- nomics toolbox” can effectively be used on sediments as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Methods Site description and sample collection The top 10 cm sediments from four hypersaline, eutrophic soda lakes located in the Kulunda Steppe (south-western Siberia, Altai, Russia) were sampled in July of 2010 and 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' General features and exact location of the sampled soda lakes are summarized in Additional file 1: Table S1; a map of the area was published previously .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Cock Soda Lake (a stand-alone lake, sampled both in 2010 and 2011) and Tanatar-3 (Tanatar system) were moderately hypersa- line (~ 100 g L−1) with sandy sediment, while Tanatar-1 and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) with sulfide-rich sapropel sediments underlined by rock trona deposits .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Especially, Bitter-1 harbors a very active microbial community, probably due to its high- organic and -mineral content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Surface sediments were col- lected by a plastic corer into sterile glass containers and transported to the laboratory in a cooler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' DNA isolation, 16S rRNA amplicon, and metagenomic sequencing The colloidal fraction of each sediment sample (~ 10% of 50 g) was separated from the course sandy fraction by several short (30–60 s) low-speed (1–2,000 rpm in 50 mL Falcon tubes) centrifugation steps and washed with 1–2 M NaCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The pelleted colloidal sedi- ment fraction was first subjected to 3 cycles of freezing in liquid nitrogen/thawing, then re-suspended in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='1 M Tris (pH 8)/10 mM EDTA, and then subjected to harsh bead beating treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Next, the samples were incu- bated with lysozyme (15 mg/mL) for 2 h at 37 °C followed by a SDS (10% w/v) and proteinase K (10 μg/ mL) treatment for 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' at 45 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' High molecular weight DNA was isolated using phenol/chloroform ex- traction, quality-checked, and sequenced as described previously .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Direct high-throughput sequencing of the DNA was performed on an Illumina HiSeq 2000 plat- form to generate 150 b paired-end reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Amplification of the V4-V6 region of prokaryote 16S rRNA genes using barcoded 926F-1392R primers, amplicon purifica- tion, quantification, and Roche (454)-sequencing was performed together in a batch with brine samples from the same sampling campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Barcodes and adapter se- quences were removed from de-multiplexed amplicon sequence reads and analyzed with the automated NGS analysis pipeline of the SILVA rRNA gene database pro- ject (SILVAngs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='3, database release version 128) using default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The OTUs (97% identity) assigned down to the genus level were only considered when they had a relative abundance ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='1% in at least one of the five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Processing metagenomics reads, assembly, binning, and post-binning Metagenomic raw reads were quality trimmed using Sickle (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='33), and only reads ≥ 21 b were retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The prokaryotic community structure at taxo- nomic top levels was extrapolated from ten million ran- domly sampled singletons from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Candidate 16S rRNA fragments > 90 b were identified and compared against the SILVA SSU database 128 (blastn, min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' length 90, min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' identity 80%, e value 1e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' To ver- ify that the microbial community composition was in- deed mostly prokaryotic, we did a more general screening of the metagenomics reads that identified also candidate 18S rRNA fragments > 90 b (see Additional file 1: Tables S4-S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The complete trimmed read sets were assembled into contigs ≥ 1 kb with MEGAHIT (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='3–6-gc3983f9) using paired-end mode, k min = 21, k max = 131, k step = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Genes were predicted using Prodigal (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='2) and RNAs with rna_hmm3 and tRNAscan-SE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Assembled 16S rRNA sequences were compared to a manually curated version from the SILVA SSU database (e value ≥ 1e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Predicted protein sequences were annotated against KEGG with GhostKOALA (genus_prokaryotes + family_eukaryotes + viruses) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Marker genes for central metabolic pathways and key environmental element transforma- tions were identified based on K number assignments [15, 69–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Contigs ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='5 kb were binned with METABAT (superspecific mode) based on differential coverage Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 13 of 18 values obtained by mapping all five trimmed readsets to all five contig sets with Bowtie2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The bins were sub- jected to post-binning (an overview of the workflow is given in Additional file 2: Figure S13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Bins were assessed with lineage-specific single copy genes using CheckM and further processed with the metage- nomics workflow in Anvi’o (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Since Candidate Phyla Radiant (CPR) is not included in the CheckM ref- erence trees and are likely to have low-genome com- pleteness, we used an existing training file of 797 CPR genomes to identify putative CPR bins .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Bins with CheckM-completeness ≥ 50% (884 out of 1778) and an additional four CPR bins were further processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Coding sequences were annotated for taxonomy against NCBI-nr (July, 2017) with USEARCH (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='32) to verify that most hits in each bin were to prokaryotic ref- erences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Phage or viral contigs were manually removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Genome contamination (redundancy) was estimated based on marker sets of universal single copy genes identified for Bacteria and Archaea as imple- mented in Anvi’o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Genome coverage was obtained by mapping trimmed reads with BBMap v36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='x (kfilter 31, subfilter 15, maxindel 80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Bins with ≥ 5% redun- dancy were further refined with Anvi’o using circle phy- lograms (guide trees tnf-cov: euclidian ward) and scanned again for CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Post-binning resulted in a total of 2499 metagenome-assembled genomes (MAGs), of which 871 were either medium-quality genome drafts (CheckM estimated completeness ≥ 50% and contamin- ation ≤ 10% , Additional file 4) or lower quality draft genomes from CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Phylogeny of the MAGs was assessed based on 16 single-copy ribosomal proteins and representative refer- ence genomes of major prokaryote lineages across the tree of life .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Individual ribosomal proteins in our MAGs were identified by K number assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Only ribosomal proteins ≥ 80 aa were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Initial maximum-likelihood (ML) trees were constructed to de- termine which organisms belonged to the Archaea, Bac- teria, or CPR with FastTree 2 (WAG + CAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Final separate trees for the three distant evolutionary groups were constructed in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Each ribosomal protein set was aligned separately with MAFFT (v7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='055b, − auto) and concatenated only if a MAG encoded at least 8 out of 16 proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' For all trees, a 100× posterior bootstraps analysis was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Phylogenetic trees were visualized together with gen- ome statistics and abundance information using iTOL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We cross-checked the taxonomic assignments based on the phylogeny of the ribosomal protein cas- sette with the top hit contig annotations against NCBI-nr and with the reference lineage obtained with CheckM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Lastly, we manually corrected the MAGs for misplaced 16S rRNA genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The final trees presented in the manuscript were redrawn using FigTree v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Detailed genome analyses CPR MAGs were re-annotated more thoroughly: genes were predicted with Prokka , and functional predictions were performed by running InterProScan 5 locally on the supplied COG, CDD, TIGRFAMs, HAMAP, Pfam, and SMART databases .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' BLAST Koala was used for KEGG pathway predictions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' To find putative carbohydrate-active enzymes in all final MAGs, we used the web-resource dbCAN to annotate all predicted proteins ≥ 80 aa against CAZy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' To identify the top ten abundant MAGs from each re- spective dataset, ten million randomly sampled single- tons were mapped onto each MAG with a cut-off of 95% identity in minimum of 50 bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Coverage values were additionally normalized for genome size and expressed as reads per kilobase of sequence per gigabase of mapped reads (RPKG) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' A positive score (from 871 to 1) was assigned to each MAG according to the rank- ing of the summed RPKG of MAGs in the high-salinity datasets (B1Sed10 and T1Sed) and a negative score ac- cording to the ranking of the summed RPKGs in the moderate salinity datasets (CSSed10, CSSed11, T3Se d10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Both scores were summed to get a “salinity prefer- ence score” with MAGs recruiting preferably from high salinity datasets on the positive end, moderate salinity datasets in the negative end, and those without prefer- ence in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We determined species delineation for the most abundant MAGs and their closest reference genomes (NCBI-nr) by Average Nucleotide Identity (ANI) and conserved DNA-matrices, as follows : ANI ≥ 95%, conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA < 69% = might be same species, ANI < 95%, condDNA < 69% = different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Single gene trees based on maximum likelihood were constructed with un- trimmed alignments (MAFFT, L-INS-i model) and FastTree 2 (WAG + CAT, increased accuracy, -spr4 mlacc 2 -slownni) using 100× bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' References were pulled from eggNOG (v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='1) and supple- mented with sequences from NCBI-nr or refined according to [7, 33, 46, 92–94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The curated MAGs were scanned for the presence of rhodopsin sequences with the hmmsearch software and a profile hidden Markov model (HMM) of the bacteriorhodopsin-like protein family (Pfam accession number PF01036).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The identified sequences with significant similarity were aligned together with a curated database composed of a collection of type-1 rhodopsins, using MAFFT (L-INS-i accuracy model) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' This protein alignment was further utilized to Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 14 of 18 construct a maximum likelihood tree with 100× boot- strap with FastTree 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All other genes were identified using the KEGG annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Additional files Additional file 1: Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' General features of the four sampled soda lakes at time of sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' SILVA classification of the 16S rRNA gene sequences found in all ≥1 kb contigs of five soda sediment metagenomic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Enzymes involved in lipopolysaccharide biosynthesis found among different members of the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sub-kingdom classification of candidate SSU rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Top-level taxonomic classification of the 18S rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Description of the metagenomic datasets, NCBI Sequence Read Archive (SRA) accession numbers and general statistics of the assembled contigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (PDF 740 kb) Additional file 2: Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Taxonomic fingerprints determined by 16S rRNA gene amplicon sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Genome statistics of the 871 MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Phylogeny of MAGs belonging to “Candidatus Aenigmarchaeota” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Nanohaloarchaeota”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Phylogeny of MAGs related to “Candidatus Acetothermia”, candidate division WS1 and “Candidatus Lindowbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Phylogeny of MAGs related to candidate division KSB3 and “Candidatus Schekmanbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Multiple sequence alignment of the V-type ATPase subunits K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Multiple sequence alignment of the F-type ATPase subunits c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- like proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Maximum likelihood tree of the putative rhodopsins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Predicted isoelectric points (pI) profiles of all MAGs from CPR members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Predicted isoelectric points profiles for members of the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Nealsonbacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Vogelbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Multiple sequence alignment of the dissimilatory cytochrome c nitrite reductases (nrfA/TvNiR, K03385).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Overview of the post-binning workflow used for genome recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (PDF 6548 kb) Additional file 3: Dataset S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Relative abundance of the OTUs assigned to the genus-level within the Archaea, Bacteria and organelles from Eukaryota detected by 16S rRNA gene amplicon sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The OTUs with less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='1% abundance accross all five datasets are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The names of highly abundant genera (≥1% in at least one of the data- sets) are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (XLSX 24 kb) Additional file 4: Dataset S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Organism names, statistics and general description incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Completeness and contamination estimates, phylogeny and DDBJ/EMBL/Genbank accession numbers of the metagenome assembled genomes (MAGs) described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All submitted versions described in this paper are version XXXX01000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Size = recovered genome size, Completeness (Compl1), contamination (Cont), strain heterogenity (Str het) and Taxon CheckM were inferred from lineage-specific marker sets and a reference tree build with CheckM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Additional completeness (compl2) and redundancy (red) estimates were inferred based on the presence of universal single copy genes for Bacteria and Archaea .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Decision and confidence intervals from the Candidate Phyla Radiation (CPR) scan are given, as well as the taxonomy of the besthit in SILVA when 16S rRNA genes were present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Phylum/class 16 ribosomal proteins is the taxonomy derived from our ribosomal protein trees (see main text: Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' OTU gives the inferred link of a population genome with our 16S rRNA gene amplicon dataset (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (XLSX 253 kb) Additional file 5: Dataset S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Estimated abundance and derived salinity preference from each MAG in each metagenomic dataset expressed as Reads per Kilobase of MAG per Gigabase of mapped reads (RPKG) and “salinity preference score” (see Methods section), basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (XLSX 143 kb) Additional file 6: Dataset S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Average Nucleotide Identity (ANI) and conserved DNA (condna) matrices to determine species delineation between the most abundant MAGs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 4, closely related (less abundant) MAGs and NCBI reference genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Decision matrix shows: 1 = same species, − 1 = might be same species, 0 = different species (see Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (XLSX 1161 kb) Additional file 7: Dataset S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sheet 1 Presence and absence of marker genes and putative carbohydrate-active enzymes in the MAGs to infer putative roles in C, N and S element cycles based on K-number assignments and CAZy annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sheet 2 Summary basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (XLSX 41 kb) Additional file 8: Information S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' More detailed description of the main metabolisms encoded by Thioalkalivibrio-related MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Information S2 More detailed description of the main metabolisms encoded by Deltaproteobacterial-related MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (PDF 219 kb) Additional file 9: Dataset 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sheet 1 shows the MAGs positive for the marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 6, namely presence and absence of key genes involved in the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis and pyruvate to CO2 conversion is shown for each MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sheet 2 shows the MAGs positive for the marker gene cdhC (K00193) encoding for the beta subunit of an acetyl-CoA decarboxylase synthase complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' While acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- type (methanogens) enzymes with the same function, we found few discrepancies between marker gene and genome phylogeny within the Methanomassiliicoccaceae and Chloroflexi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' (XLSX 52 kb) Acknowledgments We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Nikolai Chernych for his technical assistance during the isolation and purification of metagenomics DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' We also thank the Department of Energy Joint Genome Institute for sequencing the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Funding CDV and GM were supported by the ERC Advanced Grant PARASOL (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 322551).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' A-SA and RG were supported by the research grant 17-04828S from the Grant Agency of the Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' MM was supported by the Czech Academy of Sciences (Postdoc program PPPLZ application number L200961651).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' DYS was supported by the SIAM/Gravitation Program (Dutch Ministry of Education and Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- 00121).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sequencing was performed by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Department of Energy Joint Genome Institute, a DOE Office of Science User Facility, as part of the Community Sequencing Program (contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' DE-AC02- 05CH11231).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Availability of data and materials The raw sequence reads of the five metagenomes have been deposited to the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession numbers and read and contig statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The final 871 MAGs described in this paper have been deposited as Whole Genome Shotgun projects at DDBJ/ EMBL/GenBank, and accession numbers are listed in Additional file 4 (BioProject ID PRJNA434545).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All versions described in this paper are version XXXX01000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' The cleaned and dereplicated amplicon sequence datasets are available in FigShare (https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='com/s/7684627445e3621aba24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Maximum likelihood trees based on the concatenated alignment of 16 ribosomal proteins, basis for Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2 and 3, in newick format (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='tre file) and complementary datasets (used to plot completeness, contamination, genome recovery size, G + C mol% and RPKG in iTOL), as well as K number assignments for the predicted proteins of all MAGs (KEGG-orthologues, Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions of this article are also available in FigShare (https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content='com/s/ 7684627445e3621aba24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Authors’ contributions GM and DYS initiated this study and were responsible for the fieldwork, sample preparation, and sequencing effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' CDV conceptualized the research goals under supervision of DYS and GM, and performed the bioinformatics analysis under close guidance of A-SA and RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' CDV is the primary author of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' MM, RG, and CDV prepared the main figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' All authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Ethics approval and consent to participate Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbiome (2018) 6:168 Page 15 of 18 Consent for publication Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Author details 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2Department of Aquatic Microbial Ecology, Institute of Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 3Winogradsky Institute of Microbiology, Research Centre of Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2, Moscow, Russian Federation117312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 4Environmental Biotechnology, Department of Biotechnology, Delft University of Technology, Van der Maasweg 9, 2629, HZ, Delft, the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Received: 23 June 2018 Accepted: 3 September 2018 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Microbial diversity and biogeochemical cycling in soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' Extremophiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2014;18:791–809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} +page_content=' 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_50/content/kb_50.pdf'} diff --git a/kdFIT4oBgHgl3EQfrCu8/content/tmp_files/2301.11330v1.pdf.txt b/kdFIT4oBgHgl3EQfrCu8/content/tmp_files/2301.11330v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f479348d54a47f3699d6907664feaa67c5eee92 --- /dev/null +++ b/kdFIT4oBgHgl3EQfrCu8/content/tmp_files/2301.11330v1.pdf.txt @@ -0,0 +1,948 @@ +Conservative Safety Monitors of Stochastic +Dynamical Systems +Matthew Cleaveland1, Ivan Ruchkin2, Oleg Sokolsky1, and Insup Lee1 +1 University of Pennsylvania, Philadelphia, PA, USA 19104 +2 University of Florida, Gainesville, FL, USA, 32611 +Abstract. Generating accurate runtime safety estimates for autonomous +systems is vital to ensuring their continued proliferation. However, accu- +rately reasoning about future system behaviors is generally too complex +to do at runtime. To better reason about system safety at runtime, we +propose a method for leveraging design time model checking results at +runtime. Specifically, we model the system as a probabilistic automa- +ton (PA) and compute bounded time reachability probabilities over the +states of the PA at design time. At runtime, we combine distributions of +state estimates with the safety probabilities from design time to produce +a bounded time safety estimate. We argue that our approach produces +well calibrated safety probabilities, assuming the estimated state distri- +butions are well calibrated. We evaluate our approach using a case study +of simulated water tanks. +Keywords: Runtime Monitoring · Probabilistic Model Checking · Cal- +ibrated Prediction +1 +Introduction +As autonomous systems see increased use and perform critical tasks in an open +world, reasoning about their safety and performance is critical. In particular, it +is vital to know if a system is likely to reach an unsafe state in the near future. +The field of predictive runtime monitoring offers ways for performing this rea- +soning. The basic idea behind predictive runtime monitoring is to reason about +the expected future behaviors of the system and their properties. However, accu- +rately computing future system states is computationally infeasible at runtime, +as it requires running expensive reachability analysis on complex models. Previ- +ous works have computed libraries of reachability analysis results at design time +and used them at runtime [9]. But these approaches require the system dynamics +to have certain invariances to reduce the number of times reachability analysis +must be called offline. +Other lines of work use system execution data to learn discrete probabilistic +models of the system, which are then used to perform predictive runtime mon- +itoring, as there is rich literature for runtime monitoring of discrete automata. +These models range from discrete-time Markov chains (DTMCs) [2] to hidden +Markov models (HMMs) [4] to Bayesian networks [17]. However, it is difficult +arXiv:2301.11330v1 [cs.LO] 26 Jan 2023 + +2 +M. Cleaveland et al. +to provide guarantees relating the performance of the automata models to the +real system, due to the fact that they are fit using finite data. Of particular +interest is ensuring the models are conservative: it is essential to avoid run-time +overconfidence in the safety of the dynamical system. +In this paper, we propose a method for predictive run-time monitoring of +safety probabilities that builds on the strengths of the existing works. We use a +mix of conservative modeling techniques and data-driven modeling techniques to +transform the dynamical system into a probabilistic automaton (PA).3 We then +employ probabilistic model checking (PMC) to compute the safety of the model +over all its states offline. Finally, we synthesize lightweight monitors that rely on +the model checking results and a well-calibrated state estimator to compute the +probability of system safety at runtime. +Under the assumption that the PA model is conservative and that the state +estimator is well-calibrated, we prove that our runtime monitors are conserva- +tive. We argue that our modeling technique is likely to result in conservative PA +models — and empirically demonstrate this fact in the experiments. Finally, we +demonstrate that our method produces well-calibrated, accurate, and conserva- +tive monitors on a case study using water tanks. +The contributions of this paper are threefold: +– We present a method for conservatively modeling dynamical systems as PAs +and using PMC results at runtime to monitor the system’s safety. +– We prove that if our PA models are conservative then the monitor safety +estimates will be conservative. +– We demonstrate our approach on a case study of water tanks. We empirically +show that our PA models and runtime monitors are both conservative. +The rest of the paper is structured as follows. We give an overview of the +related work in Section 2, provide the necessary formal background in Section 3, +and formulate the problem in Section 4. Section 5 goes over our proposed ap- +proach and Section 6 provides formal conservatism guarantees for the approach. +We describe the results of our case study in Section 7 and conclude in Section 8. +2 +Related Work +We divide the previous works in the area of predictive runtime monitoring into +two bins. The first bin analyzes dynamical system models, while the second +analyzes automata models. +Dynamical systems approaches +A large portion of the predictive monitor- +ing for dynamical systems literature focuses on reasoning about the safety of +autonomous vehicles. Prior work has employed reachability analysis to estimate +the future positions of other cars to estimate the safety of a proposed vehicle +3 In our scope, PAs are equivalent to Markov Decision Processes (MDPs) without +rewards: both have finite states with probabilistic and non-deterministic transitions. + +Conservative Safety Monitors of Stochastic Dynamical Systems +3 +trajectory [1]. In [18], the authors develop techniques to estimate the probabil- +ity of a proposed trajectory resulting in a collision with other vehicles, which +are given as distributions of states predicted by neural networks (NNs) [18]. In +[9], the authors use precomputed reachability analysis and Bayesian inference to +compute the probability of an autonomous vehicle colliding with static obsta- +cles. However, this approach requires the system dynamics to have translational +and rotation invariances to ensure the reachability analysis can be feasibly run +at design time. This approach is conceptually similar to ours, using computa- +tional power at design time to precompute system behaviors. But our approach +employs automata-based abstractions instead of making invariance assumptions +about the system dynamics. +Previous works have also addressed the problem of synthesizing runtime +monitors for signal temporal logic (STL) properties of dynamical systems. Ap- +proaches range from conformal prediction [8,25], design time forward reachability +analysis [34], computing safe envelopes of control commands [33], online linear +regression [15], and uncertainty aware STL monitoring [26]. +Automata approaches The first works of this type developed predictive LTL +semantics, also called LTL3 [35,24], for discrete automata. The LTL3 semantics +allowed to the system to determine if every infinite extension of an observed +finite trace would satisfy or not satisfy a specification. Recent work has extended +these ideas to timed systems [28], multi-model systems [12], and systems with +assumptions [10]. Another approach uses neural networks to classify if unsafe +states of a hybrid automata can be reached from the current state of the HA +[5,6,7]. They additionally use conformal prediction to get guarantees about the +accuracy of their predictions [32]. However, these frameworks give very coarse +predictions, as they can only determine if a system is guaranteed to be safe, +guaranteed to be unsafe, or not sure. +Another thread of works uses data to learn probabilistic models from data. +These models can then be used in conjunction with predictive monitoring tech- +niques. In [4], the authors learn an HMM model of the system from simulation +data and perform bounded reachability analysis to determine the probability of +an LTL specification being violated from each state of the HMM. This work +was extended using abstraction techniques to simplify the learned models [3]. In +[2], the same authors employ importance sampling to efficiently learn discrete +time Markov chain (DTMC) models from data, which they then use to synthe- +size predictive monitors. In [17], the authors use Bayesian networks to model +temporal properties of stochastic timed automata. The Bayesian networks are +updated online to improve their performance. Finally, in [13] the authors use +process mining techniques to learn predictive models of systems, which are in +turn used to synthesize predictive runtime monitors. An interesting line of future +work for us is exploring applying our runtime monitoring technique using these +models as they are updated from new observations online. +The most similar work to ours presents two methods for synthesizing predic- +tive monitors for partially observable Markov decision processes (POMDPs) [19]. + +4 +M. Cleaveland et al. +The first approach combines precomputed safety probabilities of each state with +POMDP state estimators to estimate the probability that the system will remain +safe. The main downside of this approach is that state estimation of POMDPs is +computationally expensive since the set of potential state distributions increases +exponentially due to the non-determinism in the model. The second approach +uses model checking of conditional probabilities to directly compute the safety +of the system based on the observation trace. A downside of this approach is +that it requires running model checking at runtime. Our method, on the other +hand, avoids expensive computations at run time while maintaining design-time +scalability through abstraction. +3 +Background +In the following Definitions 1 to 3, borrowed from Kwiatkowska et al. [23], we +use Dist(S) to refer to the set of probability distributions over a set S, ηs as +the distribution with all its probability mass on s ∈ S, and µ1 × µ2 to be the +product distribution of µ1 and µ2. +Definition 1. A probabilistic automaton (PA) is a tuple M = (S, ¯s, α, δ, L), +where S is a finite set of states, ¯s ∈ S is the initial state, α is an alphabet +of action labels, (S, α, Dist(S)) ∈ δ is a probabilistic transition relation, and +L : S → 2AP is a labeling function from states to sets of atomic propositions +from the set AP. +If (s, a, µ) ∈ δ then the PA can make a transition in state s with action label +a and move based on distribution µ to state s′ with probability µ(s′), which is +denoted by s +a−→ µ. If (s, a, ηs′) ∈ δ then we say the PA can transition from state +s to state s′ via action a. A state s is terminal if no elements of δ contain s. A +path in M is a finite/infinite sequence of transitions π = s0 +a0,µ0 +−−−→ s1 +a1,µ1 +−−−→ . . . +with s0 = ¯s and µi(si+1) > 0. A set of paths is denoted as Π. We use M(s) to +denote the PA M with initial state s. +Reasoning about PAs also requires the notion of schedulers, which resolve +the non-determinism during an execution of a PA. For our purposes, a scheduler +σ maps each state of the PA to an available action label in that state. We use +Πσ +M for the set of all paths through M when controlled by scheduler σ and SchM +for the set of all schedulers for M. Finally, given a scheduler σ, we define a +probability space Prσ +M over the set of paths Πσ +M in the standard manner. +Given PAs M1 and M2, we define parallel composition as follows: +Definition 2. The parallel composition of PAs M1 = (S1, ¯s1, α1, δ1, L1) and +M2 = (S2, ¯s2, α2, δ2, L2) is given by the PA M1 || M2 = (S1 × S2, (¯s1, ¯s2), α1 ∪ +α2, δ, L), where L(s1, s2) = L1(s1)∪L2(s2) and δ is such that (s1, s2) +a−→ µ1 ×µ2 +iff one of the following holds: (i) s1 +a−→ µ1, s2 +a−→ µ2 and a ∈ α1 ∩ α2, (ii) +s1 +a−→ µ1, µ2 = ηs2 and a ∈ (α1 \ α2), (iii) µ1 = ηs1, s2 +a−→ µ2 and a ∈ (α2 \ α1). +In this paper, we are concerned with probabilities that the system will not +enter an unsafe state within a bounded amount of time. These are represented + +Conservative Safety Monitors of Stochastic Dynamical Systems +5 +as bounded-time safety properties, which we express using linear temporal logic +(LTL) [29]. Following the notation from [20], we denote these properties as +□≤T s /∈ Sunsafe, +where Sunsafe ⊂ S is the set of unsafe states and T ≥ 0 is the time bound. +Definition 3. For LTL formula ψ, PA M, and scheduler σ ∈ SchM, the prob- +ability of ψ holding is: +Prσ +M(ψ) := Prσ +M{π ∈ Πσ +M | π |= ψ}, +where π |= ψ indicates that the path π satisfies ψ in the standard LTL seman- +tics [29]. We specifically consider LTL safety properties, which are LTL speci- +fications that can be falsified by a finite trace though a model. Both ψnocol and +ψflow are LTL safety properties. +Probabilistically verifying an LTL formula ψ against M requires checking +that the probability of satisfying ψ meets a probability bound for all schedulers. +This involves computing the minimum or maximum probability of satisfying ψ +over all schedulers: +Prmin +M +(ψ) := infσ∈SchM Prσ +M(ψ) +Prmax +M +(ψ) := supσ∈SchM Prσ +M(ψ) +We call σ a min scheduler of M if Prσ +M(ψ) = Prmin +M +(ψ). We use Schmin +M +to +denote the set of min schedulers of M. +Remark: For the rest of this paper, we use Pr when referring to model check- +ing probabilities and P for all other probabilities. +Calibration and Conservatism Consider a scenario where a probability es- +timator is predicting probability ˆp that a (desirable) event E will occur (e.g., a +safe outcome). We define the calibration for the probability estimates (adapted +from Equation (1) of [16]): +Definition 4 (Calibration). The probability estimates ˆp of event E are well- +calibrated if +P(E | ˆp = p) = p, +∀p ∈ [0, 1] +(1) +Next, we define conservatism for the probability estimates: +Definition 5 (Conservative Probability). The probability estimates ˆp of a +desirable event E are conservative if +P(E | ˆp = p) ≥ p, +∀p ∈ [0, 1] +(2) + +6 +M. Cleaveland et al. +In other words, the estimates ˆp are conservative if they underestimate the +true probability of event E. Note that any monitor that is well-calibrated (Def- +inition 4) is guaranteed to be conservative (Definition 5), but not vice versa. +Two standard metrics for assessing the calibration of the ˆp estimates are +expected calibration error (ECE) [16] and Brier score [30]. The ECE metric +is computed by dividing the ˆp values into equally spaced bins in [0, 1], within +each bin taking the absolute difference between the average ˆp and the empirical +probability of event E, and weighted-averaging across bins with their sizes as +weights. So ECE penalizes discrepancies between the estimator confidence and +empirical probability of E within each bin. The Brier score is the mean squared +error of the probability estimates +� +i +( ˆpi − 1Ei)2 +4 +Problem Statement +Consider the following discrete-time stochastic system titled MOS with dynamics: +X(t + 1) = f(X(t), U(t))), +Y (t) = g(X(t), V (t)), +¯X(t), Z(t) = h(Z(t − 1), Y (t), W(t)), +U(t) = c( ¯X(t)), +(3) +where X(t) ∈ S ⊂ Rn is the system state (with bounded S); Y (t) ∈ Rp are +the observations; ¯X(t) ∈ Rn is the estimated state of the system; ¯Z(t) ∈ Rz +is the internal state of the state estimator (e.g., a belief prior in a Bayesian +filter or a set of particles in a particle filter); U(t) ∈ U ⊂ Rm is the control +output, which we discretize, resulting in a finite number |U| of control actions, +the functions f : Rn ×Rm → Rn, g : Rn ×Rv → Rp, h : Rz ×Rp ×Rw → Rn ×Rz +describe the system dynamics, perception map, and state estimator respectively; +the function c : Rp → Rm is a stateless controller; and V (t) ∈ Dv ⊆ Rv and +W(t) ∈ Dw ⊆ Rw describe perception and state estimator noise. The V (t) noise +models inexact perception, such as an object detector missing an obstacle or a +LiDAR scanner giving noisy distance estimates. The W(T) noise accounts for +state estimators that use randomness under the hood. A common example of +this is particle filters randomly perturbing their particles so that they do not +collapse to the exact same value. +Let Sunsafe ⊂ S denote the set of unsafe states of MOS. At time t, we are +interested in whether MOS will lie in Sunsafe at some point in the next T time +steps. This is represented by the bounded time reachability property +ψMOS = □≤T (X /∈ Sunsafe) +(4) +Let P(ψMOS | ¯Z(t)) denote the probability of MOS satisfying ψMOS. Our goal +is to compute calibrated (Definition 4) and conservative (Definition 5) estimates +of P(ψMOS | ¯Z(t)) at runtime, which we denote as “ +P(ψMOS | ¯Z(t)). + +Conservative Safety Monitors of Stochastic Dynamical Systems +7 +5 +Overall Approach +At a high level, our approach consists of a design time portion and a runtime +portion. At design time, a PA of the system (including its dynamics, perception, +state estimation, and controller) is constructed using standard conservative ab- +straction techniques. Then the bounded-time safety probability for each state +of the model is computed using model checking. This results in a look-up table +that maps the states of the abstract system to the safety probability of the real +system. At runtime, the estimated state (or distribution of states) from the real +system’s state estimator is used to estimate the abstract state (or distribution +of abstract states) of the abstract system. This abstract state (or distribution of +states) is used in conjunction with the lookup table to estimate the bounded-time +safety of the real system. +5.1 +Design Time +The design time aspect of our approach has two parts. First, we convert the orig- +inal system MOS into a probabilistic automaton MAS. Then we use probabilistic +model checking to compute the bounded time safety of MAS for each state in +the model. +Model Construction To convert MOS into a probabilistic automaton, MAS, we +first need to create probabilistic models of the perception g and state estimation +h components of MOS. To do this, we simulate MOS and record the perception +errors X(t) − ¯X(t). We discretize the domain of these errors and estimate a +categorical distribution over it. For example, this distribution would contain +information such as “the perception will output a value that is between 2m/s +and 3m/s greater than the true velocity of the car with probability 1/7.” +To convert the system dynamics f and controller c to a probabilistic automa- +ton, we use a standard interval abstraction technique. The high level idea is to +divide the state space S of MOS into a finite set of equally sized hyperrectangles, +denoted as S′. So every s′ +1 ∈ S′ has a corresponding region S1 ⊂ S. MAS then +has a transition from s′ +1 to s′ +2 (in MAS) if at least one state in S1 has a transi- +tion to a state in S2 (in MOS) under some control command u ∈ U. Note that +state s′ +1 can non-deterministically transition to multiple states in S′ because it +covers an entire hyperrectangle of states in MOS. This ensures that the interval +abstraction is conservative, as it overapproximates the behaviors of MOS. +Finally, the perception error model, controller, and interval abstraction are +all parallel-composed into a single model as per Definition 2. +Remark: In describing the construction of the MAS, we have not mentioned +anything about initial states: we do not keep track of a singular initial state +for MAS. Instead, we will later run model checking for the full range of initial +states of MAS to anticipate all runtime scenarios. For our purposes, the “initial +state-action space” of MAS consists of every abstract state and control action. +We include the control action in the initial state space because when using the +model’s safety probabilities online, we know what the next control action will be. + +8 +M. Cleaveland et al. +Safety Property We need to transform the bounded time safety property on +MOS given in Equation (4) into an equivalent property on MAS. To do this, we +compute the corresponding set of unsafe states on MAS, which is defined as +S′ +unsafe = {s′ | ∃s ∈ Sunsafe, s′ corresponds to s} +Letting s′ denote the state of MAS, the bounded time safety property for +MAS is +ψMAS := □≤T �s′ /∈ S′ +unsafe +� +(5) +Probabilistic Model Checking The final design-time step of our approach +computes the safety probability of MAS for every state in the model. This step +amounts to computing the below using standard model checking tools: +Prmin +MAS(s′,u)(ψMAS), ∀s′ ∈ S′, ∀u ∈ U +This requires running model checking on MAS for a range of initial states, +which can be a time-consuming process. To mitigate this, we note that MAS +is simpler to analyze than MOS, since the size of the state space gets reduced +during the interval abstraction process. Additionally, model checking this frag- +ment of LTL on PAs is NP-complete, whereas checking generic LTL formulas is +EXPTIME-complete [21]. So the time complexity of our model checking is lower +because we restrict ourselves to bounded time safety properties. +The probabilities from the model checking are stored in a lookup table, which +we denote as G(s′, u). It will be used at runtime to estimate the likelihood of +the system being unsafe in the near future. +5.2 +Runtime +At runtime, we use the lookup table G to estimate the near-future safety of +the system. At a high level, we observe the outputs of the state estimator and +controller and run them through the lookup table to compute the probability +of the system avoiding unsafe states for the next T time steps. We propose two +different ways of utilizing the state estimator. The first way is to simply use the +point estimate from the state estimator. In cases of probabilistic estimators, this +means taking the mean of the distribution. The second way uses the estimated +state distribution from the state estimator. This requires an estimator with a +probabilistic output, but most common state estimators, such as particle filters +and Bayesian filters, keep track of the distribution of the state. The second way +takes full advantage of the available state uncertainty to predict safety. +Point Estimate At time t the state estimator outputs state estimate ¯X(t). +The controller then outputs control command U(t) = c( ¯X(t)). Finally, we get a +safety estimate ˆP mon +point( ¯X(t), U(t)) by plugging ¯X(t) and U(t) into G: +ˆP mon +point( ¯X(t), U(t))) = G( ¯X(t), U(t)) +(6) + +Conservative Safety Monitors of Stochastic Dynamical Systems +9 +State Distribution Now assume that at time t state estimator additionally +outputs a state estimate ¯X(t) and a finite, discrete distribution of the state, +denoted as P ¯ +X(t). The controller still outputs control command U(t) = c( ¯X(t)). +To estimate the safety of the system, we compute a weighted sum of the safety +of each state in P ¯ +X(t) using G and U(t): +ˆP mon +dist (P ¯ +X(t), U(t)) = +� +s∈Supp(P ¯ +X(t)) +P ¯ +X(t)(s) · G(s′, U(t)) +(7) +where Supp +Ä +P ¯ +X(t) +ä +denotes the (finite) support of P ¯ +X(t), P ¯ +X(t)(s) denotes the +estimated probability of MOS being in state s according to P ¯ +X(t), and s′ ∈ S′ is +the state in MAS that corresponds to state s ∈ S in MOS. +6 +Conservatism Guarantees +This section proves that our state-distribution monitoring produces safety esti- +mates that are conservative and well-calibrated; that is, we underestimate the +probability of safety. We require two assumptions for that. The first assumption +is the conservatism of abstract model MAS, by which we mean that its probability +of being safe is always less than that of MOS for the same initial condition. The +second assumption is the calibration of the state estimator, which means that it +produces state probabilities that align with the frequencies of these states. Below +we formalize and discuss these assumptions before proceeding to our proof. +Definition 6 (Model Conservatism). Abstraction MAS is conservative with +respect to system MOS if +PMOS(s,u)(ψ) ≥ Prmin +MAS(s′,u)(ψ) ∀s ∈ S, u ∈ U +(8) +where s′ ∈ S′ is the state in MAS that corresponds to state s in MOS. +In general, it is difficult to achieve provable conservatism of MAS by con- +struction: the model parameters of complex components (e.g., vision-based per- +ception) are estimated from data, and they may have complicated interactions +with the safety chance. Instead, we explain why our approach is likely to be +conservative in practice and validate this assumption in the next section. +Consider MOS and MAS as compositions of two sub-models: dynamics/control +and perception/state estimation. We construct MAS such that its dynamics/control +component always overapproximates the dynamics/control portion of MOS. That +means that any feasible sequence of states and control actions from MOS is also +feasible in MAS. This follows from the use of reachability analysis over the in- +tervals of states to compute the transitions of MAS. +It is unclear how to formally compare the conservatism of perception/state +estimation portions of MAS and MOS when they are created from simulations of +the perception/state estimation component of MOS. First, these components are + +10 +M. Cleaveland et al. +not modeled explicitly due to the high dimensionality of learning-based percep- +tion. Thus, when estimating probabilities from samples, we essentially approxi- +mate the average-case behavior of the component. Second, it is often unknown +in which direction the probabilities need to be shifted to induce a conservative +shift to the model. One opportunity is to use monotonic safety rules [11]; for +now, this remains a promising and important future research direction. +To summarize, the dynamics/control portion of MAS overapproximates that +of MOS, while the perception/state estimation portion of MAS approximates the +average-case behavior of MOS. So one would expect, on average, MAS to be +conservative with respect to MOS, even though we cannot formally prove that. +Next, we define the calibration for the state estimator (adapted from Equa- +tion (1) of [16]): +Definition 7 (Calibration). Given the dynamical system from Equation (3) +and state estimator h that outputs a discrete, finite distribution of the estimated +state, denoted P¯x(t), we say that h is well-calibrated if +P(x(t) = s | P¯x(t)(s) = p) = p, +∀p ∈ [0, 1] +(9) +Intuitively, what this definition means is that if the state estimator says that +there is probability p that the system is in state s, then the system will be in +state s with probability p. Calibration is an increasingly common requirement +for learning-based detectors [16,27,14,31], and we validate it in our experiments. +Now we are ready for our main theoretical result: assuming that MAS is +conservative with respect to MOS and that the state estimator is well-calibrated, +we show that the safety estimates of our monitoring are conservatively calibrated. +Theorem 1. Let the system MOS in Equation (3) be given with a well-calibrated +state estimator (Definition 7). Let MAS be a conservative model of MOS (Defini- +tion 6). Finally, assume that the safety of MOS conditioned on the true state of +the system is independent of the safety estimate from the monitor. Given state +estimator distribution P ¯ +X(t) and control command U ∈ U, the safety estimates +from the state distribution monitor (Equation (7)) are conservative: +P(ψMOS | ˆP mon +dist (P ¯ +X(t), U(t)) = p) ≥ p +∀p ∈ [0, 1] +(10) +Proof. We start with conditioning the safety of the system on the state of the +system and proceed with equivalent transformations: +P(ψMOS | ˆP mon +dist (P ¯ +X(t), U(t)) = p) = +� +s∈S +P +� +ψMOS | X(t) = s, ˆP mon +dist +�P ¯ +X(t), U(t)� = p +� +· P +� +X(t) = s | ˆP mon +dist +�P ¯ +X(t), U(t)� = p +� +ds = +� +s∈S +P(ψMOS | X(t) = s) · P ¯ +X(t)(s)ds = +� +s∈P ¯ +X(t) +P(ψMOS | X(t) = s) · P ¯ +X(t)(s) = +� +s∈P ¯ +X(t) +PMOS(s,U(t))(ψ) · P ¯ +X(t)(s) = + +Conservative Safety Monitors of Stochastic Dynamical Systems +11 +� +s∈P ¯ +X(t) +PMOS(s,U(t))(ψ) · P ¯ +X(t)(s) ≥ +� +s∈P ¯ +X(t) +Prmin +MAS(s↓,U(t))(ψ) ∗ P ¯ +X(t)(s) = p +The first step comes from marginalizing the state X(t) into the left side of +Equation (10). The second step comes from the assumption that the safety of the +system given the state is independent of the monitor output and the assumed +calibration of the monitor from Equation (9). The third step follows from the +discrete, finite support of the state estimator output and the calibration. The +fifth and sixth steps come from substituting and rearranging terms. The final +step comes from the assumed conservatism of MAS in Definition 6. +7 +Case Study +Our experimental evaluation aims to demonstrate that the safety estimates from +our monitoring approach are conservative and accurate. Additionally, we com- +pare the effect of using the point-wise and distribution-wise state estimation. We +perform the evaluation on a simulated water tank system and use the PRISM +model checker [22] to perform the probabilistic model checking. +7.1 +Water Tanks +Consider a system consisting of J water tanks, each of size TS, draining over +time, and a central controller that aims to maintain some water level in each +tank. With wi[t] as the water level in the ith tank at time t, the discrete-time +dynamics for the water level in the tank is given by: +wi[t + 1] = wi[t] − outi[t] + ini[t], +(11) +where ini[t] and outi[t] are the amounts of water entering (“inflow”) and leaving +(“outflow”) respectively the ith tank at time t. The inflow is determined by the +controller and the outflow is a constant determined by the environment. +Each tank is equipped with a noisy sensor to report its current perceived +water level, ˆw, which is a noisy function of the true current water level, w. The +noise on the sensor outputs is a Gaussian with 0 mean and known variance. In +addition, with constant probability, the perception can output ˆw =0 or ˆw =TS. +Each water tank uses a standard Bayesian filter as a state estimator. The +filter maintains a discrete distribution over the state of the system. On each +perception reading, the filter updates its state distribution using a standard +application of Bayes rule. The mean of the state distribution at this point is the +estimator’s point prediction, which is sent to the controller. Once the control +action is computed, the filter updates its state distribution by applying the +system dynamics. + +12 +M. Cleaveland et al. +The central controller has a single source of water to fill one tank at a time +(or none at all) based on the estimated water levels. Then this tank receives a +constant value in > 0 of water, whereas the other tanks receive 0 water. Each +tank has a local controller that requests itself to be filled when its water level +drops below the lower decision threshold LT and stops requesting to be filled +after its water level reaches the upper threshold UT. If several tanks request to +be filled, the controller fills the one with the lowest water level (or, if equal, it +flips a coin to decide). +At runtime, we are interested in the probability that a tank will neither be +empty or overflowing, represented by the bounded-time safety property: +ψwt := □≤10 ∨i=1..J (wli > 0 ∧ wli < TS) +Model Construction We construct the MAS model for J = 2 water tanks, +in = 13.5, outi[t] = 4.3, TS = 100, LT = 10, UT = 90, and water level intervals +of size 1 by following the description in Section 5.1. To model the combination of +perception and state estimation, we estimated the state distributions with 100 +trials of 50 time steps. Figure 1 shows a histogram of the state estimation errors. +Fig. 1: Histogram of state estimation errors for the water tanks. +Model Checking The initial state of MAS comprises the water level of each +tank, the low-level control command of each tank, and the filling command of the +central controller. There are 101 discrete water levels in each tank and 5 possible +configurations of the 3 control commands, for a total of 51005 different initial +states of MAS. We model-checked ψwt in these initial states on a server with +80 Intel(R) Xeon(R) Gold 6148 CPU @ 2.40GHz CPUs by running 50 parallel + +700 +600 +500 +5 +in Count +400 +B300 +200 +100- +0 +10 +-5 +0 +5 +10 +StateEstimatorErrorConservative Safety Monitors of Stochastic Dynamical Systems +13 +PRISM instances at a time. The full verification process took approximately 24 +hours, which is acceptable for the design-time phase. +7.2 +Results +To test our approach, we ran 500 trials of the water tanks starting from water +levels between 40 and 60. Each trial lasted for 50 time steps (recall that the model +checking checked 10 time steps into the future) and 74 trials resulted in a water +tank either over- or underflowing. We evaluated three different monitors in our +approach. One used the point estimates from the Bayesian filter (“point estimate +monitor”), another used the estimated distribution from the state estimator +(“state distribution monitor”), and the last used the true state of the system +(“true state monitor”, for comparison only). +Qualitative performance Figure 2 shows the safety estimates of the monitors +for two trials, one safe and one unsafe. The monitors keep pretty high safety +estimates for the entirety of the safe trial. In the unsafe trial, the failure occurred +at time step 42 due to a tank overflowing. The safety estimates are quite high +at first but then begin to drop around time step 30, predicting the failure with +a 10-step time horizon. +(a) Safe trial +(b) Unsafe trial +Fig. 2: Monitor safety estimates for two water tank trials. +Calibration Next, to examine the overall calibration of our safety estimates, we +bin the safety estimates into 10 bins of width 0.1 ([0−0.1, 0.1−0.2, . . . , 0.9−1]) +and compute the empirical safety chance within each bin. The results are shown +in Figure 3, with the caveat that we only plot bins with at least 50 samples to +ensure statistical significance. The point estimate monitor and true state monitor +are conservative for all of their bins. On the other hand, the state distribution + +1.0 +0.8 +Safety Estimate +0.6 +0.4 +Pointestimatemonitor +0.2 +Statedistributionmonitor +Truestatemonitor +0.0 +0 +10 +20 +30 +40 +50 +Time (s)1.0 +0.8 +Safety Estimate +0.6 +0.4 +Pointestimatemonitor +0.2 +Statedistributionmonitor +Truestatemonitor +0.0 +0 +10 +20 +30 +40 +Time (s)14 +M. Cleaveland et al. +monitor has the best overall calibration. We also computed the ECE and Brier +scores for the monitors, which are shown in Table 1. To assess the conservatism of +the monitors, we introduce a novel metric called expected conservative calibration +error (ECCE). It is similar to ECE, except that it only sums the bins where the +average monitor confidence is greater than the empirical safety probability (i.e., +the cases where the monitor is overconfident in safety). The ECCE values for the +monitors are also shown in Table 1. Note that ECE ≥ ECCE, because ECCE +only aggregates a subset of the bins that ECE does. Our results show that the +monitors are well-calibrated and conservative, and that the state distribution +monitor manages to capture the uncertainty particularly well. +(a) Point estimate monitor (b) State distribution mon- +itor +(c) True state monitor +Fig. 3: Calibration plots for the three monitors. The x-axis shows the binned +safety estimates reported by the monitor and the y-axis shows the empirical +safety probability. The diagonal dashed line denotes perfect calibration. Bars +higher than the dashed line represent under-confidence (i.e., conservatism) and +bars lower than the dashed line represent over-confidence. +Accuracy Finally, we are interested in the ability of the monitors to distinguish +safe and unsafe scenarios for the water tanks. To do this, we computed a receiver +operating characteristic (ROC) curve for the three monitors, shown in Figure 4 +and areas under curve (AUC) in Table 1. As expected, the state distribution +monitor and true state monitor outperform the point estimate monitor. One +surprising aspect is that the state distribution monitor performs about as well as +the true state monitor. We hypothesize that this is because the state distribution +contains information about how well the state estimator will perform in the near +future, which has a large affect on the safety of the system. Investigating this +potential phenomena is another area of future work. +Validation of assumptions First, we empirically validate whether MAS is +conservative with respect to MOS. Directly verifying this claim is infeasible, +since it requires computing PMOS(s,u)(ψ) for an infinite number of states s ∈ S. +However, we can examine the performance of the true state monitor as a proxy + +1.0 +Empirical Safety Probability +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +MonitorSafetyProbability1.0 +Empirical Safety Probability +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +MonitorSafetyProbability1.0 +Empirical Safety Probability +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +MonitorSafetyProbabilityConservative Safety Monitors of Stochastic Dynamical Systems +15 +Fig. 4: ROC curves for the three monitors. +Monitor Type +ECE +ECCE +Brier Score AUC +State estimate +0.0157 +0.00818 +0.0292 +0.828 +State distribution 0.00252 0.000899 +0.0275 +0.867 +True state +0.0129 +0.00459 +0.0273 +0.870 +Table 1: Calibration and classification metrics for the monitors. +for the conservatism of MAS: the true state monitor obtains the probabilities +from MAS using the true state, avoiding any sensing and state estimation noise. +The slightly underconfident true state monitor bins in Figure 3 and the very low +ECCE in Table 1 both provide strong evidence that MAS is indeed conservative. +Second, we examine the calibration assumption of the state estimator. We +computed its ECE across all water levels, resulting in the negligible value of +0.00656. We conclude that this state estimator gives calibrated results in practice. +8 +Conclusion +This paper introduced a method for synthesizing conservative and well-calibrated +predictive runtime monitors for stochastic dynamical systems. Our method ab- +stracts the system as a PA and uses PMC to verify the safety of the states of the +PA. At runtime, these safety values are used to estimate the true safety of the +system. We proved that our safety estimates are conservative provided the PA +abstraction is conservative and the system’s state estimator is well-calibrated. +We demonstrated our approach on a case study with water tanks. Future work +includes applying our method to existing approaches that learn discrete ab- +stractions directly from data, exploring how to construct conservative percep- +tion/state estimation abstractions, and investigating the effects of the estimated +state distribution’s variance on the future system safety. + +1.0 +(Sensitivity) +0.8 +0.6 +Rate or +True Positive +0.4 +0.2 +Point Estimate Monitor +State Distribution Monitor +True State Monitor +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate or (1 -Specifity)16 +M. Cleaveland et al. +Acknowledgments +This work was supported in part by DARPA/AFRL FA8750-18-C-0090 and by +ARO W911NF-20-1-0080. 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Springer (2012) + diff --git a/kdFIT4oBgHgl3EQfrCu8/content/tmp_files/load_file.txt b/kdFIT4oBgHgl3EQfrCu8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2262c37926d0f2ef008d6e96f49a4726fea3abb6 --- /dev/null +++ b/kdFIT4oBgHgl3EQfrCu8/content/tmp_files/load_file.txt @@ -0,0 +1,773 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf,len=772 +page_content='Conservative Safety Monitors of Stochastic Dynamical Systems Matthew Cleaveland1, Ivan Ruchkin2, Oleg Sokolsky1, and Insup Lee1 1 University of Pennsylvania, Philadelphia, PA, USA 19104 2 University of Florida, Gainesville, FL, USA, 32611 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Generating accurate runtime safety estimates for autonomous systems is vital to ensuring their continued proliferation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' However, accu- rately reasoning about future system behaviors is generally too complex to do at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' To better reason about system safety at runtime, we propose a method for leveraging design time model checking results at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Specifically, we model the system as a probabilistic automa- ton (PA) and compute bounded time reachability probabilities over the states of the PA at design time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' At runtime, we combine distributions of state estimates with the safety probabilities from design time to produce a bounded time safety estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We argue that our approach produces well calibrated safety probabilities, assuming the estimated state distri- butions are well calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We evaluate our approach using a case study of simulated water tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Keywords: Runtime Monitoring · Probabilistic Model Checking · Cal- ibrated Prediction 1 Introduction As autonomous systems see increased use and perform critical tasks in an open world, reasoning about their safety and performance is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In particular, it is vital to know if a system is likely to reach an unsafe state in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The field of predictive runtime monitoring offers ways for performing this rea- soning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The basic idea behind predictive runtime monitoring is to reason about the expected future behaviors of the system and their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' However, accu- rately computing future system states is computationally infeasible at runtime, as it requires running expensive reachability analysis on complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Previ- ous works have computed libraries of reachability analysis results at design time and used them at runtime [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' But these approaches require the system dynamics to have certain invariances to reduce the number of times reachability analysis must be called offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Other lines of work use system execution data to learn discrete probabilistic models of the system, which are then used to perform predictive runtime mon- itoring, as there is rich literature for runtime monitoring of discrete automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' These models range from discrete-time Markov chains (DTMCs) [2] to hidden Markov models (HMMs) [4] to Bayesian networks [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' However, it is difficult arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='11330v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='LO] 26 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Cleaveland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' to provide guarantees relating the performance of the automata models to the real system, due to the fact that they are fit using finite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Of particular interest is ensuring the models are conservative: it is essential to avoid run-time overconfidence in the safety of the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In this paper, we propose a method for predictive run-time monitoring of safety probabilities that builds on the strengths of the existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We use a mix of conservative modeling techniques and data-driven modeling techniques to transform the dynamical system into a probabilistic automaton (PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='3 We then employ probabilistic model checking (PMC) to compute the safety of the model over all its states offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Finally, we synthesize lightweight monitors that rely on the model checking results and a well-calibrated state estimator to compute the probability of system safety at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Under the assumption that the PA model is conservative and that the state estimator is well-calibrated, we prove that our runtime monitors are conserva- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We argue that our modeling technique is likely to result in conservative PA models — and empirically demonstrate this fact in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Finally, we demonstrate that our method produces well-calibrated, accurate, and conserva- tive monitors on a case study using water tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The contributions of this paper are threefold: – We present a method for conservatively modeling dynamical systems as PAs and using PMC results at runtime to monitor the system’s safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' – We prove that if our PA models are conservative then the monitor safety estimates will be conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' – We demonstrate our approach on a case study of water tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We empirically show that our PA models and runtime monitors are both conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We give an overview of the related work in Section 2, provide the necessary formal background in Section 3, and formulate the problem in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Section 5 goes over our proposed ap- proach and Section 6 provides formal conservatism guarantees for the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We describe the results of our case study in Section 7 and conclude in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 2 Related Work We divide the previous works in the area of predictive runtime monitoring into two bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The first bin analyzes dynamical system models, while the second analyzes automata models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Dynamical systems approaches A large portion of the predictive monitor- ing for dynamical systems literature focuses on reasoning about the safety of autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Prior work has employed reachability analysis to estimate the future positions of other cars to estimate the safety of a proposed vehicle 3 In our scope, PAs are equivalent to Markov Decision Processes (MDPs) without rewards: both have finite states with probabilistic and non-deterministic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Conservative Safety Monitors of Stochastic Dynamical Systems 3 trajectory [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In [18], the authors develop techniques to estimate the probabil- ity of a proposed trajectory resulting in a collision with other vehicles, which are given as distributions of states predicted by neural networks (NNs) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In [9], the authors use precomputed reachability analysis and Bayesian inference to compute the probability of an autonomous vehicle colliding with static obsta- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' However, this approach requires the system dynamics to have translational and rotation invariances to ensure the reachability analysis can be feasibly run at design time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This approach is conceptually similar to ours, using computa- tional power at design time to precompute system behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' But our approach employs automata-based abstractions instead of making invariance assumptions about the system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Previous works have also addressed the problem of synthesizing runtime monitors for signal temporal logic (STL) properties of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Ap- proaches range from conformal prediction [8,25], design time forward reachability analysis [34], computing safe envelopes of control commands [33], online linear regression [15], and uncertainty aware STL monitoring [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Automata approaches The first works of this type developed predictive LTL semantics, also called LTL3 [35,24], for discrete automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The LTL3 semantics allowed to the system to determine if every infinite extension of an observed finite trace would satisfy or not satisfy a specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Recent work has extended these ideas to timed systems [28], multi-model systems [12], and systems with assumptions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Another approach uses neural networks to classify if unsafe states of a hybrid automata can be reached from the current state of the HA [5,6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' They additionally use conformal prediction to get guarantees about the accuracy of their predictions [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' However, these frameworks give very coarse predictions, as they can only determine if a system is guaranteed to be safe, guaranteed to be unsafe, or not sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Another thread of works uses data to learn probabilistic models from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' These models can then be used in conjunction with predictive monitoring tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In [4], the authors learn an HMM model of the system from simulation data and perform bounded reachability analysis to determine the probability of an LTL specification being violated from each state of the HMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This work was extended using abstraction techniques to simplify the learned models [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In [2], the same authors employ importance sampling to efficiently learn discrete time Markov chain (DTMC) models from data, which they then use to synthe- size predictive monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In [17], the authors use Bayesian networks to model temporal properties of stochastic timed automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The Bayesian networks are updated online to improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Finally, in [13] the authors use process mining techniques to learn predictive models of systems, which are in turn used to synthesize predictive runtime monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' An interesting line of future work for us is exploring applying our runtime monitoring technique using these models as they are updated from new observations online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The most similar work to ours presents two methods for synthesizing predic- tive monitors for partially observable Markov decision processes (POMDPs) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Cleaveland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The first approach combines precomputed safety probabilities of each state with POMDP state estimators to estimate the probability that the system will remain safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The main downside of this approach is that state estimation of POMDPs is computationally expensive since the set of potential state distributions increases exponentially due to the non-determinism in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The second approach uses model checking of conditional probabilities to directly compute the safety of the system based on the observation trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' A downside of this approach is that it requires running model checking at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Our method, on the other hand, avoids expensive computations at run time while maintaining design-time scalability through abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 3 Background In the following Definitions 1 to 3, borrowed from Kwiatkowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' [23], we use Dist(S) to refer to the set of probability distributions over a set S, ηs as the distribution with all its probability mass on s ∈ S, and µ1 × µ2 to be the product distribution of µ1 and µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' A probabilistic automaton (PA) is a tuple M = (S, ¯s, α, δ, L), where S is a finite set of states, ¯s ∈ S is the initial state, α is an alphabet of action labels, (S, α, Dist(S)) ∈ δ is a probabilistic transition relation, and L : S → 2AP is a labeling function from states to sets of atomic propositions from the set AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' If (s, a, µ) ∈ δ then the PA can make a transition in state s with action label a and move based on distribution µ to state s′ with probability µ(s′), which is denoted by s a−→ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' If (s, a, ηs′) ∈ δ then we say the PA can transition from state s to state s′ via action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' A state s is terminal if no elements of δ contain s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' A path in M is a finite/infinite sequence of transitions π = s0 a0,µ0 −−−→ s1 a1,µ1 −−−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' with s0 = ¯s and µi(si+1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' A set of paths is denoted as Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We use M(s) to denote the PA M with initial state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Reasoning about PAs also requires the notion of schedulers, which resolve the non-determinism during an execution of a PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' For our purposes, a scheduler σ maps each state of the PA to an available action label in that state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We use Πσ M for the set of all paths through M when controlled by scheduler σ and SchM for the set of all schedulers for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Finally, given a scheduler σ, we define a probability space Prσ M over the set of paths Πσ M in the standard manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Given PAs M1 and M2, we define parallel composition as follows: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The parallel composition of PAs M1 = (S1, ¯s1, α1, δ1, L1) and M2 = (S2, ¯s2, α2, δ2, L2) is given by the PA M1 || M2 = (S1 × S2, (¯s1, ¯s2), α1 ∪ α2, δ, L), where L(s1, s2) = L1(s1)∪L2(s2) and δ is such that (s1, s2) a−→ µ1 ×µ2 iff one of the following holds: (i) s1 a−→ µ1, s2 a−→ µ2 and a ∈ α1 ∩ α2, (ii) s1 a−→ µ1, µ2 = ηs2 and a ∈ (α1 \\ α2), (iii) µ1 = ηs1, s2 a−→ µ2 and a ∈ (α2 \\ α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In this paper, we are concerned with probabilities that the system will not enter an unsafe state within a bounded amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' These are represented Conservative Safety Monitors of Stochastic Dynamical Systems 5 as bounded-time safety properties, which we express using linear temporal logic (LTL) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Following the notation from [20], we denote these properties as □≤T s /∈ Sunsafe, where Sunsafe ⊂ S is the set of unsafe states and T ≥ 0 is the time bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' For LTL formula ψ, PA M, and scheduler σ ∈ SchM, the prob- ability of ψ holding is: Prσ M(ψ) := Prσ M{π ∈ Πσ M | π |= ψ}, where π |= ψ indicates that the path π satisfies ψ in the standard LTL seman- tics [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We specifically consider LTL safety properties, which are LTL speci- fications that can be falsified by a finite trace though a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Both ψnocol and ψflow are LTL safety properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Probabilistically verifying an LTL formula ψ against M requires checking that the probability of satisfying ψ meets a probability bound for all schedulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This involves computing the minimum or maximum probability of satisfying ψ over all schedulers: Prmin M (ψ) := infσ∈SchM Prσ M(ψ) Prmax M (ψ) := supσ∈SchM Prσ M(ψ) We call σ a min scheduler of M if Prσ M(ψ) = Prmin M (ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We use Schmin M to denote the set of min schedulers of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Remark: For the rest of this paper, we use Pr when referring to model check- ing probabilities and P for all other probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Calibration and Conservatism Consider a scenario where a probability es- timator is predicting probability ˆp that a (desirable) event E will occur (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=', a safe outcome).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We define the calibration for the probability estimates (adapted from Equation (1) of [16]): Definition 4 (Calibration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The probability estimates ˆp of event E are well- calibrated if P(E | ˆp = p) = p, ∀p ∈ [0, 1] (1) Next, we define conservatism for the probability estimates: Definition 5 (Conservative Probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The probability estimates ˆp of a desirable event E are conservative if P(E | ˆp = p) ≥ p, ∀p ∈ [0, 1] (2) 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Cleaveland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In other words, the estimates ˆp are conservative if they underestimate the true probability of event E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Note that any monitor that is well-calibrated (Def- inition 4) is guaranteed to be conservative (Definition 5), but not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Two standard metrics for assessing the calibration of the ˆp estimates are expected calibration error (ECE) [16] and Brier score [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The ECE metric is computed by dividing the ˆp values into equally spaced bins in [0, 1], within each bin taking the absolute difference between the average ˆp and the empirical probability of event E, and weighted-averaging across bins with their sizes as weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' So ECE penalizes discrepancies between the estimator confidence and empirical probability of E within each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The Brier score is the mean squared error of the probability estimates � i ( ˆpi − 1Ei)2 4 Problem Statement Consider the following discrete-time stochastic system titled MOS with dynamics: X(t + 1) = f(X(t), U(t))), Y (t) = g(X(t), V (t)), ¯X(t), Z(t) = h(Z(t − 1), Y (t), W(t)), U(t) = c( ¯X(t)), (3) where X(t) ∈ S ⊂ Rn is the system state (with bounded S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Y (t) ∈ Rp are the observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' ¯X(t) ∈ Rn is the estimated state of the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' ¯Z(t) ∈ Rz is the internal state of the state estimator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=', a belief prior in a Bayesian filter or a set of particles in a particle filter);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' U(t) ∈ U ⊂ Rm is the control output, which we discretize, resulting in a finite number |U| of control actions, the functions f : Rn ×Rm → Rn, g : Rn ×Rv → Rp, h : Rz ×Rp ×Rw → Rn ×Rz describe the system dynamics, perception map, and state estimator respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' the function c : Rp → Rm is a stateless controller;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' and V (t) ∈ Dv ⊆ Rv and W(t) ∈ Dw ⊆ Rw describe perception and state estimator noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The V (t) noise models inexact perception, such as an object detector missing an obstacle or a LiDAR scanner giving noisy distance estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The W(T) noise accounts for state estimators that use randomness under the hood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' A common example of this is particle filters randomly perturbing their particles so that they do not collapse to the exact same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Let Sunsafe ⊂ S denote the set of unsafe states of MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' At time t, we are interested in whether MOS will lie in Sunsafe at some point in the next T time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This is represented by the bounded time reachability property ψMOS = □≤T (X /∈ Sunsafe) (4) Let P(ψMOS | ¯Z(t)) denote the probability of MOS satisfying ψMOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Our goal is to compute calibrated (Definition 4) and conservative (Definition 5) estimates of P(ψMOS | ¯Z(t)) at runtime, which we denote as “ P(ψMOS | ¯Z(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Conservative Safety Monitors of Stochastic Dynamical Systems 7 5 Overall Approach At a high level, our approach consists of a design time portion and a runtime portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' At design time, a PA of the system (including its dynamics, perception, state estimation, and controller) is constructed using standard conservative ab- straction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Then the bounded-time safety probability for each state of the model is computed using model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This results in a look-up table that maps the states of the abstract system to the safety probability of the real system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' At runtime, the estimated state (or distribution of states) from the real system’s state estimator is used to estimate the abstract state (or distribution of abstract states) of the abstract system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This abstract state (or distribution of states) is used in conjunction with the lookup table to estimate the bounded-time safety of the real system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='1 Design Time The design time aspect of our approach has two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' First, we convert the orig- inal system MOS into a probabilistic automaton MAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Then we use probabilistic model checking to compute the bounded time safety of MAS for each state in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Model Construction To convert MOS into a probabilistic automaton, MAS, we first need to create probabilistic models of the perception g and state estimation h components of MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' To do this, we simulate MOS and record the perception errors X(t) − ¯X(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We discretize the domain of these errors and estimate a categorical distribution over it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' For example, this distribution would contain information such as “the perception will output a value that is between 2m/s and 3m/s greater than the true velocity of the car with probability 1/7.” To convert the system dynamics f and controller c to a probabilistic automa- ton, we use a standard interval abstraction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The high level idea is to divide the state space S of MOS into a finite set of equally sized hyperrectangles, denoted as S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' So every s′ 1 ∈ S′ has a corresponding region S1 ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' MAS then has a transition from s′ 1 to s′ 2 (in MAS) if at least one state in S1 has a transi- tion to a state in S2 (in MOS) under some control command u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Note that state s′ 1 can non-deterministically transition to multiple states in S′ because it covers an entire hyperrectangle of states in MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This ensures that the interval abstraction is conservative, as it overapproximates the behaviors of MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Finally, the perception error model, controller, and interval abstraction are all parallel-composed into a single model as per Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Remark: In describing the construction of the MAS, we have not mentioned anything about initial states: we do not keep track of a singular initial state for MAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Instead, we will later run model checking for the full range of initial states of MAS to anticipate all runtime scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' For our purposes, the “initial state-action space” of MAS consists of every abstract state and control action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We include the control action in the initial state space because when using the model’s safety probabilities online, we know what the next control action will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Cleaveland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Safety Property We need to transform the bounded time safety property on MOS given in Equation (4) into an equivalent property on MAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' To do this, we compute the corresponding set of unsafe states on MAS, which is defined as S′ unsafe = {s′ | ∃s ∈ Sunsafe, s′ corresponds to s} Letting s′ denote the state of MAS, the bounded time safety property for MAS is ψMAS := □≤T �s′ /∈ S′ unsafe � (5) Probabilistic Model Checking The final design-time step of our approach computes the safety probability of MAS for every state in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This step amounts to computing the below using standard model checking tools: Prmin MAS(s′,u)(ψMAS), ∀s′ ∈ S′, ∀u ∈ U This requires running model checking on MAS for a range of initial states, which can be a time-consuming process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' To mitigate this, we note that MAS is simpler to analyze than MOS, since the size of the state space gets reduced during the interval abstraction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Additionally, model checking this frag- ment of LTL on PAs is NP-complete, whereas checking generic LTL formulas is EXPTIME-complete [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' So the time complexity of our model checking is lower because we restrict ourselves to bounded time safety properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The probabilities from the model checking are stored in a lookup table, which we denote as G(s′, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' It will be used at runtime to estimate the likelihood of the system being unsafe in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 Runtime At runtime, we use the lookup table G to estimate the near-future safety of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' At a high level, we observe the outputs of the state estimator and controller and run them through the lookup table to compute the probability of the system avoiding unsafe states for the next T time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We propose two different ways of utilizing the state estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The first way is to simply use the point estimate from the state estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In cases of probabilistic estimators, this means taking the mean of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The second way uses the estimated state distribution from the state estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This requires an estimator with a probabilistic output, but most common state estimators, such as particle filters and Bayesian filters, keep track of the distribution of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The second way takes full advantage of the available state uncertainty to predict safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Point Estimate At time t the state estimator outputs state estimate ¯X(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The controller then outputs control command U(t) = c( ¯X(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Finally, we get a safety estimate ˆP mon point( ¯X(t), U(t)) by plugging ¯X(t) and U(t) into G: ˆP mon point( ¯X(t), U(t))) = G( ¯X(t), U(t)) (6) Conservative Safety Monitors of Stochastic Dynamical Systems 9 State Distribution Now assume that at time t state estimator additionally outputs a state estimate ¯X(t) and a finite, discrete distribution of the state, denoted as P ¯ X(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The controller still outputs control command U(t) = c( ¯X(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' To estimate the safety of the system, we compute a weighted sum of the safety of each state in P ¯ X(t) using G and U(t): ˆP mon dist (P ¯ X(t), U(t)) = � s∈Supp(P ¯ X(t)) P ¯ X(t)(s) · G(s′, U(t)) (7) where Supp Ä P ¯ X(t) ä denotes the (finite) support of P ¯ X(t), P ¯ X(t)(s) denotes the estimated probability of MOS being in state s according to P ¯ X(t), and s′ ∈ S′ is the state in MAS that corresponds to state s ∈ S in MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 6 Conservatism Guarantees This section proves that our state-distribution monitoring produces safety esti- mates that are conservative and well-calibrated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' that is, we underestimate the probability of safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We require two assumptions for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The first assumption is the conservatism of abstract model MAS, by which we mean that its probability of being safe is always less than that of MOS for the same initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The second assumption is the calibration of the state estimator, which means that it produces state probabilities that align with the frequencies of these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Below we formalize and discuss these assumptions before proceeding to our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Definition 6 (Model Conservatism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Abstraction MAS is conservative with respect to system MOS if PMOS(s,u)(ψ) ≥ Prmin MAS(s′,u)(ψ) ∀s ∈ S, u ∈ U (8) where s′ ∈ S′ is the state in MAS that corresponds to state s in MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In general, it is difficult to achieve provable conservatism of MAS by con- struction: the model parameters of complex components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=', vision-based per- ception) are estimated from data, and they may have complicated interactions with the safety chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Instead, we explain why our approach is likely to be conservative in practice and validate this assumption in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Consider MOS and MAS as compositions of two sub-models: dynamics/control and perception/state estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We construct MAS such that its dynamics/control component always overapproximates the dynamics/control portion of MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' That means that any feasible sequence of states and control actions from MOS is also feasible in MAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' This follows from the use of reachability analysis over the in- tervals of states to compute the transitions of MAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' It is unclear how to formally compare the conservatism of perception/state estimation portions of MAS and MOS when they are created from simulations of the perception/state estimation component of MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' First, these components are 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Cleaveland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' not modeled explicitly due to the high dimensionality of learning-based percep- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Thus, when estimating probabilities from samples, we essentially approxi- mate the average-case behavior of the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Second, it is often unknown in which direction the probabilities need to be shifted to induce a conservative shift to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' One opportunity is to use monotonic safety rules [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' for now, this remains a promising and important future research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' To summarize, the dynamics/control portion of MAS overapproximates that of MOS, while the perception/state estimation portion of MAS approximates the average-case behavior of MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' So one would expect, on average, MAS to be conservative with respect to MOS, even though we cannot formally prove that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Next, we define the calibration for the state estimator (adapted from Equa- tion (1) of [16]): Definition 7 (Calibration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Given the dynamical system from Equation (3) and state estimator h that outputs a discrete, finite distribution of the estimated state, denoted P¯x(t), we say that h is well-calibrated if P(x(t) = s | P¯x(t)(s) = p) = p, ∀p ∈ [0, 1] (9) Intuitively, what this definition means is that if the state estimator says that there is probability p that the system is in state s, then the system will be in state s with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Calibration is an increasingly common requirement for learning-based detectors [16,27,14,31], and we validate it in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Now we are ready for our main theoretical result: assuming that MAS is conservative with respect to MOS and that the state estimator is well-calibrated, we show that the safety estimates of our monitoring are conservatively calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Let the system MOS in Equation (3) be given with a well-calibrated state estimator (Definition 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Let MAS be a conservative model of MOS (Defini- tion 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Finally, assume that the safety of MOS conditioned on the true state of the system is independent of the safety estimate from the monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Given state estimator distribution P ¯ X(t) and control command U ∈ U, the safety estimates from the state distribution monitor (Equation (7)) are conservative: P(ψMOS | ˆP mon dist (P ¯ X(t), U(t)) = p) ≥ p ∀p ∈ [0, 1] (10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We start with conditioning the safety of the system on the state of the system and proceed with equivalent transformations: P(ψMOS | ˆP mon dist (P ¯ X(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' U(t)) = p) = � s∈S P � ψMOS | X(t) = s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' ˆP mon dist �P ¯ X(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' U(t)� = p � P � X(t) = s | ˆP mon dist �P ¯ X(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' U(t)� = p � ds = � s∈S P(ψMOS | X(t) = s) · P ¯ X(t)(s)ds = � s∈P ¯ X(t) P(ψMOS | X(t) = s) · P ¯ X(t)(s) = � s∈P ¯ X(t) PMOS(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='U(t))(ψ) · P ¯ X(t)(s) = Conservative Safety Monitors of Stochastic Dynamical Systems 11 � s∈P ¯ X(t) PMOS(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='U(t))(ψ) · P ¯ X(t)(s) ≥ � s∈P ¯ X(t) Prmin MAS(s↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='U(t))(ψ) ∗ P ¯ X(t)(s) = p The first step comes from marginalizing the state X(t) into the left side of Equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The second step comes from the assumption that the safety of the system given the state is independent of the monitor output and the assumed calibration of the monitor from Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The third step follows from the discrete, finite support of the state estimator output and the calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The fifth and sixth steps come from substituting and rearranging terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The final step comes from the assumed conservatism of MAS in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 7 Case Study Our experimental evaluation aims to demonstrate that the safety estimates from our monitoring approach are conservative and accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Additionally, we com- pare the effect of using the point-wise and distribution-wise state estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We perform the evaluation on a simulated water tank system and use the PRISM model checker [22] to perform the probabilistic model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='1 Water Tanks Consider a system consisting of J water tanks, each of size TS, draining over time, and a central controller that aims to maintain some water level in each tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' With wi[t] as the water level in the ith tank at time t, the discrete-time dynamics for the water level in the tank is given by: wi[t + 1] = wi[t] − outi[t] + ini[t], (11) where ini[t] and outi[t] are the amounts of water entering (“inflow”) and leaving (“outflow”) respectively the ith tank at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The inflow is determined by the controller and the outflow is a constant determined by the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Each tank is equipped with a noisy sensor to report its current perceived water level, ˆw, which is a noisy function of the true current water level, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The noise on the sensor outputs is a Gaussian with 0 mean and known variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In addition, with constant probability, the perception can output ˆw =0 or ˆw =TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Each water tank uses a standard Bayesian filter as a state estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The filter maintains a discrete distribution over the state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' On each perception reading, the filter updates its state distribution using a standard application of Bayes rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The mean of the state distribution at this point is the estimator’s point prediction, which is sent to the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Once the control action is computed, the filter updates its state distribution by applying the system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Cleaveland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The central controller has a single source of water to fill one tank at a time (or none at all) based on the estimated water levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Then this tank receives a constant value in > 0 of water, whereas the other tanks receive 0 water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Each tank has a local controller that requests itself to be filled when its water level drops below the lower decision threshold LT and stops requesting to be filled after its water level reaches the upper threshold UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' If several tanks request to be filled, the controller fills the one with the lowest water level (or, if equal, it flips a coin to decide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' At runtime, we are interested in the probability that a tank will neither be empty or overflowing, represented by the bounded-time safety property: ψwt := □≤10 ∨i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='.J (wli > 0 ∧ wli < TS) Model Construction We construct the MAS model for J = 2 water tanks, in = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='5, outi[t] = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='3, TS = 100, LT = 10, UT = 90, and water level intervals of size 1 by following the description in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' To model the combination of perception and state estimation, we estimated the state distributions with 100 trials of 50 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Figure 1 shows a histogram of the state estimation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 1: Histogram of state estimation errors for the water tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Model Checking The initial state of MAS comprises the water level of each tank, the low-level control command of each tank, and the filling command of the central controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' There are 101 discrete water levels in each tank and 5 possible configurations of the 3 control commands, for a total of 51005 different initial states of MAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We model-checked ψwt in these initial states on a server with 80 Intel(R) Xeon(R) Gold 6148 CPU @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='40GHz CPUs by running 50 parallel 700 600 500 5 in Count 400 B300 200 100- 0 10 5 0 5 10 StateEstimatorErrorConservative Safety Monitors of Stochastic Dynamical Systems 13 PRISM instances at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The full verification process took approximately 24 hours, which is acceptable for the design-time phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 Results To test our approach, we ran 500 trials of the water tanks starting from water levels between 40 and 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Each trial lasted for 50 time steps (recall that the model checking checked 10 time steps into the future) and 74 trials resulted in a water tank either over- or underflowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We evaluated three different monitors in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' One used the point estimates from the Bayesian filter (“point estimate monitor”), another used the estimated distribution from the state estimator (“state distribution monitor”), and the last used the true state of the system (“true state monitor”, for comparison only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Qualitative performance Figure 2 shows the safety estimates of the monitors for two trials, one safe and one unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The monitors keep pretty high safety estimates for the entirety of the safe trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In the unsafe trial, the failure occurred at time step 42 due to a tank overflowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The safety estimates are quite high at first but then begin to drop around time step 30, predicting the failure with a 10-step time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' (a) Safe trial (b) Unsafe trial Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 2: Monitor safety estimates for two water tank trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Calibration Next, to examine the overall calibration of our safety estimates, we bin the safety estimates into 10 bins of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='1 ([0−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='9−1]) and compute the empirical safety chance within each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The results are shown in Figure 3, with the caveat that we only plot bins with at least 50 samples to ensure statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The point estimate monitor and true state monitor are conservative for all of their bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' On the other hand, the state distribution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 Safety Estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 Pointestimatemonitor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 Statedistributionmonitor Truestatemonitor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0 10 20 30 40 50 Time (s)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 Safety Estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 Pointestimatemonitor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 Statedistributionmonitor Truestatemonitor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0 10 20 30 40 Time (s)14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Cleaveland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' monitor has the best overall calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We also computed the ECE and Brier scores for the monitors, which are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' To assess the conservatism of the monitors, we introduce a novel metric called expected conservative calibration error (ECCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' It is similar to ECE, except that it only sums the bins where the average monitor confidence is greater than the empirical safety probability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=', the cases where the monitor is overconfident in safety).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The ECCE values for the monitors are also shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Note that ECE ≥ ECCE, because ECCE only aggregates a subset of the bins that ECE does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Our results show that the monitors are well-calibrated and conservative, and that the state distribution monitor manages to capture the uncertainty particularly well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' (a) Point estimate monitor (b) State distribution mon- itor (c) True state monitor Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 3: Calibration plots for the three monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The x-axis shows the binned safety estimates reported by the monitor and the y-axis shows the empirical safety probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The diagonal dashed line denotes perfect calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Bars higher than the dashed line represent under-confidence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=', conservatism) and bars lower than the dashed line represent over-confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Accuracy Finally, we are interested in the ability of the monitors to distinguish safe and unsafe scenarios for the water tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' To do this, we computed a receiver operating characteristic (ROC) curve for the three monitors, shown in Figure 4 and areas under curve (AUC) in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' As expected, the state distribution monitor and true state monitor outperform the point estimate monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' One surprising aspect is that the state distribution monitor performs about as well as the true state monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We hypothesize that this is because the state distribution contains information about how well the state estimator will perform in the near future, which has a large affect on the safety of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Investigating this potential phenomena is another area of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Validation of assumptions First, we empirically validate whether MAS is conservative with respect to MOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Directly verifying this claim is infeasible, since it requires computing PMOS(s,u)(ψ) for an infinite number of states s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' However, we can examine the performance of the true state monitor as a proxy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 Empirical Safety Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 MonitorSafetyProbability1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 Empirical Safety Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 MonitorSafetyProbability1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 Empirical Safety Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 MonitorSafetyProbabilityConservative Safety Monitors of Stochastic Dynamical Systems 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 4: ROC curves for the three monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Monitor Type ECE ECCE Brier Score AUC State estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='00818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='828 State distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='00252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='000899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='867 True state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='00459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='870 Table 1: Calibration and classification metrics for the monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' for the conservatism of MAS: the true state monitor obtains the probabilities from MAS using the true state, avoiding any sensing and state estimation noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' The slightly underconfident true state monitor bins in Figure 3 and the very low ECCE in Table 1 both provide strong evidence that MAS is indeed conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Second, we examine the calibration assumption of the state estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We computed its ECE across all water levels, resulting in the negligible value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='00656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We conclude that this state estimator gives calibrated results in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 8 Conclusion This paper introduced a method for synthesizing conservative and well-calibrated predictive runtime monitors for stochastic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Our method ab- stracts the system as a PA and uses PMC to verify the safety of the states of the PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' At runtime, these safety values are used to estimate the true safety of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We proved that our safety estimates are conservative provided the PA abstraction is conservative and the system’s state estimator is well-calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' We demonstrated our approach on a case study with water tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Future work includes applying our method to existing approaches that learn discrete ab- stractions directly from data, exploring how to construct conservative percep- tion/state estimation abstractions, and investigating the effects of the estimated state distribution’s variance on the future system safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 (Sensitivity) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 Rate or True Positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 Point Estimate Monitor State Distribution Monitor True State Monitor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content='0 False Positive Rate or (1 -Specifity)16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Cleaveland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 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verification with predictive semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' In: NASA Formal Methods Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' 418–432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} +page_content=' Springer (2012)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFIT4oBgHgl3EQfrCu8/content/2301.11330v1.pdf'} diff --git a/l9E5T4oBgHgl3EQfiw8z/content/tmp_files/2301.05650v1.pdf.txt b/l9E5T4oBgHgl3EQfiw8z/content/tmp_files/2301.05650v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..75e5dbad06b172a5ac9c47ec8c64d6c51f74f81a --- /dev/null +++ b/l9E5T4oBgHgl3EQfiw8z/content/tmp_files/2301.05650v1.pdf.txt @@ -0,0 +1,2427 @@ +Contribution of Nuclear Excitation Electromagnetic Form Factors in 12C and 16O to +the Coulomb Sum Rule +A. Bodek1 and M. E. Christy2 +1Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA +2Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA +(Dated: January 16, 2023) +We report on empirical parameterizations of longitudinal and transverse nuclear excitation elec- +tromagnetic form factors in 12C and 16O. We extract the contribution of nuclear excitations to +the Normalized Inelastic Coulomb Sum Rule (SL(q)) as a function of momentum transfer q and +find that it is significant (0.29±0.030 at q= 0.22 GeV). The total contributions of nuclear exci- +tations to SL(q) in 12C and 16O are found to be equal within the uncertainties. Since the cross +sections for nuclear excitations are significant, the radiative tails from nuclear excitations should be +included in precise calculations of radiative corrections to quasielastic electron scattering at low q +and deep-inelastic electron scattering at large energy transfers ν. The parameterizations also serve +as a benchmark in testing theoretical modeling of cross sections for excitation of nuclear states in +electron and neutrino interactions on nuclear targets at low energies. +I. +INTRODUCTION +The Normalized Inelastic Coulomb Sum Rule SL(q) [1] +in electron scattering on nuclear targets is the integral +of the longitudinal nuclear response function RL(q, ν)dν +(excluding the nuclear elastic peak and pion production +processes) divided by the square of the proton electric +form factor and by the number of protons in the nucleus. +Here, q is the momentum transfer and ν is the energy +transfer to the nuclear target. The sum rule has contri- +butions from quasielastic (QE) scattering and from the +electro-excitations of nuclear states. At high q it is ex- +pected that SL → 1 because both nuclear excitation form +factors and Pauli suppression are small. At small q it is +expected that SL → 0 because all cross sections for in- +elastic processes (e.g. QE, nuclear excitation and pion +production processes) must be zero at q=0. +In this paper we present details of empirical parame- +terizations of the q dependence of all longitudinal and +transverse excitation form factors in 12C. Since there are +fewer measurements on 16O we only parameterize the +longitudinal form factors for this nucleus. We use these +parameterizations to compute the contribution of nuclear +excitations to SL(q) for both nuclei. Our investigation +of the QE contribution to SL(q) is reported in an earlier +publication[2]. +Since the cross sections for nuclear excitations are sig- +nificant at low q, the parametrizations should be used in +precise calculations of radiative corrections to quasielas- +tic electron scattering at low q. Because of intial state +radiation, nuclear excitations also contribute to radia- +tive corrections in deep-inelastic electron scattering at +large ν. The parameterizations also serve as benchmark +in testing theoretical modeling of electron and neutrino +scattering at low energies. Because of recent advances +in theoretical methods[3–5] for the calculations of the re- +sponse functions of electron scattering on nuclear targets, +it is now possible to make theoretical predictions of the +form factors for the excitation of nuclear states in both +electron and neutrino scattering[6–8]. +Figures 1 and 2 show the relative contributions of the +cross sections for elastic scattering from the 12C nucleus, +as well as the low lying excitations of nuclear states for +several low energy data sets [9–11]. Also shown as a solid +curve is our parameterization utilizing the experimental +resolution to apply a Gaussian smearing to each state. +II. +THEORETICAL FRAMEWORK +The electron scattering differential cross section can be +written in terms of longitudinal (RL(q, ν)) and transverse +(RT (q, ν)) nuclear response functions [12]: +dσ +dνdΩ = σM[ARL(q, ν) + BRT (q, ν)] +(1) +where σM is the Mott cross section, +σM = α2 cos2)(θ/2) +4E2 +0 sin4(θ/2). +(2) +Here, E0 is the incident electron energy, E′ is energy +of the final state electron, ν = E0 − E′ is the energy +transfer to the target, q is the 3-momentum transfer, Q2 +is the square of the 4-momentum transfer (defined to be +positive such that q2 = Q2 + ν2), A = (Q2/q2)2 and +B = tan2(θ/2) + Q2/2q2. For nuclear elastic scattering +at very low q on 12C Q2 = q2 to a good approximation. +For elastic scattering and nuclear excitations the +square of the electric and magnetic form factors are ob- +tained by the integration of the measured response func- +tions over ν. +In the experimental extractions of form +factors for elastic scattering and nuclear excitations the +Mott cross section is defined with an additional factor +of Z2 because both the nuclear elastic cross section and +the cross sections for the the electro excitation of nuclear +states are proportional to Z2 times charge form factors +F 2 +iC(q). +Here, the subscript zero denotes the nuclear +arXiv:2301.05650v1 [nucl-th] 13 Jan 2023 + +2 +FIG. 1: Top panel: Radiatively corrected cross section from +Yamaguchi[10](measured with high resolution of 0.25%) for +the scattering of 250 MeV electrons from 12C at 350. Here, +the cross section for the elastic peak has been divided by 100 +and the cross section for the 4.43 MeV state by 4. Middle +panel: +Radiatively corrected cross section[9] (in arbitrary +units) for the scattering of 250 MeV electrons from 12C at +550. Bottom panel: Radiatively corrected cross section[9] +(in arbitrary units) for the scattering of 600 MeV electrons +from 12C at 330. The peaks for elastic scattering and for the +first three nuclear excitations at 4.43, 7.66 and 9.64 MeV are +clearly visible. The solid curve is the predicted radiatively +corrected cross section using our fits to the form factors and +QE cross sections. The fit is normalized to the elastic cross +section for the E=250 MeV and 550 data. For the E= 600 +MeV and 330 data we normalize to the cross section for the +4.43 MeV state. +elastic peak and subscripts 1-N denote nuclear excita- +tions. The charge form factors can be thought of as the +product[11, 13] of the proton electric form factor and the +form factors of the spatial distribution of protons in the +nucleus. +FIG. 2: Radiatively uncorrected cross section (in arbitrary +units) from the LEDEX experiment[11] on 12C. The solid red +line is the radiatively uncorrected cross section from our fit to +the form factors and QE cross sections. It is normalized to +the elastic cross section at zero excitation energy for the 12.5 +and 17.0 degree data, and to the cross section for the 4.43 +MeV state for the 30.5 and 61.0 degree data). +III. +COULOMB SUM RULE +The inelastic Coulomb Sum Rule is the integral of +RL(q, ν)dν, excluding the elastic peak and pion produc- +tion processes. It has contributions from QE scattering +and from electro-excitations of nuclear states: +CSR(q) = +� +RL(q, ν)dν +(3) += +� +RQE +L (q, ν)dν + G′2 +E(Q2) × Z2 +L +� +all +F 2 +i (q) += G′2 +E(Q2) × +� +Z +� +V QE +L +(q, ν)dν + Z2 +L +� +all +F 2 +i (q) +� +. +We define V QE(q, ν) as the reduced longitudinal QE re- +sponse, which integrates to unity in the absence of any +suppression (e.g. Pauli blocking). The charge form fac- + +Yamaguchi 71 +q = 0.75 fm-1 (0.15 GeV) +30 +Elastic +7.65 MeV +da/dQ/dE' [μb/(sr GeV)) + 4.43 MeV +25 +12C +E = 250 MeV +20 +0 = 35 deg +9.64 MeV +10 +5 +0 +5 +10 +15 +20 +25 +30 +Excitation Energy (Mev)Crannell 64 q = 1.15 fm-1 (0.23 GeV) +10 +Elastic +8 +12C +E = 250 MeV +@ = 55 deg +Counts +6 +4.43 MeV +9.64 MeV +7.65 MeV +2 +2.5 +0 +2.5 +5 +7.5 +10 +12.5 +15 +17.5 +20 +Excitation Energy (Mev)Crannell 66 +q=1.74 fm-1 (0.34 GeV) +1000 +12C +4.43 MeV +E = 600 MeV +@ = 33 deg +800 +9.64 MeV +7.65 Mel +400 +不 +200 +Elastic +2.5 +0 +2.5 +5 +7.5 +10 +12.5 +15 +17.5 +20 +Excitation Energy (MeV)10000 +E = 0.362 0 = 12.5 +5000 +q = 0.4 fm-1 = 0.08 MeV +0 +Events /charge +40000 +E = 0.362 0 = 61 +q = 1.85 fm-1 = 0.36 MeV +20000 +75000 +E = 0.685 0 = 17 +50000 +q = 1.03 fm-1 = 0.2 MeV +25000 +0 +× 10 +2000 +E = 0.685 0 = 30.5 +q = 1.85 fm-1 = 0.36 Mev +1000 +0 +0 +2 +4 +6 +8 +101214 +16 +18 +20 +Excitation Energy (MeV)3 +1.E-8 +1.E-7 +1.E-6 +1.E-5 +1.E-4 +1.E-3 +1.E-2 +1.E-1 +1.E+0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 13 14 +qeff2 (fm-2 ) +Carbon Elastic (Form Factor)2 +80 MeV +150 MeV +187 MeV +300 MeV +374.5 MeV +600 MeV +747.2 MeV +800 MeV +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 13 14 +qeff2 +(fm-2 ) +Carbon Elastic +(Form Factor)2 ratio to fit +80 MeV +150 MeV +187 MeV +300 MeV +374.5 MeV +600 MeV +747.2 MeV +800 MeV +1.0 +10- 1 +10- 2 +10- 3 +10- 4 +10- 5 +10- 6 +10- 7 +10- 8 +FIG. 3: Top panel: Measurements[17] of the nuclear elastic +longitudinal charge form factor (squared) for 12C versus q2 +eff. +Bottom panel: Ratio to our fit. +tors for the electro-excitation of nuclear states F 2 +iC(q) is +related to F 2 +i (q) by the expression F 2 +iC(q) = G′2 +Ep(Q2) × +F 2 +i (q). +In order to account for the small contribution +of the neutron and relativistic effects G′2 +E(Q2) is given +by[12]: +G′2 +E(Q2) = [G2 +Ep(Q2) + N +Z G2 +En(Q2)] 1 + τ +1 + 2τ , +(4) +where, GEp and GEn are the electric form factors [14] of +the proton and neutron, respectively and τ = Q2/4M 2 +p. +By dividing Eq. 3 by ZG′2 +Eq) we obtain the normalized +inelastic Coulomb Sum Rule SL(q) : +SL(q) = +� +V QE +L +(q, ν)dν + Z +L +� +all +F 2 +i (q). +(5) +IV. +PARAMETERIZATION OF 12C NUCLEAR +ELASTIC AND NUCLEAR EXCITATION FORM +FACTORS +A. +12C elastic form factor versus q2 +eff +The 12C nucleus has a spin parity of 0+. We fit the +measured 12C elastic longitudinal (charge) form factor +with the following functional form: +F 2 +oC(q2 +eff) = 1 + 1.5 × 10−3q4 +eff +1 + eP ower +[H2(q2 +eff) + G(q2 +eff)] (6) +Here, Power = +q2 +eff−12.0 +1.4 +is included to better describe +the form factor at very large q. The effective[15] q2 is +q2 +eff = q2(1 + 4Zα/(3⟨r2⟩E) +Which for carbon is q2 +eff = q2(1 + 0.00465/E)2 (where E +is in GeV). The function H(qeff 2) is the harmonic well +shape with (α= 1.21, and a0=1.65). It is is given by[16]: +H(q2 +eff) = [1 − +αq2 +effa2 +0 +2(2 + 3α)]exp[−q2 +effa2 +0 +4 +], +(7) +The function G(q2 +eff) fills in the dip in the location of the +diffraction minimum. +G(q2 +eff) = 5.0 × 10−5e−[(q2 +eff−3.1)/0.66)]2 +In the above parametrization q2 +eff is in units of fm−2. A +comparison of the parametrization of the nuclear elas- +tic charge form factor for +12C to experimental data[17] +is shown on the top panel of Fig. 3. The ratio of the +measurments to the fit is shown on the bottom panel. +B. +Form factors for nuclear excitations in 12C +We begin by parameterizing the measurements of the +longitudinal and transverse form factors for the electro- +excitation of all nuclear states in 12C with excitation en- +ergies (Ex) less than 16.0 MeV (the approximate proton +removal energy from 12C). For these states the measure- +ments are straightforward since the QE cross section is +zero for Ex < 16 MeV. +1. +12C excitation form factors for the 4.44 MeV and 9.64 +MeV states +The longitudinal form factors (squared) for the electro +excitation of the 4.44 and 9.64 MeV nuclear excited states +are parametrized as F 2 +iC(q2 +eff) where +F 2 +iC(q2 +eff) = +(q2 +eff)3 +(q2eff)3 + d +j=3 +� +j=1 +Nje−[(q2 +eff−Cj)/σ]2 +(8) +Here q2 is in units of fm−2. +The parameters for the +4.44 and 9.64 MeV states are given in Table I. Com- +parisons of our parametrizations of the excitation form +factors (squared) for the 4.44, and 9.64 MeV states to +experimental data[17] are shown in Figures 4 and 5. + +4 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +qeff (fm-1) +Carbon 4.43 MeV State +(Form Factor)2 +80 MeV +150 MeV +187 MeV +250 MeV +300 MeV +420 MeV +600 MeV +800 MeV +10- 1 +10- 2 +10- 3 +10- 4 +10- 5 +10- 6 +1.E-06 +1.E-05 +1.E-04 +1.E-03 +1.E-02 +1.E-01 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +qeff (fm-1) +Carbon 4.43 MeV State +(Form Factor)2 +80 MeV +150 MeV +187 MeV +250 MeV +300 MeV +420 MeV +600 MeV +800 MeV +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +qeff (fm-1) +Carbon 4.43 MeV State +(Form Factor)2 +80 MeV +150 MeV +187 MeV +250 MeV +300 MeV +420 MeV +600 MeV +800 MeV +10- 1 +10- 2 +10- 3 +10- 4 +10- 5 +10- 6 +0.000 +0.002 +0.004 +0.006 +0.008 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +qeff (fm-1) +Carbon 9.64 MeV State +(Form Factor)2 +150 MeV +187 MeV +250 GeV +300 MeV +600 MeV +800 MeV +FIG. 4: +Measurements[17] of the longitudinal charge form factors (squared) for the 4.43 MeV state (left) and the 9.64 MeV +state (right) in 12C. The form factors (squared) are shown on linear scales and logarithmic scales on the top and bottom panels, +respectively. +State +N1 +C1 +σ1 +N2 +C2 +σ2 +N3 +C3 σ3 +d +Data from Ref. +4.44 MeV 2+ L (q2 +eff) 1.41 × 10−2 1.125 1.71 7.2 × 10−4 3.00 2.0 7.0 × 10−6 7.6 5.0 0.10 +[17] +9.64 MeV 3−L (q2 +eff) +5.00 × 10−3 +1.46 1.70 6.6 × 10−4 3.46 1.9 2.1 × 10−5 7.0 2.5 0.20 +[17] +TABLE I: +Parameters of our fits (eq. 8) to the 12C longitudinal charge form factors (squared) for the 4.44 and 9.64 MeV +nuclear excited states in 12C. +For these states, the parametrizations are in terms of q2 +eff in units of fm−2. +Here q2 +eff = +q2 × (1 + 0.00465/E)2, where E is in GeV[15]. +2. +12C form factors for the 7.65 MeV state and states with +excitation energies above 10 MeV +Measurements of the square of the longitudinal form +factor verses q (in units of fm−1) for the 7.65 MeV state +in 12C (from Chernykh et. al. [18]) are shown on the +left panel of Fig. 6. A comparison of the nuclear elastic +form factor to the form factors of the first three nuclear +excitations versus q (in units of GeV) is shown on the +right panel of Fig. 6. +The charge form factors (squared) for the electro- +excitation of the 7.65 MeV state and for states with +excitation energies above 10 MeV are parameterized as +F 2 +iC(q) = Max(0.0, g2 +i ) where +g2 +i (qeff) = +j=3 +� +j=1 +Nje−[(qeff −Cj)/σ]2 − ae−bqeff . +(9) +Here, qeff is in units of fm−1. +The parameters are +given in Table II. (Note that these states are parame- +terized versus qeff, while the 4.44 and 9.64 MeV states +are parametrized versus qeff 2). As shown on the right +panel of Fig. 6, for q near the diffraction minimum for +elastic scattering on 12C the cross sections for the three +nuclear excitations below 10 MeV are larger than the nu- +clear elastic cross section. Note that unlike the nuclear +elastic form factor which is equal to 1.0 at q=0, all lon- +gitudinal form factors for the nuclear excitations must +vanish at q=0. + +10-2 ++ 150MeV + 187 MeV +10-3 +^ 250 GeV +. 300 MeV ++ 600 MeV + 800 MeV +10-4 +10-5 +Carbon 9.64 Mev state +(Form Factor)2 +10-6 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +qeff (fm-1)5 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +qeff (fm-1) +Carbon 4.43 MeV State +(Form Factor)2 ratio to fit +80 MeV +150 MeV +187 MeV +250 MeV +300 MeV +420 MeV +600 MeV +800 MeV +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +qeff (fm-1) +Carbon 9.64 MeV State +(Form Factor)2 ratio to fit +150 MEV +187 MeV +250 MeV +300 MeV +FIG. 5: Ratios of the measured[17] longitudinal charge form factors (squared) to our parametrizations for the 4.43 MeV state +(left) and the 9.64 MeV state (right) in 12C. +1.E-6 +1.E-5 +1.E-4 +1.E-3 +1.E-2 +1.E-1 +1.E+0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +q (GeV) +Carbon (Form Factor)2 +Elastic +4.43 MeV +9.61 MeV +7.68 MeV +1.0 +10- 1 +10- 2 +10- 3 +10- 4 +10- 5 +10- 6 +5B_all_ff.pdf +FIG. 6: Left panel: Measurements[18] of the longitudinal charge form factor (squared) for the 7.65 MeV state in 12C. Right +panel: A comparison of the nuclear elastic form factor (squared) to the form factors (squared) of the first three nuclear +excitations versus q (in units of GeV). +3. +12C form factors for states with excitation energies +above 10 MeV and below 16 MeV +We use equation 9 to parameterize the form factor for +excitation energy of 10.84 MeV[19], and also for excita- +tion energies of 12.71, 14.09 and 15.11 MeV[20, 21]. In +addition, we find that published differential cross sec- +tion measurements indicate that there is an additional +longitudinal continuum in the region between 12 to 15 +MeV. We parameterize this longitudinal continuum as +one broad state at 13.7 MeV (σ=1.25 MeV). For the +transverse form factors in this region we parametrize the +data of Hicks84[22]. +4. +12C form factors for states with excitation energies +above 16 MeV +Initially, we parameterize the longitudinal and trans- +verse form factors measured by Yamaguchi71[10] for +states with excitation energies above 16 MeV. How- +ever, in the Yamaguchi71 analysis the contributions from +quasielatic (QE) scattering are not subtracted. There- +fore, We perform a reanalysis of the Yamaguchi71 data +in combination of all published cross sections with 16 < +Ex < 55 MeV. We subtract the QE contribution using +our QE model[2] (which includes superscaling[23] with +Rosenfelder[24] Pauli Suppression) and extract updated +longitudinal and transverse form factors. For Ex > 20 + +2 +10 +12 +12 +C(e,e')"C +10° +E, = 7.654 MeV +10 +0, +(b) +F. +10" +0 +2 +3 +q [fm'l]6 +State +N1 +C1 +σ1 +N2 +C2 +σ2 +N3 +C3 +σ3 +a +b +Ref. +7.65 MeV 0+L +2.8 × 10−3 +0.93 +0.42 +3.0 × 10−4 +1.45 0.24 2.0 × 10−5 2.48 0.53 1.0 × 10−4 1 +[18] +10.84 MeV 1−L +5.0 ×10−4 +1.0 +0.3 +8.0 × 10−4 +1.4 +0.4 +− +- +- +- +- +[19] +11.83 MeV 2− T +3.9 ×10−5 +1.2 +0.5 +1.2 × 10−5 +2.0 +0.4 +- +- +- +- +- +[22] +12.71 MeV 1+T +3.0 ×10−6 +0.63 +0.4 +1.0 × 10−8 +1.0 +0.1 +2.0 × 10−6 +1.8 +0.6 +2.5 × 10−5 1 +[20] +13.7 MeV 4−L +4.0 ×10−4 +1.0 +0.35 +1.0 × 10−3 +1.75 0.45 4.0 × 10−4 0.85 0.65 +- +[9–11] +σ=1.25 MeV +- +14.08 MeV 4+L +2.4 ×10−5 +1.8 +0.6 +- +- +- +- +- +- +- +[21] +15.1 MeV 1+L +6.0 × 10−4 +0.85 +0.7 +- +- +- +- +- +- +- +- +[20] +15.1 MeV-T +2.5 ×10−4 +0.63 +0.4 +2.8 × 10−4 +0.84 0.2 +2.4 × 10−5 +2.0 +0.5 +2.5 × 10−5 1 +[20] +16.1 MeV 2+L +12.0 × 10−4 1.05 +0.6 +- +- +- +- +- +- +- +- +[10] +16.1 MeV 2+T +5.9 × 10−4 +1.2 +0.55 +2.4 × 10−4 +2.2 +0.6 +- +- +- +- +- +16.6 MeV 2−T +2.6 × 10−4 +1.6 +0.6 +5.0 × 10−5 +2.5 0.35 +- +- +- +- +- +[10][22] +18.1 MeV 1+T +1.9 × 10−4 +0.8 +0.35 +1.8 × 10−4 +1.25 0.45 +- +- +- +- +- +[10][22] +18.6 MeV-L +3.2 × 10−4 +1.3 +0.5 +- +- +- +- +- +- +- +- +[10] +19.3 MeV 2−T +1.02 × 10−3 +1.32 +0.77 +3.75 × 10−4 +1.7 +0.6 +1.0 × 10−4 +2.2 +0.3 +3.6 × 10−4 1 [10][22] +20.0 MeV 2+L +1.6 × 10−4 +1.2 +0.42 +1.6 × 10−5 +1.8 +0.4 +- +- +- +- +- +[10] +20.6 MeV 3−T +1.9 × 10−4 +1.45 +0.5 +5.5 × 10−5 +2.1 +0.4 +- +- +- +- +- +[10][22] +(21-26 MeV) +- +23.0 MeV-L +2.8 × 10−3 +0.60 +0.15 +6.9 × 10−3 +0.84 0.55 +- +- +- +- +- +[10] +σ=4.75 MeV +- +23.0 MeV-T +1.83 × 10−3 +0.8 +0.36 +1.0 × 10−4 +1.5 +0.5 +- +- +- +- +- +[10] +(26-37 MeV) +- +31.5 MeV-L +4.7 × 10−3 +1.0 +0.48 +- +- +- +- +- +- +- +[10] +σ=9 MeV +- +31.5 MeV-T +9.0 × 10−4 +0.35 +0.3 +- +- +- +- +- +- +- +- +[10] +(30-50 MeV) +- +42 MeV-L +2.6 × 10−3 +1.49 +0.7 +- +- +- +- +- +- +- +- +σ=12 MeV +- +Extra Strength +- +TABLE II: Parameterizations of the Longitudinal (L) and Transverse (T) 12C nuclear excitation form factors (squared) for the +7.65 MeV state and for states with excitation energy above 10 MeV. Unlike the parametrizations in Table I for the 4.44 and +9.64 MeV states which are functions of the square of the 3-momentum transfer q2 +eff in units of fm−2, the parametrizations for +the states in this table are functions of qeff in units of fm−1. Here q2 +eff = q2 × (1 + 0.00465/E)2, where E is in GeV[15]. +MeV (region of the Giant Dipole resonances) we group +the strength from multiple excitations into a three states +with a large width in Ex and extract effective form factors +accounting for all states in these regions. The updated +parameters are given in Table II. +The longitudinal and transverse response functions +for +12C, +RL(q, Ex) +and +RT (q, Ex), +extracted +by +Yamaguchi71[10] for excitation energies above 16 MeV +and less than 40 MeV are shown in Figure 7 (black +points). Also shown are RL(q, Ex) and RT (q, Ex) ex- +tracted from our universal fit to all electron scattering +cross section data on 12C (solid red line). The QE con- +tribution to the total response functions is shown as the +dashed red line. An estimated resolution smearing of 600 +keV has been applied to the excitations in the fit to match +the data. While individual states are well reproduced at +low excitation energy, above Ex of 20 MeV the effect of +grouping several excitations together into three broad ef- +fective states in the fit can be seen. While the fit does +not capture the structure from individual states above 20 +MeV, the total strength is seen to be well reproduced. +C. +Comparison to 12C experimental data for +excitation energies below 50 MeV +Experimental radiatively corrected inelastic electron +scattering cross sections on 12C for excitation energies +less than 50 MeV are shown in Figure 8. Also shown are +the corresponding cross sections from our universal fit to +all 12C data. The cross sections for excitation energies +less than 12 MeV are multiplied by (1/6). The pink solid +line is the predicted total cross section from our univer- +sal fit[2] which include the contributions from all sources +(nuclear excitation form factors, quasilelastic scattering +and pion production processes). +The QE contribution +is shown as the dashed blue line and the ”Transverse +Enhancement/Meson Exchange Currents” contribution + +7 +FIG. 7: Comparison of the longitudinal (RL, left) and transverse (RT , right) response functions for 12C extracted by Yamaguchi +71[10] (black squares) to the response functions extracted from our universal fit to all available electron scattering cross section +data on 12C (solid red line). The contributions from excitation energies less than 12 MeV are multiplied by (1/6). The QE +contribution to the total response functions is represented by the red dashed line. In our fit, we model the response functions +for all states the region of the Giant Dipole Resonance (20-30 MeV) region as one average broad excitation. +is shown as the dot-dashed line. Details of the fit are +described in reference[2]. Most of the cross section mea- +surements are from Yamaguchi71[10]. The cross sections +for Eo=54 MeV at 1800 are from Goldemberg64[25] and +the the cross sections for Eo=65 MeV at 1800 are from +deForest65[26]. The measurements at 1800 are only sen- +sitive to the transverse form factors. +V. +ANALYSIS OF 16O EXCITED STATES +A. +16O excited states with Ex < 12.5MeV +In order to minimuze correlations between our param- +eterizations of the form factors for the nuclear excitations +in 12C and 16O we parameterize the form factors for 16O +states using a somewhat different functional form. The +form factors for the nuclear excited states in 16O are pa- +rameterized as F 2 +iC(q2) = Max(0.0, g2 +i ) where +g2 +i (q2 +eff) = q2 +eff × +� j=3 +� +j=1 +Nje−[(q2 +eff −Cj)/σ]2 − ae−bq2 +qeff � +. +(10) +Here, q2 +eff is in units of fm−2. +The form factors for nuclear excitations in 16O with ex- +citation energies below proton removal threshold (about +12 MeV) are easily measured because there are is no con- +tribution from QE scattering in this region. +The nine +longitudinal form factors (squared) in 16O for excitation +energies below 12.5 MeV are shown in Fig. 9 on linear +and logarithmic scales. The data in the figures are from +Buti-86[27]. The solid blue lines are our parameteriza- +tions using the parameters listed in Table III. + +Fit total +Fit QE +0.002 +q = 0.148 GeV +0.002 +q = 0.167 GeV +(Mev-1) +0 +q = 0.205 GeV +0.002 +R +0 +q = 0.24 GeV +0.002 +0 +q = 0.307 GeV +0.001 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Excitation Energy (MeV)Fit total +Fit QE +0.001 +q = 0.148 GeV +0 +0.001 +q = 0.167 GeV +Rt (Mev-1) +0 +0.001 +q = 0.205 GeV +0 +0.001 +q = 0.24 GeV +0 +q = 0.307 GeV +0.001 +0 +10 +15 +20 +25 +30 +35 +40 +Excitation Energy (Mev)8 +FIG. 8: Radiatively corrected inelastic electron scattering cross sections on 12C for excitation energies less than 50 MeV. The +cross sections for excitation energies less than 12 MeV are multiplied by (1/6). The pink solid line is the predicted total cross +section from our universal fit[2] to all electron scattering data on 12C. The fit include nuclear excitations, a superscaling QE +model[23] with Rosenfelder Pauli suppression[24] (dashed blue line), ”Transverse Enhancement/Meson Exchange Currents” +(dot-dashed line) and pion production processes (at higher excitation energies). The data are from Yamaguchi71[10] except +for the cross sections for Eo=54 MeV and 1800 (from Goldemberg64[25]) and the cross sections for for Eo=65 MeV and 1800 +(from deForest65[26]). The measurements at 1800 are only sensitive to the transverse form factors. + +Total +Trans. Enhancement +QE +E = 0.054 +@ = 180 +750 +E = 025 @ = 35 +20 +9 = 0.102 GeV +q=0.147 GeV +500 +10 +X +250 +/(sr GeV)) +A.A +30 +E = 0.0888 +0 = 135 +20 +E = 0.065 0 = 180 +20 +q= 0.152 GeV +q=0.122GeV +10 +10 +E = 0.25 @ = 40 +E = 0.1 0 = 135 +400 +q= 0.166 GeV +20 + = 0.17 GeV +200 +10 +A.AA +200 +E = 0.25 +0 = 50 +20 += 0.12 = 135 +q = 0,202 GeV +q = 0.201 GeV +100 +10 +100 +20 +E = 0.25 @ = 60 +E = 0.1405 @ = 135 +q = 0.235 GeV +q = 0.232 GeV +50 +10 +20 +E=0.250=80 +10 +E = 0.1776 +0= 135 +q = 0,291 GeV +q = 0.285 GeV +10 +5 +0 +10 +20 +30 +40 +50 +10 +20 +30 +4050 +Excitation Energy (MeV) +Excitation Energy (MeV)9 +0.E+00 +1.E-04 +2.E-04 +3.E-04 +4.E-04 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 6.05 MeV +10- 4 +3 +2 +1 +0 +Longitudinal +(Form Factor)2 +16O 6.05 MeV +1.E-07 +1.E-06 +1.E-05 +1.E-04 +1.E-03 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +FF2 +qeff (fm-1 ) +16O 6.05 MeV +10- 3 +10- 4 +10- 5 +10- 6 +10- 7 +0.0E+00 +1.0E-03 +2.0E-03 +3.0E-03 +4.0E-03 +5.0E-03 +6.0E-03 +7.0E-03 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff +(fm-1 ) +16O 6.13 MeV +Longitudinal +(Form Factor)2 +6 +4 +2 +0 +16O 6.13 MeV +1.E-06 +1.E-05 +1.E-04 +1.E-03 +1.E-02 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +10- 2 +10- 3 +10- 4 +10- 5 +10- 6 +10- 3 +0.0E+00 +5.0E-04 +1.0E-03 +1.5E-03 +2.0E-03 +2.5E-03 +3.0E-03 +3.5E-03 +4.0E-03 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 6.92 MeV +10- 2 +10- 3 +10- 4 +10- 5 +10- 6 +10- 7 +Longitudinal +(Form Factor)2 +10- 3 +3 +2 +1 +0 +16O 6.92 MeV +1.E-06 +1.E-05 +1.E-04 +1.E-03 +1.E-02 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 6.92 MeV +10- 2 +10- 3 +10- 4 +10- 5 +10- 6 +0.0E+00 +1.0E-03 +2.0E-03 +3.0E-03 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 7.12 MeV +3 +2 +1 +0 +Longitudinal +(Form Factor)2 +10- 3 +16O 7.12 MeV +1.E-06 +1.E-05 +1.E-04 +1.E-03 +1.E-02 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 7.12 MeV +10- 2 +10- 3 +10- 4 +10- 5 +10- 6 +0.0E+00 +1.0E-04 +2.0E-04 +3.0E-04 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 9.84 MeV +3 +2 +1 +0 +Longitudinal +(Form Factor)2 +10- 4 +16O 9.84 MeV +1.E-06 +1.E-05 +1.E-04 +1.E-03 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 9.84 MeV +10- 3 +10- 4 +10- 5 +10- 6 +0.E+00 +1.E-04 +2.E-04 +3.E-04 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 10.35 MeV +3 +2 +1 +0 +Longitudinal +(Form Factor)2 +10- 4 +16O 10.35 MeV +1.0E-06 +1.0E-05 +1.0E-04 +1.0E-03 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff +(fm-1 ) +16O 10.35 MeV +10- 3 +10- 4 +10- 5 +10- 6 +0.E+00 +1.E-04 +2.E-04 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +FF2 +qeff (fm-1 ) +16O 11.09 MeV +10- 4 +1 +0 +Longitudinal +(Form Factor)2 +16O 11.09 MeV +1.E-06 +1.E-05 +1.E-04 +1.E-03 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 11.09 MeV +10- 3 +10- 4 +10- 5 +10- 6 +0.E+00 +1.E-03 +2.E-03 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 11.52 MeV +10- 3 +1 +0 +Longitudinal +(Form Factor)2 +16O 11.52 MeV +1.E-06 +1.E-05 +1.E-04 +1.E-03 +1.E-02 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 11.52 MeV +10- 2 +10- 3 +10- 4 +10- 5 +10- 6 +1 +0 +1.E-06 +1.E-05 +1.E-04 +1.E-03 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 12.05 MeV +10- 3 +10- 4 +10- 5 +10- 6 +0.E+00 +2.E-04 +4.E-04 +6.E-04 +8.E-04 +0.0 +1.0 +2.0 +3.0 +FF2 +qeff (fm-1 ) +16O 12.05 MeV +Longitudinal +(Form Factor)2 +6 +4 +2 +0 +10- 4 +16O 12.05 MeV +FIG. 9: +The square of the longitudinal form factors in 16O for nuclear excitations below 12.5 MeV on linear and logarithmic +scales. The data are from Buti-86[27]. The blue solid lines are our parameterizations from Table III. + +10 +O16_0.55_fb.pdf +0 +0.5 +1 +1.5 +2 +10 +20 +30 +RL 10-3/MeV +Excitation Energy (MeV) +RL q = 0.108 GeV (0.55 fb-1) +Hotta Data +Hotta QE Estimate +QE superscaling fit +16O +O16_0.85_fb.pdf +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +10 +20 +30 +RL 10-3/MeV +Excitation Energy (MeV) +RL q = 0.167 GeV (0.85 fb-1) +Hotta Data +Hotta QE Estimate +QE superscaling fit +16O +O16_1.05_fb.pdf +0 +0.5 +1 +1.5 +2 +2.5 +3 +10 +20 +30 +RL 10-3/MeV +Excitation Energy (MeV) +RL q = 0.207 GeV (1.05 fb-1) +Hotta Data +Hotta QE Estimate +QE superscaling fit +16O +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +q (fb-1 ) +O16 Hotta +O16 Fit +C12 Fit 10-20 MeV +12.5-20 MeV +1.000E-04 +1.000E-03 +1.000E-02 +1.000E-01 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +q (fb-1 ) +O16 Hotta +O16 Fit +C12 Fit 10-20 MeV +12.5-20 MeV +10- 1 +10- 2 +10- 3 +10- 4 +16O Hotta74 +16O Fit +12C Fit 10-20 MeV +16O Hotta74 +16O Fit +12C Fit 10-20 MeV +States_Figure.pptx +O16_paper_125-20.pdf +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +q (fb-1 ) +20-35 MeV +O16 Hotta +O16 Goldman +O16 fit +C12 fit +1.E-04 +1.E-03 +1.E-02 +1.E-01 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +FF2 +q (fb-1 ) +20-35 MeV +O16 Hotta +O16 Goldman +O16 fit +C12 fit +10- 1 +10- 2 +10- 3 +10- 4 +16O Hotta74 +16O Goldman70 +16O Fit +12C Fit +16O Hotta74 +16O Goldman70 +16O Fit +12C Fit +States_Figure.pptx +O16_paper_20-35.pdf +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +q (fb-1 ) +35-55 MeV +O16 Fit +C12 Fit +1.E-04 +1.E-03 +1.E-02 +1.E-01 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +FF2 +q (fb-1 ) +35-55 MeV +O16 Fit +C12 Fit +10- 1 +10- 2 +10- 3 +10- 4 +16O Fit +12C Fit +16O Fit +12C Fit +States_Figure.pptx +O16_paper_35-55.pdf +FIG. 10: +Top row: The longitudinal response function 12C, RL(q, Ex) for 16O from Hotta74[28] for three values of q. The +red dashed line is the original estimate of the QE contribution used in Hotta74. The green solid line is the QE contribution +determined using our superscaling model. We use these data to extract the longitudinal form factors for nuclear excitations in +16O for the 12.5-20 MeV and the 20-35 MeV regions in excitation energy. Middle and bottom rows: The q dependence of +the longitudinal form factor for the 12.5-20 MeV, 20-35 MeV and 35-55 MeV regions in excitation energy. +State +MeV +N1 +C1 +σ1 +N2 +C2 +σ2 +N3 +C3 +σ3 +a +b +Ref. +0+ +2 L +6.0494 +0.70 0.35 0.58 0.120 1.20 0.500 0.0050 4.30 1.60 0.220 +6.00 +Buti-86 +3− +1 L +6.1299 +1.60 1.00 0.45 4.100 1.07 1.700 0.2000 2.55 2.40 3.100 +1.10 +Buti-86 +2+ +1 L +6.9171 +6.50 0.20 0.75 1.000 1.10 0.955 0.0032 5.30 1.35 5.000 +2.50 +Buti-86 +1− +1 L +7.1169 +0.95 0.94 0.68 0.800 1.50 1.200 0.1000 2.40 1.55 0.400 +3.00 +Buti-86 +2+ +2 L +9.8445 +0.10 0.70 0.65 0.080 1.70 1.200 0.0120 2.50 2.0 +0.007 2.000 +Buti-86 +4+ +1 L +10.3560 +0.09 1.30 0.70 0.087 2.05 1.200 0.0140 3.30 1.7 +0.007 2.000 +Buti-86 +4+ +2 L +11.0967 +0.04 1.10 0.85 0.042 2.20 1.100 0.0110 3.20 1.9 +0.000 10.000 +Buti-86 +2+ +3 L +11.5200 +2.00 0.50 0.60 0.600 1.00 1.050 0.0040 5.30 1.4 +0.007 1.657 +Buti-86 +0+ +3 L +12.0490 +1.15 0.20 0.95 0.050 1.50 0.850 0.0015 5.20 1.3 +1.00 +4.00 +Buti-86 +12.5-20.0 +2.00 0.35 0.30 8.000 0.0 +1.90 +- +- +- +0.007 +1.66 +Hotta74 +20.0-35.0 18.00 0.00 0.40 7.000 0.6 +0.80 +2.5000 1.0 +1.5 +- +- +Hotta74 +35.0-55.0 +0.8 +0.00 0.30 1.300 0.6 +3.00 +- +- +- +0.004 +1.66 +use carbon +TABLE III: Parameterizations of the square of the longitudinal (L) nuclear excitation form factors in 16O (in units of 10−3). +Data taken from Buti-86[27] and Hotta74[28]. The parameterizations are functions of q2 +eff in units of fm−2. + +11 +12/9/22 +0.0 +0.1 +0.2 +0.3 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +ZS F2i (q) +q (GeV) +12C : Contribution to SL inelastic +0.1 MeV < Ex <55.0 MeV +0.1 MeV < Ex <35.0 MeV +0.1 MeV < Ex< 20.5 MeV +0.1 MeV < Ex < 10.0 MeV +C12_CSR_States_q.pdf +CSR5 copy3_specia; +C12_CSR_States_q.pdf +CSR5 copy3_specia; +C12_CSR_States_q.pdf +12/9/22 +C12_CSR_States_fit.pdf +0.0 +0.1 +0.2 +0.3 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Z SF2i (q) +q (GeV) +ZSFi2 (q) 0.1< Ex<55 MeV +(longitudinal) +Carbon +Carbon fit +FIG. 11: Left panel: The contribution of longitudinal nuclear excitations (between 2 and 55 MeV) to SL(q) in 12C. Right +panel: Our fit to the total contributions of all nuclear excitations below 55 MeV to SL(q) in 12C. The uncertainty is (shown +as a green band) is 0.01 plus 10% added in quadrature. +12/9/22 +O16_CSR_States_q.pdf +CSR5 copy3_specia; +O16_CSR_States_q.pdf} +0.0 +0.1 +0.2 +0.3 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +ZS F2i (q) +q (GeV) +16O: Contribution to SL inelastic +0.1 MeV < Ex < 55 MeV +0.1 MeV < Ex < 36 MeV +0.1 MeV < Ex < 20 MeV +0.1 MeV < Ex < 12.5 MeV +O16_CSR_States_q.pdf +AL27-O15 copy3_specia; +12/9/22 +0.0 +0.1 +0.2 +0.3 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Z SF2i (q) +q (GeV) +ZSFi2 (q) 0.1< Ex<55 MeV +(longitudinal) +Oxygen +Oxygen Fit +O16_CSR_States_fit.pdf +FIG. 12: +Left panel: The contribution of longitudinal nuclear excitations (between 2 and 55 MeV) to SL(q) in 16O. Right +panel: Our fit to the total contributions of all nuclear excitations below 55 MeV to SL(q) for 16O. The uncertainty (shown as +a green band) is 0.01 plus 10% added in quadrature. +B. +16O excited states with Ex > 12.5MeV +For excitation energy above 12.5 MeV there is a signifi- +cant contribution from QE scattering. Here we group the +states in two regions of excitation energy (12.5-20 MeV +and 20-35 MeV). +The top row in Fig. 10 shows the longitudinal response +function RL(q, Ex) for 16O from Hotta74[28] for three +values of q. The solid red line is the original estimate +of the QE contribution used in the Hotta74 publication. +The solid green line is the QE contribution determined +using the QE parameters from our universal fit to all +12C data. We find that the QE cross section predictions +for 16O using the parameters from the 12C fit also de- +scribe all (but limited) available data on 16O as shown in +[2]. We use the Hotta74 data to extract the longitudinal +form factors for the nuclear excitation in 16O in the 12.5- +20 MeV and the 20-35 MeV groupings in in excitation +energy. +The middle and bottom rows in Fig. 10 show the ex- +tracted longitudinal form factor for the 12.5-20 MeV and +20-35 MeV groupings in excitation energy on linear (mid- +dle) and logarithmic (bottom) scales. Also shown is the +form factor measurement from Goldman70[29]. The lon- +gitudinal form factor measured by Goldman70 for the 20– +30 MeV grouping in excitation energy has been corrected +by subtracting the QE contribution (from our universal +fit) and extending the excitation range to 20-35 MeV. +Since no data are available for the form factor for nu- +clear excitations the 36-55 MeV region in 16O we assume + +12 +0.0 +0.1 +0.2 +0.3 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Z SF2i (q) +q (GeV) +ZSFi +2 (q) 0.1< Ex<55 MeV +(longitudinal) +Oxygen +Carbon +12/9/22 +C12_O16_states.pdf +FIG. 13: A comparison of the contributions of nuclear excita- +tions to SL(q) in 12C and 16O. The uncertainty (green band) +in the total contribution of the excited states is 0.01 plus 10% +added in quadrature. +that the form factor for 16O is the same as the form factor +for 12C in this region. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Contributions to SL Inelastic +q (GeV) +QE Pauli suppressed +SL Inelastic +QE Supressed (Pauli+Quench) +C12 Excitated States +SL_components.pdf +12C +SL_components.pdf +FIG. 14: +The various contributions[2] to SL(q) for 12C (dot- +ted blue with yellow error band) including: QE with Pauli +suppression only (dotted-purple), QE suppressed by both +”Pauli” and ”Longitudinal Quenching” (solid-green), and the +contribution of nuclear excitations (red-dashed with green er- +ror band). +VI. +CONTRIBUTION OF NUCLEAR +EXCITATIONS TO SL(q) IN 12C AND 16O +The contributions of nuclear excitation to SL(q) (Eq. +5) in 12C and 16O are calculated using the form factor +parameterizations given in Tables I, II and III. The left +side panels of Figures 11 and 12 show the contributions +of nuclear excitations (with excitation energies below 10 +MeV, 20.5 MeV, 30 MeV and 55 MeV) to SL(q) for 12C +and 16O, respectively. +The total contribution to SL(q) can be parametrized +as follows: +Z +L +� +all +F 2 +i (q) = N1exp((x − C1)2/D2 +1) ++ N2exp(−(x − C2)2/D2 +2) ++ N3exp(−(x − C3)2/D2 +3) +(11) +where x= q/KF . For 12C KF =0.228 GeV, N1= 0.260, +C1=1.11, D1=0.50, N2= 0.075, C2=0.730, D2=0.30, and +N3= 0.01, C3=2.0, D3=0.30. The fit and the data are +shown on the right side panel of Fig. 11. +For +16O +KF =0.228 +GeV, +N1= +0.240, +C1=1.07, +D1=0.48, N2= 0.073 C2=0.70, D2=0.37, and N3= 0.039, +C3=1.55, D3=0.50. The fit and the data are shown on +the right side panel of Fig. 12. +Fig. 13 shows a comparison of the contributions of all +excited states to the SL(q) for 12C and 16O. The uncer- +tainty in the total contribution of the excited states for +both nuclei is 0.01 plus 10% added in quadrature. These +data indicate that the contribution of nuclear excitations +to SL(q) in 16O is consistent with being equal to the con- +tribution of the nuclear excitations in 12C within errors. +The total contribution of all states with excitation energy +below 55 MeV is largest at q=0.22 GeV, where it reaches +a maximum of 0.29± 0.03. +VII. +UPDATED EXTRACTION OF SL(q) FOR +12C AND 16O +In our previous paper[2] we performed a fit to all elec- +tron scattering data on 12C and 16O. +We found that +the QE transverse response function is enhanced at in- +termediate q and the longitudinal response function is +quenched at low q. We used the fits in combination with +the fits to nuclear excitations to extract SL(q) for 12C +and 16O. In our previous paper we used a very conserva- +tive estimate of the uncertainty in the total contribution +of the excited states (0.01 plus 15% added in quadra- +ture). In this paper we have updated our fits to the form +factors for individual nuclear excitations. We find that +the updated total contribution of nuclear excitations to +SL(q) for 12C and 16O is unchanged, but a smaller con- +servative estimate (0.01 plus 10% added in quadrature) +is more appropriate. +Fig. 14 shows the various contributions to the ex- +tracted SL(q) for 12C (dotted blue line with yellow er- +ror band). +Shown are the QE contribution with only +Pauli suppression (dotted-purple), the QE contribution +suppressed by both ”Pauli Suppression” and the longi- +tudinal quenching factor F L +quench(q) labeled as QE sup- + +13 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Normalized SL Inelastic +q (GeV) +Lavato 2016 Green's Function MC +SL Inelastic) +Mihaila 2000 (Carbon) AV18+UIX +Cloet 2016 (Carbon) RPA +Lovato_C12_compare +12C +Lovato_C12_compare +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Normalized SL Inelastic +q (GeV) +Sobczyk 2020CCSD: NNLO +SL Inelastic +Mihaila 2000 AV18+UIX +Potential +June 3, 2022 +Erro GO +16O +O16_SL.pdf +O16_SL.pdf +FIG. 15: +Left panel: SL(q) for 12C (dotted-blue with yellow error band) compared to theoretical calculations including +Lovato 2016 [3] (solid-purple), (Mihaila 2000[4] (dashed-green), and RPA Cloet 2016[30] (solid-red). Right panel: SL(q) for +16O (dotted-black with green error band) compared to theoretical calculations of Sobczyk 2020[5] (red-dashed) and Mihaila +2000 (dotted-dashed). +pressed (Pauli+Quench) (solid-green), and the contribu- +tion of nuclear excitations (red dashed line). +The left panel of Fig. 15 shows a comparison of the +extracted SL(q) for +12C (dotted-blue curve with yel- +low error band) to theoretical calculations. +These in- +clude the Lovato 2016[3] ”First Principle Green’s Func- +tion Monte Carlo” (GFMC) calculation (solid-purple +line), Mihaila[4] 2000 Coupled-Clusters based calcula- +tion (AV18+UIX potential, dashed-green), and Cloet +2016[30] RPA calculation (RPA solid-red). Our measure- +ment for 12C are in disagreement with Cloet 2016 RPA, +and in reasonable agreement with Lovato 2016 except +near q ≈ 0.30 GeV where the contribution from nuclear +excitations is significant. +The right panel of Fig. 15 shows SL(q) for 16O (dotted- +blue with green error band) compared to theoretical cal- +culations. These include the Sobczyk 2020[5] ”Coupled- +Cluster with Singles-and Doubles (CCSD) NNLOsat” +(red-dashed line), and Mihaila 2000[4] Coupled-Cluster +calculation with (AV18+UIX potential, dashed green +line). The data are in reasonable agreement with Sobczyk +2020. +VIII. +SUMMARY +We report on empirical parameterizations of longitu- +dinal and transverse nuclear excitation electromagnetic +form factors in 12C and 16O and extract the contribution +of nuclear excitations to the Normalized Inelastic Sum +Rule SL(q) as a function momentum transfer q. We find +that the total contribution is significant (0.29±0.030) at +q= 0.22 GeV. The total contributions of nuclear exci- +tations in 12C and 16O are consistent with being equal +within errors. Since the cross sections for nuclear exci- +tations are significant at low q, the radiative tails from +nuclear excitations should be included in precise calcula- +tions of radiative corrections to quasielastic electron scat- +tering at low q and deep-inelastic electron scattering at +large ν. +The parameterization also serves as a benchmark in +testing theoretical modeling of electron and neutrino +scattering cross sections at low energies. +Theoretical +studies of the excitation of nuclear states in electron and +neutrino scattering[6–8] indicate that both are equally +significant at low values of q. Therefore, nuclear excita- +tions should be included in both electron and neutrino +MC generators. +We note that for excitation energies +above proton removal threshold (about 16 MeV in 12C +and 12 MeV in 16O) the decays of nuclear excitations can +have a proton in the final state and therefore cannot be +distinguished experimentally from QE scattering in low +resolution neutrino experiments. +IX. +ACKNOWLEDGEMENTS +Research supported by the U.S. Department of En- +ergy under University of Rochester grant number DE- +SC0008475, and the Office of Science, Office of Nuclear +Physics under contract DE-AC05-06OR23177. +[1] D. Drechsel and M M Giannini 1989 Rep. Prog. Phys. +52 1083 (eq. 7.9); T. de Forest Jr. and J.D. Walecka, +Advances in Physics, 15:57, 1-109 (1966) (eq. 6.8). + +14 +[2] A. Bodek and M. E. 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Udias, Phys. +Rev. D 89, 093002 (2014); G.D. Megias Vazquez (Tesis +Doctoral). Universidad de Sevilla, Sevilla (2017). +[25] J. Goldemberg and W. C. Barber, Phys. Rev. 134, B963 +(1964). +[26] T. de Forest, J. D. Walecka, G. Vanpraet, and W. C. +Barber, Phys. Letters 16, 311 (1965). +[27] T. N. Buti et al., Phys. Rev., C33, 755 (1986). +[28] T. A. Hotta, K. Itoh and T. Saito., Phys. Rev. Lett. 33, +790 (1974). +[29] A. Goldmann and M. Stroetzel, Z. Phys. 239, 235 (1970). +[30] Ian C. Cloet, Wolfgang Bentz, Anthony W. Thomas, +Phys. Rev. Lett. 116, 032701 (2016). + diff --git a/l9E5T4oBgHgl3EQfiw8z/content/tmp_files/load_file.txt b/l9E5T4oBgHgl3EQfiw8z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..09f5ef5cdf972124ca6b30be6c45946288f468e2 --- /dev/null +++ b/l9E5T4oBgHgl3EQfiw8z/content/tmp_files/load_file.txt @@ -0,0 +1,1657 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf,len=1656 +page_content='Contribution of Nuclear Excitation Electromagnetic Form Factors in 12C and 16O to the Coulomb Sum Rule A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Bodek1 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Christy2 1Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA 2Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA (Dated: January 16, 2023) We report on empirical parameterizations of longitudinal and transverse nuclear excitation elec- tromagnetic form factors in 12C and 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We extract the contribution of nuclear excitations to the Normalized Inelastic Coulomb Sum Rule (SL(q)) as a function of momentum transfer q and find that it is significant (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='030 at q= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='22 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The total contributions of nuclear exci- tations to SL(q) in 12C and 16O are found to be equal within the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Since the cross sections for nuclear excitations are significant, the radiative tails from nuclear excitations should be included in precise calculations of radiative corrections to quasielastic electron scattering at low q and deep-inelastic electron scattering at large energy transfers ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The parameterizations also serve as a benchmark in testing theoretical modeling of cross sections for excitation of nuclear states in electron and neutrino interactions on nuclear targets at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' INTRODUCTION The Normalized Inelastic Coulomb Sum Rule SL(q) [1] in electron scattering on nuclear targets is the integral of the longitudinal nuclear response function RL(q, ν)dν (excluding the nuclear elastic peak and pion production processes) divided by the square of the proton electric form factor and by the number of protons in the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Here, q is the momentum transfer and ν is the energy transfer to the nuclear target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The sum rule has contri- butions from quasielastic (QE) scattering and from the electro-excitations of nuclear states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' At high q it is ex- pected that SL → 1 because both nuclear excitation form factors and Pauli suppression are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' At small q it is expected that SL → 0 because all cross sections for in- elastic processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' QE, nuclear excitation and pion production processes) must be zero at q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' In this paper we present details of empirical parame- terizations of the q dependence of all longitudinal and transverse excitation form factors in 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Since there are fewer measurements on 16O we only parameterize the longitudinal form factors for this nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We use these parameterizations to compute the contribution of nuclear excitations to SL(q) for both nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Our investigation of the QE contribution to SL(q) is reported in an earlier publication[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Since the cross sections for nuclear excitations are sig- nificant at low q, the parametrizations should be used in precise calculations of radiative corrections to quasielas- tic electron scattering at low q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Because of intial state radiation, nuclear excitations also contribute to radia- tive corrections in deep-inelastic electron scattering at large ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The parameterizations also serve as benchmark in testing theoretical modeling of electron and neutrino scattering at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Because of recent advances in theoretical methods[3–5] for the calculations of the re- sponse functions of electron scattering on nuclear targets, it is now possible to make theoretical predictions of the form factors for the excitation of nuclear states in both electron and neutrino scattering[6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Figures 1 and 2 show the relative contributions of the cross sections for elastic scattering from the 12C nucleus, as well as the low lying excitations of nuclear states for several low energy data sets [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Also shown as a solid curve is our parameterization utilizing the experimental resolution to apply a Gaussian smearing to each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' THEORETICAL FRAMEWORK The electron scattering differential cross section can be written in terms of longitudinal (RL(q, ν)) and transverse (RT (q, ν)) nuclear response functions [12]: dσ dνdΩ = σM[ARL(q, ν) + BRT (q, ν)] (1) where σM is the Mott cross section, σM = α2 cos2)(θ/2) 4E2 0 sin4(θ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' (2) Here, E0 is the incident electron energy, E′ is energy of the final state electron, ν = E0 − E′ is the energy transfer to the target, q is the 3-momentum transfer, Q2 is the square of the 4-momentum transfer (defined to be positive such that q2 = Q2 + ν2), A = (Q2/q2)2 and B = tan2(θ/2) + Q2/2q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' For nuclear elastic scattering at very low q on 12C Q2 = q2 to a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' For elastic scattering and nuclear excitations the square of the electric and magnetic form factors are ob- tained by the integration of the measured response func- tions over ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' In the experimental extractions of form factors for elastic scattering and nuclear excitations the Mott cross section is defined with an additional factor of Z2 because both the nuclear elastic cross section and the cross sections for the the electro excitation of nuclear states are proportional to Z2 times charge form factors F 2 iC(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Here, the subscript zero denotes the nuclear arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='05650v1 [nucl-th] 13 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 1: Top panel: Radiatively corrected cross section from Yamaguchi[10](measured with high resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='25%) for the scattering of 250 MeV electrons from 12C at 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Here, the cross section for the elastic peak has been divided by 100 and the cross section for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV state by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Middle panel: Radiatively corrected cross section[9] (in arbitrary units) for the scattering of 250 MeV electrons from 12C at 550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Bottom panel: Radiatively corrected cross section[9] (in arbitrary units) for the scattering of 600 MeV electrons from 12C at 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The peaks for elastic scattering and for the first three nuclear excitations at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='66 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The solid curve is the predicted radiatively corrected cross section using our fits to the form factors and QE cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The fit is normalized to the elastic cross section for the E=250 MeV and 550 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' For the E= 600 MeV and 330 data we normalize to the cross section for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' elastic peak and subscripts 1-N denote nuclear excita- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The charge form factors can be thought of as the product[11, 13] of the proton electric form factor and the form factors of the spatial distribution of protons in the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 2: Radiatively uncorrected cross section (in arbitrary units) from the LEDEX experiment[11] on 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The solid red line is the radiatively uncorrected cross section from our fit to the form factors and QE cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' It is normalized to the elastic cross section at zero excitation energy for the 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 degree data, and to the cross section for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV state for the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 and 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 degree data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' COULOMB SUM RULE The inelastic Coulomb Sum Rule is the integral of RL(q, ν)dν, excluding the elastic peak and pion produc- tion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' It has contributions from QE scattering and from electro-excitations of nuclear states: CSR(q) = � RL(q, ν)dν (3) = � RQE L (q, ν)dν + G′2 E(Q2) × Z2 L � all F 2 i (q) = G′2 E(Q2) × � Z � V QE L (q, ν)dν + Z2 L � all F 2 i (q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We define V QE(q, ν) as the reduced longitudinal QE re- sponse, which integrates to unity in the absence of any suppression (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Pauli blocking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The charge form fac- Yamaguchi 71 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='75 fm-1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='15 GeV) 30 Elastic 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content="65 MeV da/dQ/dE' [μb/(sr GeV)) 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV 25 12C E = 250 MeV 20 0 = 35 deg 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV 10 5 0 5 10 15 20 25 30 Excitation Energy (Mev)Crannell 64 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='15 fm-1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='23 GeV) 10 Elastic 8 12C E = 250 MeV @ = 55 deg Counts 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65 MeV 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 10 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 15 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 20 Excitation Energy (Mev)Crannell 66 q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='74 fm-1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='34 GeV) 1000 12C 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV E = 600 MeV @ = 33 deg 800 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65 Mel 400 不 200 Elastic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 10 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 15 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 20 Excitation Energy (MeV)10000 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='362 0 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 5000 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 fm-1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='08 MeV 0 Events /charge 40000 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='362 0 = 61 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='85 fm-1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='36 MeV 20000 75000 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='685 0 = 17 50000 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='03 fm-1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 MeV 25000 0 × 10 2000 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='685 0 = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='85 fm-1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='36 Mev 1000 0 0 2 4 6 8 101214 16 18 20 Excitation Energy (MeV)3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E+0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 qeff2 (fm-2 ) Carbon Elastic (Form Factor)2 80 MeV 150 MeV 187 MeV 300 MeV 374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 MeV 600 MeV 747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 MeV 800 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 qeff2 (fm-2 ) Carbon Elastic (Form Factor)2 ratio to fit 80 MeV 150 MeV 187 MeV 300 MeV 374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 MeV 600 MeV 747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 MeV 800 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 10- 1 10- 2 10- 3 10- 4 10- 5 10- 6 10- 7 10- 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 3: Top panel: Measurements[17] of the nuclear elastic longitudinal charge form factor (squared) for 12C versus q2 eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Bottom panel: Ratio to our fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' tors for the electro-excitation of nuclear states F 2 iC(q) is related to F 2 i (q) by the expression F 2 iC(q) = G′2 Ep(Q2) × F 2 i (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' In order to account for the small contribution of the neutron and relativistic effects G′2 E(Q2) is given by[12]: G′2 E(Q2) = [G2 Ep(Q2) + N Z G2 En(Q2)] 1 + τ 1 + 2τ , (4) where, GEp and GEn are the electric form factors [14] of the proton and neutron, respectively and τ = Q2/4M 2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' By dividing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 3 by ZG′2 Eq) we obtain the normalized inelastic Coulomb Sum Rule SL(q) : SL(q) = � V QE L (q, ν)dν + Z L � all F 2 i (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' (5) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' PARAMETERIZATION OF 12C NUCLEAR ELASTIC AND NUCLEAR EXCITATION FORM FACTORS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 12C elastic form factor versus q2 eff The 12C nucleus has a spin parity of 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We fit the measured 12C elastic longitudinal (charge) form factor with the following functional form: F 2 oC(q2 eff) = 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 × 10−3q4 eff 1 + eP ower [H2(q2 eff) + G(q2 eff)] (6) Here, Power = q2 eff−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 is included to better describe the form factor at very large q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The effective[15] q2 is q2 eff = q2(1 + 4Zα/(3⟨r2⟩E) Which for carbon is q2 eff = q2(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='00465/E)2 (where E is in GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The function H(qeff 2) is the harmonic well shape with (α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='21, and a0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' It is is given by[16]: H(q2 eff) = [1 − αq2 effa2 0 2(2 + 3α)]exp[−q2 effa2 0 4 ], (7) The function G(q2 eff) fills in the dip in the location of the diffraction minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' G(q2 eff) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−5e−[(q2 eff−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1)/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='66)]2 In the above parametrization q2 eff is in units of fm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' A comparison of the parametrization of the nuclear elas- tic charge form factor for 12C to experimental data[17] is shown on the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The ratio of the measurments to the fit is shown on the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Form factors for nuclear excitations in 12C We begin by parameterizing the measurements of the longitudinal and transverse form factors for the electro- excitation of all nuclear states in 12C with excitation en- ergies (Ex) less than 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 MeV (the approximate proton removal energy from 12C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' For these states the measure- ments are straightforward since the QE cross section is zero for Ex < 16 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 12C excitation form factors for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='44 MeV and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV states The longitudinal form factors (squared) for the electro excitation of the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='44 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV nuclear excited states are parametrized as F 2 iC(q2 eff) where F 2 iC(q2 eff) = (q2 eff)3 (q2eff)3 + d j=3 � j=1 Nje−[(q2 eff−Cj)/σ]2 (8) Here q2 is in units of fm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The parameters for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='44 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV states are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Com- parisons of our parametrizations of the excitation form factors (squared) for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='44, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV states to experimental data[17] are shown in Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 qeff (fm-1) Carbon 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV State (Form Factor)2 80 MeV 150 MeV 187 MeV 250 MeV 300 MeV 420 MeV 600 MeV 800 MeV 10- 1 10- 2 10- 3 10- 4 10- 5 10- 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-06 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 qeff (fm-1) Carbon 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV State (Form Factor)2 80 MeV 150 MeV 187 MeV 250 MeV 300 MeV 420 MeV 600 MeV 800 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='025 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 qeff (fm-1) Carbon 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV State (Form Factor)2 80 MeV 150 MeV 187 MeV 250 MeV 300 MeV 420 MeV 600 MeV 800 MeV 10- 1 10- 2 10- 3 10- 4 10- 5 10- 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 qeff (fm-1) Carbon 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV State (Form Factor)2 150 MeV 187 MeV 250 GeV 300 MeV 600 MeV 800 MeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 4: Measurements[17] of the longitudinal charge form factors (squared) for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV state (left) and the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV state (right) in 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The form factors (squared) are shown on linear scales and logarithmic scales on the top and bottom panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' State N1 C1 σ1 N2 C2 σ2 N3 C3 σ3 d Data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='44 MeV 2+ L (q2 eff) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='41 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='71 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='10 [17] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV 3−L (q2 eff) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='00 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='70 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 × 10−5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='20 [17] TABLE I: Parameters of our fits (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 8) to the 12C longitudinal charge form factors (squared) for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='44 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV nuclear excited states in 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' For these states, the parametrizations are in terms of q2 eff in units of fm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Here q2 eff = q2 × (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='00465/E)2, where E is in GeV[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 12C form factors for the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65 MeV state and states with excitation energies above 10 MeV Measurements of the square of the longitudinal form factor verses q (in units of fm−1) for the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65 MeV state in 12C (from Chernykh et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' [18]) are shown on the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' A comparison of the nuclear elastic form factor to the form factors of the first three nuclear excitations versus q (in units of GeV) is shown on the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The charge form factors (squared) for the electro- excitation of the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65 MeV state and for states with excitation energies above 10 MeV are parameterized as F 2 iC(q) = Max(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0, g2 i ) where g2 i (qeff) = j=3 � j=1 Nje−[(qeff −Cj)/σ]2 − ae−bqeff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' (9) Here, qeff is in units of fm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The parameters are given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' (Note that these states are parame- terized versus qeff, while the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='44 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV states are parametrized versus qeff 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' As shown on the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 6, for q near the diffraction minimum for elastic scattering on 12C the cross sections for the three nuclear excitations below 10 MeV are larger than the nu- clear elastic cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Note that unlike the nuclear elastic form factor which is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 at q=0, all lon- gitudinal form factors for the nuclear excitations must vanish at q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 10-2 + 150MeV 187 MeV 10-3 ^ 250 GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 300 MeV + 600 MeV 800 MeV 10-4 10-5 Carbon 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 Mev state (Form Factor)2 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 qeff (fm-1)5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 qeff (fm-1) Carbon 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV State (Form Factor)2 ratio to fit 80 MeV 150 MeV 187 MeV 250 MeV 300 MeV 420 MeV 600 MeV 800 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 qeff (fm-1) Carbon 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV State (Form Factor)2 ratio to fit 150 MEV 187 MeV 250 MeV 300 MeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 5: Ratios of the measured[17] longitudinal charge form factors (squared) to our parametrizations for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV state (left) and the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV state (right) in 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E+0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 q (GeV) Carbon (Form Factor)2 Elastic 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='43 MeV 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='61 MeV 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='68 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 10- 1 10- 2 10- 3 10- 4 10- 5 10- 6 5B_all_ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 6: Left panel: Measurements[18] of the longitudinal charge form factor (squared) for the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65 MeV state in 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Right panel: A comparison of the nuclear elastic form factor (squared) to the form factors (squared) of the first three nuclear excitations versus q (in units of GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 12C form factors for states with excitation energies above 10 MeV and below 16 MeV We use equation 9 to parameterize the form factor for excitation energy of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='84 MeV[19], and also for excita- tion energies of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='71, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='09 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='11 MeV[20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' In addition, we find that published differential cross sec- tion measurements indicate that there is an additional longitudinal continuum in the region between 12 to 15 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We parameterize this longitudinal continuum as one broad state at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 MeV (σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='25 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' For the transverse form factors in this region we parametrize the data of Hicks84[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 12C form factors for states with excitation energies above 16 MeV Initially, we parameterize the longitudinal and trans- verse form factors measured by Yamaguchi71[10] for states with excitation energies above 16 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' How- ever, in the Yamaguchi71 analysis the contributions from quasielatic (QE) scattering are not subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' There- fore, We perform a reanalysis of the Yamaguchi71 data in combination of all published cross sections with 16 < Ex < 55 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We subtract the QE contribution using our QE model[2] (which includes superscaling[23] with Rosenfelder[24] Pauli Suppression) and extract updated longitudinal and transverse form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' For Ex > 20 2 10 12 12 C(e,e\')"C 10° E, = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='654 MeV 10 0, (b) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 10" 0 2 3 q [fm\'l]6 State N1 C1 σ1 N2 C2 σ2 N3 C3 σ3 a b Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65 MeV 0+L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−4 1 [18] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='84 MeV 1−L 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 ×10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 − [19] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='83 MeV 2− T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 ×10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 [22] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='71 MeV 1+T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 ×10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 × 10−5 1 [20] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 MeV 4−L 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 ×10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65 [9–11] σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='25 MeV 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='08 MeV 4+L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 ×10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 [21] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 MeV 1+L 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 [20] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 MeV-T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 ×10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 × 10−5 1 [20] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 MeV 2+L 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 [10] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 MeV 2+T 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 MeV 2−T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='35 [10][22] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 MeV 1+T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='45 [10][22] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 MeV-L 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 [10] 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 MeV 2−T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='02 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='77 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='75 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 × 10−4 1 [10][22] 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 MeV 2+L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 [10] 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 MeV 3−T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 [10][22] (21-26 MeV) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 MeV-L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='55 [10] σ=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='75 MeV 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 MeV-T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='83 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 [10] (26-37 MeV) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 MeV-L 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='48 [10] σ=9 MeV 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 MeV-T 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 [10] (30-50 MeV) 42 MeV-L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 σ=12 MeV Extra Strength TABLE II: Parameterizations of the Longitudinal (L) and Transverse (T) 12C nuclear excitation form factors (squared) for the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='65 MeV state and for states with excitation energy above 10 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Unlike the parametrizations in Table I for the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='44 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='64 MeV states which are functions of the square of the 3-momentum transfer q2 eff in units of fm−2, the parametrizations for the states in this table are functions of qeff in units of fm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Here q2 eff = q2 × (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='00465/E)2, where E is in GeV[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' MeV (region of the Giant Dipole resonances) we group the strength from multiple excitations into a three states with a large width in Ex and extract effective form factors accounting for all states in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The updated parameters are given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The longitudinal and transverse response functions for 12C, RL(q, Ex) and RT (q, Ex), extracted by Yamaguchi71[10] for excitation energies above 16 MeV and less than 40 MeV are shown in Figure 7 (black points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Also shown are RL(q, Ex) and RT (q, Ex) ex- tracted from our universal fit to all electron scattering cross section data on 12C (solid red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The QE con- tribution to the total response functions is shown as the dashed red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' An estimated resolution smearing of 600 keV has been applied to the excitations in the fit to match the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' While individual states are well reproduced at low excitation energy, above Ex of 20 MeV the effect of grouping several excitations together into three broad ef- fective states in the fit can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' While the fit does not capture the structure from individual states above 20 MeV, the total strength is seen to be well reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Comparison to 12C experimental data for excitation energies below 50 MeV Experimental radiatively corrected inelastic electron scattering cross sections on 12C for excitation energies less than 50 MeV are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Also shown are the corresponding cross sections from our universal fit to all 12C data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The cross sections for excitation energies less than 12 MeV are multiplied by (1/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The pink solid line is the predicted total cross section from our univer- sal fit[2] which include the contributions from all sources (nuclear excitation form factors, quasilelastic scattering and pion production processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The QE contribution is shown as the dashed blue line and the ”Transverse Enhancement/Meson Exchange Currents” contribution 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 7: Comparison of the longitudinal (RL, left) and transverse (RT , right) response functions for 12C extracted by Yamaguchi 71[10] (black squares) to the response functions extracted from our universal fit to all available electron scattering cross section data on 12C (solid red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The contributions from excitation energies less than 12 MeV are multiplied by (1/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The QE contribution to the total response functions is represented by the red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' In our fit, we model the response functions for all states the region of the Giant Dipole Resonance (20-30 MeV) region as one average broad excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' is shown as the dot-dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Details of the fit are described in reference[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Most of the cross section mea- surements are from Yamaguchi71[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The cross sections for Eo=54 MeV at 1800 are from Goldemberg64[25] and the the cross sections for Eo=65 MeV at 1800 are from deForest65[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The measurements at 1800 are only sen- sitive to the transverse form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' ANALYSIS OF 16O EXCITED STATES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 16O excited states with Ex < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5MeV In order to minimuze correlations between our param- eterizations of the form factors for the nuclear excitations in 12C and 16O we parameterize the form factors for 16O states using a somewhat different functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The form factors for the nuclear excited states in 16O are pa- rameterized as F 2 iC(q2) = Max(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0, g2 i ) where g2 i (q2 eff) = q2 eff × � j=3 � j=1 Nje−[(q2 eff −Cj)/σ]2 − ae−bq2 qeff � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' (10) Here, q2 eff is in units of fm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The form factors for nuclear excitations in 16O with ex- citation energies below proton removal threshold (about 12 MeV) are easily measured because there are is no con- tribution from QE scattering in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The nine longitudinal form factors (squared) in 16O for excitation energies below 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 MeV are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 9 on linear and logarithmic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The data in the figures are from Buti-86[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The solid blue lines are our parameteriza- tions using the parameters listed in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Fit total Fit QE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='002 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='148 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='002 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='167 GeV (Mev-1) 0 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='205 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='002 R 0 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='24 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='002 0 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='307 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='001 0 5 10 15 20 25 30 35 40 Excitation Energy (MeV)Fit total Fit QE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='001 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='148 GeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='001 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='167 GeV Rt (Mev-1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='001 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='205 GeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='001 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='24 GeV 0 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='307 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='001 0 10 15 20 25 30 35 40 Excitation Energy (Mev)8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 8: Radiatively corrected inelastic electron scattering cross sections on 12C for excitation energies less than 50 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The cross sections for excitation energies less than 12 MeV are multiplied by (1/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The pink solid line is the predicted total cross section from our universal fit[2] to all electron scattering data on 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The fit include nuclear excitations, a superscaling QE model[23] with Rosenfelder Pauli suppression[24] (dashed blue line), ”Transverse Enhancement/Meson Exchange Currents” (dot-dashed line) and pion production processes (at higher excitation energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The data are from Yamaguchi71[10] except for the cross sections for Eo=54 MeV and 1800 (from Goldemberg64[25]) and the cross sections for for Eo=65 MeV and 1800 (from deForest65[26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The measurements at 1800 are only sensitive to the transverse form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Total Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Enhancement QE E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='054 @ = 180 750 E = 025 @ = 35 20 9 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='102 GeV q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='147 GeV 500 10 X 250 /(sr GeV)) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='A 30 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0888 0 = 135 20 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='065 0 = 180 20 q= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='152 GeV q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='122GeV 10 10 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='25 @ = 40 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0 = 135 400 q= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='166 GeV 20 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='17 GeV 200 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='AA 200 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='25 0 = 50 20 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='12 = 135 q = 0,202 GeV q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='201 GeV 100 10 100 20 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='25 @ = 60 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1405 @ = 135 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='235 GeV q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='232 GeV 50 10 20 E=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='250=80 10 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1776 0= 135 q = 0,291 GeV q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='285 GeV 10 5 0 10 20 30 40 50 10 20 30 4050 Excitation Energy (MeV) Excitation Energy (MeV)9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='E-04 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 10: Top row: The longitudinal response function 12C, RL(q, Ex) for 16O from Hotta74[28] for three values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The red dashed line is the original estimate of the QE contribution used in Hotta74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The green solid line is the QE contribution determined using our superscaling model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We use these data to extract the longitudinal form factors for nuclear excitations in 16O for the 12.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 Hotta74 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Data taken from Buti-86[27] and Hotta74[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The parameterizations are functions of q2 eff in units of fm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 11 12/9/22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 Z SF2i (q) q (GeV) ZSFi2 (q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1< Ex<55 MeV (longitudinal) Carbon Carbon fit FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 11: Left panel: The contribution of longitudinal nuclear excitations (between 2 and 55 MeV) to SL(q) in 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Right panel: Our fit to the total contributions of all nuclear excitations below 55 MeV to SL(q) in 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The uncertainty is (shown as a green band) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='01 plus 10% added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 12/9/22 O16_CSR_States_q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf CSR5 copy3_specia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' O16_CSR_States_q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf} 0.' metadata={'source': 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+page_content='1 MeV < Ex < 36 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 MeV < Ex < 20 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 MeV < Ex < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 MeV O16_CSR_States_q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf AL27-O15 copy3_specia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 12/9/22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 Z SF2i (q) q (GeV) ZSFi2 (q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1< Ex<55 MeV (longitudinal) Oxygen Oxygen Fit O16_CSR_States_fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 12: Left panel: The contribution of longitudinal nuclear excitations (between 2 and 55 MeV) to SL(q) in 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Right panel: Our fit to the total contributions of all nuclear excitations below 55 MeV to SL(q) for 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The uncertainty (shown as a green band) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='01 plus 10% added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 16O excited states with Ex > 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5MeV For excitation energy above 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 MeV there is a signifi- cant contribution from QE scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Here we group the states in two regions of excitation energy (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5-20 MeV and 20-35 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The top row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 10 shows the longitudinal response function RL(q, Ex) for 16O from Hotta74[28] for three values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The solid red line is the original estimate of the QE contribution used in the Hotta74 publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The solid green line is the QE contribution determined using the QE parameters from our universal fit to all 12C data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We find that the QE cross section predictions for 16O using the parameters from the 12C fit also de- scribe all (but limited) available data on 16O as shown in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We use the Hotta74 data to extract the longitudinal form factors for the nuclear excitation in 16O in the 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5- 20 MeV and the 20-35 MeV groupings in in excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The middle and bottom rows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 10 show the ex- tracted longitudinal form factor for the 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5-20 MeV and 20-35 MeV groupings in excitation energy on linear (mid- dle) and logarithmic (bottom) scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Also shown is the form factor measurement from Goldman70[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The lon- gitudinal form factor measured by Goldman70 for the 20– 30 MeV grouping in excitation energy has been corrected by subtracting the QE contribution (from our universal fit) and extending the excitation range to 20-35 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Since no data are available for the form factor for nu- clear excitations the 36-55 MeV region in 16O we assume 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 Z SF2i (q) q (GeV) ZSFi 2 (q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1< Ex<55 MeV (longitudinal) Oxygen Carbon 12/9/22 C12_O16_states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 13: A comparison of the contributions of nuclear excita- tions to SL(q) in 12C and 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The uncertainty (green band) in the total contribution of the excited states is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='01 plus 10% added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' that the form factor for 16O is the same as the form factor for 12C in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 Contributions to SL Inelastic q (GeV) QE Pauli suppressed SL Inelastic QE Supressed (Pauli+Quench) C12 Excitated States SL_components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf 12C SL_components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 14: The various contributions[2] to SL(q) for 12C (dot- ted blue with yellow error band) including: QE with Pauli suppression only (dotted-purple), QE suppressed by both ”Pauli” and ”Longitudinal Quenching” (solid-green), and the contribution of nuclear excitations (red-dashed with green er- ror band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' CONTRIBUTION OF NUCLEAR EXCITATIONS TO SL(q) IN 12C AND 16O The contributions of nuclear excitation to SL(q) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 5) in 12C and 16O are calculated using the form factor parameterizations given in Tables I, II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The left side panels of Figures 11 and 12 show the contributions of nuclear excitations (with excitation energies below 10 MeV, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 MeV, 30 MeV and 55 MeV) to SL(q) for 12C and 16O, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The total contribution to SL(q) can be parametrized as follows: Z L � all F 2 i (q) = N1exp((x − C1)2/D2 1) + N2exp(−(x − C2)2/D2 2) + N3exp(−(x − C3)2/D2 3) (11) where x= q/KF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' For 12C KF =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='228 GeV, N1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='260, C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='11, D1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='50, N2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='075, C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='730, D2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='30, and N3= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='01, C3=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0, D3=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The fit and the data are shown on the right side panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' For 16O KF =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='228 GeV, N1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='240, C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='07, D1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='48, N2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='073 C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='70, D2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='37, and N3= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='039, C3=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='55, D3=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The fit and the data are shown on the right side panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 13 shows a comparison of the contributions of all excited states to the SL(q) for 12C and 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The uncer- tainty in the total contribution of the excited states for both nuclei is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='01 plus 10% added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' These data indicate that the contribution of nuclear excitations to SL(q) in 16O is consistent with being equal to the con- tribution of the nuclear excitations in 12C within errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The total contribution of all states with excitation energy below 55 MeV is largest at q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='22 GeV, where it reaches a maximum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='29± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' UPDATED EXTRACTION OF SL(q) FOR 12C AND 16O In our previous paper[2] we performed a fit to all elec- tron scattering data on 12C and 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We found that the QE transverse response function is enhanced at in- termediate q and the longitudinal response function is quenched at low q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We used the fits in combination with the fits to nuclear excitations to extract SL(q) for 12C and 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' In our previous paper we used a very conserva- tive estimate of the uncertainty in the total contribution of the excited states (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='01 plus 15% added in quadra- ture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' In this paper we have updated our fits to the form factors for individual nuclear excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We find that the updated total contribution of nuclear excitations to SL(q) for 12C and 16O is unchanged, but a smaller con- servative estimate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='01 plus 10% added in quadrature) is more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 14 shows the various contributions to the ex- tracted SL(q) for 12C (dotted blue line with yellow er- ror band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Shown are the QE contribution with only Pauli suppression (dotted-purple), the QE contribution suppressed by both ”Pauli Suppression” and the longi- tudinal quenching factor F L quench(q) labeled as QE sup- 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content="9 Normalized SL Inelastic q (GeV) Lavato 2016 Green's Function MC SL Inelastic) Mihaila 2000 (Carbon) AV18+UIX Cloet 2016 (Carbon) RPA Lovato_C12_compare 12C Lovato_C12_compare 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='9 Normalized SL Inelastic q (GeV) Sobczyk 2020CCSD: NNLO SL Inelastic Mihaila 2000 AV18+UIX Potential June 3, 2022 Erro GO 16O O16_SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf O16_SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='pdf FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 15: Left panel: SL(q) for 12C (dotted-blue with yellow error band) compared to theoretical calculations including Lovato 2016 [3] (solid-purple), (Mihaila 2000[4] (dashed-green), and RPA Cloet 2016[30] (solid-red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Right panel: SL(q) for 16O (dotted-black with green error band) compared to theoretical calculations of Sobczyk 2020[5] (red-dashed) and Mihaila 2000 (dotted-dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' pressed (Pauli+Quench) (solid-green), and the contribu- tion of nuclear excitations (red dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 15 shows a comparison of the extracted SL(q) for 12C (dotted-blue curve with yel- low error band) to theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' These in- clude the Lovato 2016[3] ”First Principle Green’s Func- tion Monte Carlo” (GFMC) calculation (solid-purple line), Mihaila[4] 2000 Coupled-Clusters based calcula- tion (AV18+UIX potential, dashed-green), and Cloet 2016[30] RPA calculation (RPA solid-red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Our measure- ment for 12C are in disagreement with Cloet 2016 RPA, and in reasonable agreement with Lovato 2016 except near q ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='30 GeV where the contribution from nuclear excitations is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' 15 shows SL(q) for 16O (dotted- blue with green error band) compared to theoretical cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' These include the Sobczyk 2020[5] ”Coupled- Cluster with Singles-and Doubles (CCSD) NNLOsat” (red-dashed line), and Mihaila 2000[4] Coupled-Cluster calculation with (AV18+UIX potential, dashed green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The data are in reasonable agreement with Sobczyk 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' SUMMARY We report on empirical parameterizations of longitu- dinal and transverse nuclear excitation electromagnetic form factors in 12C and 16O and extract the contribution of nuclear excitations to the Normalized Inelastic Sum Rule SL(q) as a function momentum transfer q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We find that the total contribution is significant (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='030) at q= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='22 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The total contributions of nuclear exci- tations in 12C and 16O are consistent with being equal within errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Since the cross sections for nuclear exci- tations are significant at low q, the radiative tails from nuclear excitations should be included in precise calcula- tions of radiative corrections to quasielastic electron scat- tering at low q and deep-inelastic electron scattering at large ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' The parameterization also serves as a benchmark in testing theoretical modeling of electron and neutrino scattering cross sections at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Theoretical studies of the excitation of nuclear states in electron and neutrino scattering[6–8] indicate that both are equally significant at low values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Therefore, nuclear excita- tions should be included in both electron and neutrino MC generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' We note that for excitation energies above proton removal threshold (about 16 MeV in 12C and 12 MeV in 16O) the decays of nuclear excitations can have a proton in the final state and therefore cannot be distinguished experimentally from QE scattering in low resolution neutrino experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Research supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Department of En- ergy under University of Rochester grant number DE- SC0008475, and the Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content='7:231-316;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' ibid Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Modern Physics, 28, 214 (1956);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Fregeau and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfiw8z/content/2301.05650v1.pdf'} +page_content=' Hofstadter, Phys Rev.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..d89a5b71a328e3683b0d2146ac481f6d049991b4 --- /dev/null +++ b/ndE1T4oBgHgl3EQfOgNK/content/tmp_files/2301.03015v1.pdf.txt @@ -0,0 +1,1458 @@ +1 + +A Modelling Framework for Regression with Collinearity + + +Takeaki KARIYA１, Nagoya University of Commerce and Business, Graduate School +of Management, Nisshin, Aichi, Japan 470-0193 +The corresponding author: thekariya70@gmail.com +Hiroshi KURATA2 University of Tokyo, Graduate School of Arts and Sciences, +Meguroku, Tokyo, Japan 153-8902 kurata@waka.c.u-tokyo.ac.jp +Takaki HAYASHI3 Keio University, Graduate School of Business Administration, +Yokohama, Kanagawa, Japan 223-8526 hayashi.kbs@gmail.com + +Abstract +This study addresses a fundamental, yet overlooked, gap between standard theory and +empirical modelling practices in the OLS regression model 𝒚 = 𝑿𝜷 + 𝒖 with +collinearity. In fact, while an estimated model in practice is desired to have stability and +efficiency in its “individual OLS estimates”, 𝒚 itself has no capacity to identify and +control the collinearity in 𝑿 and hence no theory including model selection process +(MSP) would fill this gap unless 𝑿 is controlled in view of sampling theory. In this paper, +first introducing a new concept of “empirically effective modelling“ (EEM), we propose +our EEM methodology (EEM-M) as an integrated process of two MSPs with data (𝒚𝒐, 𝑿) +given. The first MSP uses 𝑿 only, called the XMSP, and pre-selects a class 𝐷 of models +with individually inefficiency-controlled and collinearity-controlled OLS estimates, +where the corresponding two controlling variables are chosen from predictive standard +error of each estimate. Next, defining an inefficiency-collinearity risk index for each +model, a partial ordering is introduced onto the set of models to compare without using +𝒚𝒐, where the better-ness and admissibility of models are discussed. The second MSP is +a commonly used MSP that uses (𝒚𝒐, 𝑿), and evaluates total model performance as a +whole by such AIC, BIC, etc. to select an optimal model from 𝐷. Third, to materialize the +XMSP, two algorithms are proposed. + +Key words: OLS, model selection process, collinearity effect, empirically modelling, t- +test +AMS Classification: 62J05, 62J20 + + + + +2 + +1 Introduction + Within the traditional ordinary least squares (OLS) framework in the linear regression +model with collinearity +(1.1) +𝒚 = 𝑿𝜷 + 𝒖 with 𝐸(𝒖) = 𝟎 and 𝑉𝑎𝑟(𝒖) = 𝜎"𝑰, +this paper first proposes a new concept of “Empirically Effective Modelling” (EEM) +together with the EEM methodology (EEM-M) for finding an empirically effective model +from an observation (𝒚𝒐, 𝑿) for model in (1.1). The EEM-M is based on the EEM concept +and forms a new integrated model selection process (MSP) consisting of bundle model +concept, two different MSPs, new model comparison methodology and algorithms for +selecting inefficiency-controlled and collinearity-controlled models, where the two MSPs +are the XMSP with using X only and 𝑦#MSP with using (𝒚𝒐, 𝑿). +As notation, let 𝑿 = (𝑥$%) = (𝒙𝟏, 𝒙𝟐, ⋯ , 𝒙𝑲) with 𝒙𝟏 = (1, ⋯ ,1)) ≡ 𝒆 be an 𝑁 × +𝐾(𝑁 > 𝐾) explanatory matrix of rank K, and 𝑿 is assumed to contain all possible +variables for MSP and to be used as they are. We call each vector 𝒙𝒌 a “variable”, and X +a “model”. And X is interchangeably used as the set consisting of K vectors {𝒙𝒌} as well. +Similarly, a sub-set 𝑿∗ of 𝑿 is called a sub-model or simply a model formed as a matrix +and it is assumed that 𝑿∗ always includes a constant term, i.e., 𝒙𝟏 = 𝒆. + Our concept of collinearity is confirmed. Since models always include 𝒙𝟏 = 𝒆, so that +the coefficient of determination (CD) in regression can be used as a measure of +collinearity. Let 𝑿M𝒌 = N𝒙𝒋 ∈ 𝑋, 𝑗 ≠ 𝑘T in 𝑅- be the set X with the k-th variable deleted, +where 𝒙𝟏 = 𝒆 ∈ 𝑿M𝒌 and 𝑘 = 2, ⋯ , 𝐾. We also use 𝑿M𝒌 as the 𝑁 × (𝐾 − 1) matrix formed +by its columns. Let 𝑅X% +" is the CD when 𝒙𝒌 is regressed on 𝑿M𝒌 in 𝑅-. In our terminology, +each 𝒙𝒌 (𝑘 ≥ 2) is said to be collinear with 𝑿M𝒌 unless 𝑅X% +" = 0 or 𝑅X% +" = 1, and it is said to +be perfectly collinear with 𝑿M𝒌 if 𝑅X% +" = 1, and zero-collinear with 𝑿M𝒌 if 𝑅X% +" = 0. When 𝑅X% +" +is close to 1, 𝒙𝒌 is said to be strongly collinear with 𝑿M𝒌. By assuming rank(X)=K, the case +of 𝑅X% +" = 1 is excluded. +In this definition, collinear relations always exist among the variables in each model +unless 𝑅X% +" = 0 for any 𝑘 ≥ 2, and model performance comes with the collinearity. What +to do in modelling is not to avoid it but to exclude each variable 𝒙𝒌 that has “strong” +collinearity with 𝑿M𝒌 in each model. In the EEM-M, the XMSP will screen such variables +out and select models with 𝑅X% +" ≤ 𝑑 for any 𝑘 ≥ 2, where 𝑑 is a control parameter. + Concerning collinearity, as its vast literature shows, 𝒚𝒐 itself does not have a capacity +to identify the collinearity structure of 𝑿 via a 𝑦#MSP. In fact, there will be no literature +on such a 𝑦#MSP that mitigates the ill-effect within the OLS framework even if 𝒚𝒐 is +really generated from it, though a lot of hybrid remedies such as shrinkage-type +estimation have been provided outside of the OLS framework. Besides, under strong + +3 + +collinearity, 𝒚𝒐 loses its own capacity to evaluate model performances by a 𝑦#MSP, +because OLSEs and maximum likelihood estimates suffer from the ill-effect, which +invalidates such a 𝑦# MSP as the ones with Akaike’s information criterion (AIC), +Bayesian information criterion (BIC), adjusted coefficient of determination (CD), etc. +In our terminology, MSP is defined as a procedure to select a model 𝑿∗ from X that is +estimated by the OLS method as 𝜷]∗ = 𝜷]∗(𝑿∗, 𝒚), where +(1.2) 𝜷] ≡ 𝜷](𝑿, 𝒚) = (𝛽_%) = (𝑿)𝑿)./𝑿)𝒚. +By the OLSE we mean either OLS estimator 𝜷]∗ or OLS estimate 𝜷]∗𝒐 = 𝜷](𝑿∗, 𝒚𝒐) for +given (𝒚𝒐, 𝑿∗) along its context. Here it is noted that collinearity does not affect the +optimality of 𝜷] in (1.2) as a whole. In fact, if the initial model in (1.1) is supposedly +“true” with rank(X) =K, 𝜷] is the best linear unbiased estimator in the 𝐾 × 𝐾 nonnegative +definite ordering, in which no shrinkage-type estimator will beat the OLS estimator in +risk matrix. This optimality holds no matter how strong the collinearity in X may be, +implying no collinearity effect on this basic optimality. However, strong collinearity +seriously affects some individual OLSEs so that a selected model will be sensitive to +small changes of predictive variables and lose stability or trust. + Our results do not much depend on other authors’ work. Hence, in the sequel, we +summarize our work in some details. +1.1 Empirically Effective Modelling Methodology (EEM-M) +From a practical viewpoint, we first define a new concept of “Empirically Effective +Modelling (EEM)” in the OLS framework, where it leads to an empirically effective +model from an observation (𝒚𝒐, 𝑿) in its own sense. Here we let 𝑿 also represents all +the cases of sub-model 𝑿∗. +Definition 1.1 A model estimated by the OLS with data (𝒚𝒐, 𝑿) is defined to be +empirically effective if it is judged to hold the two properties [1] and [2]. +[1] Efficiency and stability of each “individual” OLS estimate in the estimated model. +[2] Goodness or optimality in total model performance. +To make this EEM concept implemented, we develop the EEM-M with two MSPs. +The first MSP is the XMSP applied to data 𝑿 without using 𝒚𝒐 for [1] in order to pre- +select a class 𝐷 of inefficiency-controlled and collinearity-controlled models, or shortly +“IC-controlled models”. In this paper, the property [1] is measured by the pair (𝐼%, 𝐶%) for +the k-th OLS estimate 𝛽_%, which is given by +(1.3) 𝐼% ≡ 𝑧̅% +" = 𝑥̅% +"/𝑠% +" ≥ 0 and 𝐶% is 𝑅X% +" or equivalently 𝑉𝐼𝐹% = 1/(1 − 𝑅X% +") +where mean 𝑥̅% of 𝒙𝒌 = {𝑥%$} and 𝑠0% +" = ∑ +(𝑥$% − 𝑥̄%)" +- +$1/ +/𝑁. These are taken from + +4 + +the individual predictive sampling variance (IndPSV) of the k-th OLS “estimator” 𝛽_% (see +(1.5) in 1.3). For the OLS “estimate” 𝛽_% given, 𝐼% will be called the inefficient factor (I- +factor) of 𝛽_%, while 𝐶% ≥ 0 is called the collinearity factor (C-factor) of 𝛽_% (𝑘 ≥ 2). It is +noted that values of (𝐼%, 𝐶%) are determined only by 𝑿 as in (1.3). By this fact, the XMSP +aims to control the two objective variables (𝐼%(𝑝%𝑿), 𝑅X% +"(𝑿)) with respect to 𝑿 to get a +class 𝐷 of “ IC-controlled models” with control level (c, d) before 𝒚𝒐 is observed, where +𝑝%𝑿 += 𝒙𝒌 and (c,d) is prespecified. The VIF in (1.3) is the well-known variance +inflation factor (VIF) (Jolliffe, 1986 ). In general, in empirical analysis, it will be desirable +to compute the values of I-factor and C-factor of each estimate 𝛽_% as such performance +statistics of each estimate as standard errors (SD) and t-value, because these factors +provide the information on how much of the property [1] is guaranteed. In this paper, the +XMSP provides two algorithms to pre-select a class 𝐷23 of IC-controlled models with +𝐼% ≤ 𝑐 and 𝑅X% +" ≤ 𝑑 for any 𝑘 ≥ 2 in it’s OLS estimates. Such a class 𝐷23 (or simply 𝐷) +will be called an IC-controlled class or “ 𝒚𝒐-accommodating” class for 𝑦#MSPs. +The second MSP is a 𝑦#MSP applied to data (𝒚𝒐, 𝑿) for [2] in order to select an optimal +model from the class 𝐷23. Since there are many useful 𝑦#MSPs available, any reasonable +𝑦#MSP in the EEM-M can be used for finding an optimal model by its own criterion so +long as it independently evaluates the total performance of each IC-controlled model in +𝐷23. In this sense, we will take it for granted that an optimal model can be obtained by +such a 𝑦#MSP, and so the optimality of an IC-controlled model depends on the choice of +𝑦#MSP, implying that the optimal model is not unique. Examples of 𝑦#MSP are those +with AIC, BIC, adjusted CD, or hybrids, etc. +However, we will not include the 𝑦#MSPs that have a process of pre-testing, model +selection and estimation in selecting models within one sample "𝒚𝒐" (Saleh, 2006). In +view of sampling theory, these MSPs intrinsically entail a nonlinear and conditional +structure in a selected model and so it is out of the OLS framework. Besides, those MSPs +mostly aim to select significant variables via testing and do not seek an optimal model as +a whole. +Thus, the XMSP fills the overlooked gap between the problem that an estimated model +needs to have the property [1] for individual OLS estimates and the problem that the +traditional 𝑦#MSP cannot identify and control the ill-effect of collinearity on individual +OLS estimates, because 𝒚𝒐 itself have no capacity to do it. In fact, the XMSP redeems the +weak point in selecting a class 𝐷 of models with property [1] before 𝒚𝒐 is observed, and +then a 𝑦#MSP comes into own capacity in 𝐷 to select an optimal model with property [2]. +Therefore, the EEM-M will give us an empirically effective model. + Before some more details are described below, we need to add some more components + +5 + +to the EEM-M. The whole picture of the EEM-M is symbolically described as an +integrated process of 𝐵ℳ([𝑿]) −XMSP —𝑦# MSP. Here 𝐵ℳ([𝑿]) denotes a bundle +model replacing (1.1) and it makes the two MSP concepts consistent with the EEM (see +1.2). In addition, the XMSP includes a framework of comparing models in terms of +(𝐼%, 𝑅X% +"). In fact, the maximums 𝑚𝑎𝑥%4"𝐼% and 𝑚𝑎𝑥%4"𝐶% in a given model 𝑿∗ are +respectively defined to be the inefficiency risk (I-risk) and the collinearity risk (C-risk) +of the model 𝑿∗. The pair of the maximums will be called inefficiency-collinearity risk +index (ICRI) (see 1.4 and 3.3). The ICRI gives a decision theoretic framework for +comparing models because it introduces a partial ordering onto the set of sub-models. In +particular, in comparing models, they provide the concept of better-ness and admissibility +of a model in terms of the ICRI (See 3). As a part of the EEM-M, the two algorithms for +the XMSP are given for selecting a 𝐷 of IC-controlled models (See 4). +1.2 Bundle Model 𝐵ℳ([𝑿]) + As a part of the EEM-M, we replace the traditional model concept expressed in (1.1) by +our bundle model concept for the EEM-M with the XMSP and 𝑦#MSP. This is because +the EEM-M starts with observed data (𝒚𝒐, 𝑿) and pursues the effectiveness of empirical +model in the sense of Definition 1.1 with some evaluation criteria derived in the OLS +framework. As an example, let us consider the problem of selecting one of the models: +(1) 𝒚 = 𝑿∗𝜷∗ + 𝑿∗∗𝜷∗∗ + 𝒖 in (1.1) with 𝑿 = (𝑿∗, 𝑿∗∗) +(2) 𝒚 = 𝑿∗𝜷∗ + 𝒖. +Such a problem is often treated as a binary decision problem where the F-testing scheme +is used within the so-called frequentist’s framework. This approach leads us to the MSP +with a procedure of preliminary test, model selection, and estimation. As has been stated, +we exclude this MSP because it includes an internal inconsistency so long as the models +in the procedure are estimated by the OLS. While, in the EEM-M, 𝒚 is realized as 𝒚𝒐, +implying that it has to be generated by either (1) or (2). Hence the error terms in (1) and +(2) should be different because the two models cannot generate 𝒚𝒐 under the same 𝒖. +Taking this view into the EEM-M framework, we use the bundle model defined by +(1.1a) 𝐵ℳ([𝑿]) = { 𝒚𝝉 = 𝑿𝝉𝜷𝝉 + 𝒖𝝉 | 𝑿𝝉 ∈ [𝑿], 𝒖𝝉 ∈ [𝒖]}. +Here [𝑿] is the set of all the sub-models {𝑿𝝉: 𝜏 ∈ Λ} with 𝒙𝝉/ = 𝒆 and Λ = +{1, 2, ⋯ , 2"#$ − 1}, where 𝜏 is a parameter that distinguishes models, and [𝒖] is the set of +error terms corresponding to the set [𝑿] +(1.1b) [𝒖] = {𝒖𝝉: 𝜏 ∈ Λ, 𝐸(𝒖𝝉) = 𝟎, 𝑉𝑎𝑟(𝒖𝝉) = 𝜎"𝑰}. +Then 𝒚𝒐 is regarded as realized from one of the sub-models in 𝐵ℳ([𝑿]), not from a “true +model”. It is noted that no specific stochastic structure is specified among 𝒖𝝉′𝑠 except for + +6 + +its own two moments as in (1.1b). This is because any 𝑦#MSP in the EEM-M is assumed +to be able to evaluate the total performance of each individual model in 𝐵ℳ([𝑿]) with its +own criterion when (𝒚𝒐, 𝑿) is given. +In the EEM-M, the XMSP will replace [𝑿] in 𝐵ℳ([𝑿]) by a class 𝐷 of IC-controlled +models, which makes 𝑦#MSP better accommodated or equivalently more effectively +functioned. This can be done in advance before 𝒚𝒐 is observed. + The above argument shows that in our analytical framework, there is no concept of “true” +model. After all, a finally selected empirically effective model via a 𝑦#MSP will have to +be regarded as the model having generated 𝒚𝒐 in EEM-M or even in any empirical +analysis, so long as 𝒚𝒐 is regressed on the final model and it is used in applications. +1.3 Individual Predictive Sampling Variance as for [1] +To describe the two factors in (1.3) for [1] in a sub-model 𝒚∗ = 𝑿∗𝜷∗ + 𝒖∗, let us first +consider the individual sampling variance (IndSV) 𝑉𝑎𝑟(𝛽_% +∗) of each OLSE 𝛽_% +∗. Then, as +is well known, it is decomposed as +(1.4) 𝑉𝑎𝑟r𝛽_% +∗s = +6! +778" +! = +6! +-9#" +! × 𝑉𝐼𝐹% with 𝐸𝐸𝐹% +" = 𝑁𝑠𝑥𝑘 +2 (1 − 𝑅M𝑘 +2), +where 𝑘 = 2, ⋯ , 𝐾∗. Here 𝐸𝐸𝐹% +." is the (k,k) element of (𝑿∗)𝑿∗)./ and 𝐸𝐸𝐹% is called +empirically effective factor (EEF) that controls 𝑉𝑎𝑟r𝛽_% +∗s. It is desirable for 𝐸𝐸𝐹% to be +larger. Also, 𝑉𝑎𝑟r𝛽_% +∗s in (1.4) is linearly affected by 𝑉𝐼𝐹% in (1.3), which is used as one +of the equivalent alternatives for the C-factor in (1.3). + Clearly each 𝐸𝐸𝐹% +" consists of the two components: +1) the variate-own effect 𝑁𝑠0% +" of predictive variable 𝒙𝒌 +∗ and +2) the collinearity effect 𝑉𝐼𝐹%. +In this case, for given 𝑉𝐼𝐹%, the larger the 𝑁𝑠0% +" is, the smaller 𝑉𝑎𝑟r𝛽_% +∗s is, and the more +efficient the k-th OLSE is. This is inappropriate in comparing the standard errors (SEs) +√𝑉𝑎𝑟 +u r𝛽_% +∗s′𝑠 of individual OLSEs in an estimated model. Besides, the physical units of +𝒙% +∗ ’s in measurement are different in general and so IndSVs in (1.3) are not comparable. +To overcome the incomparability of IndSVs, we pay attention to individual terms +𝑥$%𝛽_% +∗)𝑠 in each model. Since these terms have a common physical unit with 𝑦$, we adopt +the predictive sampling variance (IndPSV) of each 𝛽_% +∗ that is defined by +(1.5) 𝐼𝑛𝑑𝑃𝑆𝑉% ≡ ∑ +𝑉𝑎𝑟r𝑥$%𝛽_% +∗s +- +$1/ += 𝜎"(1 + 𝑧̅% +")𝑉𝐼𝐹% ≡ 𝜎"𝐻(𝑧̅% +", 𝑉𝐼𝐹%), +where 𝑧$% = 𝑥$% 𝑠0% +⁄ + (see (2.3)). Then it is clear that the two variables 𝑧̅% +" and 𝑉𝐼𝐹% in +(1.5) are measurement-unit-free since they are scale-invariant (see 3). Note that 𝜎 carries +the same physical unit with 𝑦$. And the 𝐼𝑛𝑑𝑃𝑆𝑉% of the k-th term {𝑥$%𝛽_% +∗: 𝑛 = 1, ⋯ , 𝑁} +consists of the two effects: + +7 + +3) the variate-own inefficiency effect ||𝒙𝒌 +∗||"/ 𝑁𝑠0% +" = 1 + 𝑧̅% +" and +4) the collinearity effect 𝑉𝐼𝐹%. +In the expression (1.5), it is desirable for both 𝑧̅% +" ≥ 0 and 𝑉𝐼𝐹% ≥ 1 to be smaller so that +𝐼𝑛𝑑𝑃𝑆𝑉% becomes smaller. In (1.5), for the k-th OLS estimate in model 𝑿∗, 𝐼% ≡ 𝑧̅% +" is +the I-factor to be controlled as in (1.3), while as one of the two equivalents in (1.3), the +𝑉𝐼𝐹% is called the C-factor 𝐶% to be controlled. In fact, 𝑉𝐼𝐹% ≤ 𝑑 is equivalent to 𝑅X% +" ≤ +𝑑: = 1 − 1/𝑑, and so 𝑉𝐼𝐹% and 𝑅X% +" are interchangeably used in this paper. + In Section 2, for reference, the standard error (SE) of each estimate based on IndSV is +compared to the SE based on Ind PSV after model is estimated with 𝒚𝒐. Also, we give a +necessary and sufficient condition for a model 𝑿∗ to attain the lower bound for the +IndPSVs of all the estimates. +1.4 𝒚𝒐-Accommodating Class, IC Risk Index and Admissibility of Model +In Section 3, we will formulate a decision theoretic framework for comparing models +by the ICRI of 𝑿∗. Since the degree of collinearity of model 𝑿∗ is not greater than that of +𝑿∗∗ if 𝑿∗ ⊂ 𝑿∗∗, it is necessary to take the column size into account in comparing +models by 𝑉𝐼𝐹%. Hence, letting 𝐽(𝑿∗) be the column size of 𝑿∗, the distinguishability of +models {𝑿∗} with 𝐽(𝑿∗) = 𝐽 is made only through the set of I-factors and C-factors of +each OLS estimate in 𝑿∗; +(1.6) {(𝑧̅% +"(𝑝%𝑿∗), 𝑉𝐼𝐹%(𝑿∗))|𝑘 = 2, ⋯ , 𝐽}, +where 𝑝%𝑿∗ = 𝒙𝒌 +∗. And, let 𝐷; +23 denote the class of IC-controlled models of column size +J with control level (c, d), where +(1.7a) 𝐷; +23 = {𝑿∗ ∈ [𝑿: 𝐽]| 𝑧̅% +"(𝑝%𝑿∗) ≤ 𝑐, 𝑉𝐼𝐹%(𝑿∗) ≤ 𝑑, (𝑘 = 2, ⋯ , 𝐽)}, +where [𝑿: 𝐽] denotes the set of models whose column sizes are J. Also let + (1.7b) 𝐷23 =∪𝐽=2 +𝐾 +𝐷; +23 +be the set of all the IC-controlled models with control level (c, d). To materialize the +XMSP to find 𝐷23, two algorithms are proposed in Section 4. In this paper, the problem +of how to choose (c, d) is left open (see Sec. 3.2). + Also in Section 3, a partial ordering on models with column size J is introduced based +on the ICRI of model 𝑿𝝉; +∗ , which is defined by +(1.8a) 𝑟r𝑿𝝉; +∗ s = r𝑐<=;, 𝑑<=;s for each 𝑿𝝉; +∗ , and +(1.8b) 𝑐<=; = 𝑚𝑎𝑥%𝑧̅% +"(𝑝%𝑿𝝉𝑱 +∗ ) and 𝑑<=; = 𝑚𝑎𝑥%𝑉𝐼𝐹%(𝑿𝝉; +∗ ). +The model 𝑿𝝉; +∗ with (1.8b) is also denoted as 𝑿𝝉∗r𝑐<=;, 𝑑<=;s. Then for each J fixed, +𝑿𝝉∗r𝑐<=;, 𝑑<=;s is said to better accommodate 𝒚𝒐 than 𝑿𝝉) +∗ r𝑐<=; +) +, 𝑑<=; +) +s if 𝑐<=; ≤ 𝑐<=; +) + +and 𝑑<=; ≤ 𝑑<=; +) + hold with one of the inequalities strict. Importantly, this can be + +8 + +determined before 𝒚𝒐 is observed. From this set-up, the concept of admissibility naturally +follows (Sec. 3), though the characterization of admissible 𝒚𝒐-accommodating class of +models is made only in the case of J=2. +1.5 Two Algorithms for materializing the XMSP +In Section 4, to make the XMSP practically feasible, we develop two computational +algorithms: variable-increasing and variable-reducing algorithms. In the latter case, +principal component analysis is used, and it is shown that 𝑅X% +"(X) = 𝑅X% +"(Z) for the matrix +Z of standardized variates of 𝒙+ (𝑘 ≥ 2). In particular, we focus on the algorithm to make +models satisfy the condition 𝑅"! +" ≤ 𝑑𝑅 ≡ 1 − 1/𝑑 for any k, since it is easy to find models +that satisfy 𝑧̅% +" ≤ 𝑐. Then a set {𝑿∗} of IC-controlled models is obtained by combining +them as the intersection of these sets. +1.6 A brief Literature Review on Collinearity in Model Selection +Research history on collinearity in regression is very long and has been still +accumulating a vast amount of literature, though no clear-cut solution exists. In our OLS +context, a recent example is Tsao (2019) proposing estimating a linear combination of +regression parameters in a strongly correlated model X. While, because they are not in the +OLS framework, we do not treat such hybrid methods and procedures of the GLS-type +(Kariya and Kurata, 2003) , LASSO-type (Zou and Hastie, 2005), ridge-type (Hoerl and +Kennard, 1988), Stein-type (Kubokawa and Srivastava, 2004), principal component-type +(Jolliffe, 1986), and standardization-type (Aitken and West, 1996) methods. These +sophisticated methods basically aim to secure the stability of estimates in the context of +collinearity. Stewart (1987) mathematically clarified the algebraic structure of collinearity +in X. Fox and Monette (1991) generalized the concept of VIF to the case where 𝑿 is +divided into three categories of variables, including dummy variables. Lavery, Acharya, +Sivo and Xu (2019) studied on the relation between the number of variables and +collinearity. +Some 𝑦#MSPs are associated with a pre-testing, variable selection and estimation +procedure and often used as those of stepwise forward, backward or hybrid methods. In +their textbook, Draper and Smith (1998) well describe them with various empirical +examples. But these MSPs do not lead to an optimal model as a whole. In addition, the +ill-effect of collinearity may let us select a wrong model. +2 Sampling Variance and Predictive Sampling Variance of OLSE +In this section, first, using the individual predictive sampling variance (IndPSV) of +each individual OLS estimate in (1.5), we study some important relations between I-C + +9 + +factors (𝐼%, 𝐶%) in (1.3) and the structure of model 𝑿∗that carries ({(𝐼%, 𝐶%): 𝑘 = 2, ⋯ , 𝐾∗}. +In 2.2, we compare the traditional standard errors (SDs) of individual OLS estimates +based on the IndSV and the corresponding predictive standard errors (PSDs) based on the +IndPSV). In the sequel, we let X represent all the cases of 𝑿∗ ∈ [𝑿]. + The effectiveness of the XMSP relies on the fact that the predictive sampling variance +(IndPSV) of each term {𝑥$%𝛽_%} = {𝑥$%𝛽_%: 𝑛 = 1, ⋯ , 𝑁} with 𝛽_% being the OLS +“estimator” is expressed as 𝜎"𝐻(𝑧̅% +", 𝑉𝐼𝐹%) in (1.5), which is a function of I-factor 𝑧̅% +" and +C-factor 𝑉𝐼𝐹%. Since the 𝐻 function depends only on 𝑿, an “estimated” model is made +IC-controlled by controlling the I-factor and the C-factor separately. Hence, the IndPSVs +as a sampling property of {𝑥$%𝛽_%} is connected with the effectiveness of the OLS +estimates in the estimated model before 𝒚𝒐 is observed. + +2.1 Relations between (𝐼%, 𝐶%) and the Structure of 𝑿 +More specifically, let us consider an estimated predictive model: + (2.1) 𝑦•$ = 𝛽_/ + 𝑥$"𝛽_" + ⋯ + 𝑥$?𝛽_?, +which is the n-th element 𝑦•$ of 𝒚€ = ∑ +𝒙𝒌𝛽_% +? +%1/ += 𝑿𝜷]. This is also viewed as pre- +sampled version as 𝑦•$ ≡ 𝑦•$(𝒚) for given X. Then, for {𝑥$%} given, the sampling +distribution of 𝑦•$(𝒚) is dependent on the whole covariance matrix 𝑿𝑉𝑎𝑟r𝜷]s𝑿′, but in +view of the EEM, what matters in the post-sampled predictive model with 𝒚𝒐 given is the +efficiency and stability of individual terms {𝑥$%𝛽_%}′𝑠 by Definition 1.1. By the above +observation, it suffices to control the inefficiency factor (I-factor) and the collinearity +factor (C-factor) of each term, since it controls 𝑉𝑎𝑟(𝑥$%𝛽_%) for each 𝑥$% whether or not +𝒚𝒐 is observed. If the IndPSV is small, the contribution of 𝑥$%𝛽_% to 𝑦•$ will be stable and +𝑥$%𝛽_% is likely to be realized in a neighborhood of its mean 𝑥$%𝛽% for fixed 𝑥$%, though +the variation is that of 𝛽_%. Since 𝑥$% varies over {𝑥$%}, summing 𝑉𝑎𝑟(𝑥$%𝛽_%) up over +𝑛 = 1, ⋯ , 𝑁 yields the IndPSV as +(2.2) 𝐼𝑛𝑑𝑃𝑆𝑉% ≡ ∑ +𝑉𝑎𝑟(𝑥$%𝛽_%) +- +$1/ += +6! ∑ +𝑥𝑛𝑘2 +𝑁 +𝑛=1 +778" +! + ≡ 𝜎"𝐻%. + Here letting 𝑧$% = 𝑥$% 𝑠0% +⁄ + and using ||𝒙𝒌||" = ∑ +𝑥$% +" +- +$1/ +, + ||𝒙𝒌||"/𝑁𝑠0% +" = +/ +- ∑ +• +0$" +9#"‚ +" +- +$1/ += 1 + 𝑧̅% +", + and hence each IndPSV is determined by the measurement-unit-free variable; +(2.3) 𝐻% = 𝐻(𝑧̅% +", 𝑉𝐼𝐹%) ≡ (1 + 𝑧̅% +") × 𝑉𝐼𝐹% (≥ 1). +This 𝐻% is clearly free from the physical unit of 𝑥$% so that its size is comparable with +those of the others, and the physical unit of y is carried over to its population standard +deviation 𝜎 in (2.2). In (2.3), the IndPSV is separated into the inefficiency factor 𝑧̅% +" and + +10 + +the collinearity factor 𝑉𝐼𝐹%. +From (2.3), it is easy to observe the following facts. +(a) 𝑧̅% +"=0 if and only if 𝑥̄% = 0, and 𝑉𝐼𝐹% =1 if and only if 𝑅X% +" = 0. +(b) 𝐻% = 1 if and if 𝑥̄% = 0 and 𝑅X% +" = 0. +When 𝑥̄% = 0, then 𝐻% = 𝑉𝐼𝐹%, and so controlling 𝑉𝐼𝐹% is equivalent to controlling the +IndPSV in (2.2). While, when 𝑉𝐼𝐹%=1, then 𝐻% = 1 + 𝑧̅% +" so that it is desirable for 𝑧̅% +" to +be small. When 𝐻%(≥ 1) is small, the term {𝑥$%𝛽_%} in (2.1) is stabilized in the sense of +𝐼𝑛𝑑𝑃𝑆𝑉% and the collinearity factor is not large whether or not y is observed. + The next proposition gives the condition for the attainment of the lower bound of 𝐻%. +Proposition 2.1 In (2.3), for given 𝑘(≥ 2), 𝐻% = 1 if and only if 𝑥̄% = 0 and 𝒙𝒊′𝒙𝒌 = +0 for any 𝑖 ≠ 𝑘 (𝑖 ≥ 1). Hence, 𝐻% = 1 for any 𝑘(≥ 2) if and only if +(1) 𝑥̄% = 0 for any 𝑘(≥ 2) and (2) 𝑿-𝑿 = 𝑑𝑖𝑎𝑔{𝑎$, 𝑎., ⋯ , 𝑎"}, + where 𝑎/ = 𝑁 and 𝑎B = 𝒙𝒊′𝒙𝒊. If model satisfies (2) only, 𝐻% = 1 + 𝑧̅% +" only denotes the +effect of inefficiency value. Here 𝑑𝑖𝑎𝑔{𝑎$, ⋯ , 𝑎"} denotes diagonal matrix. +Proof. By the definition of 𝑅X% +", letting 𝑴M 𝒌 = 𝑿M𝒌(𝑿M𝒌′𝑿M𝒌).𝟏𝑿M𝒌′ with 𝑿M𝒌 = N𝒙𝒋 ∈ 𝑿, 𝑗 ≠ +𝑘T, 𝒙€𝒌 = 𝑴M 𝒌𝒙𝒌 and 𝑴𝒆 = 𝒆(𝒆)𝒆).𝟏𝒆) with 𝑴M 𝒌𝒆 = 𝒆. Then + 𝑅X% +" = 𝑠0D% +" /𝑠0% +" = 𝒙€𝒌 +) (𝑰 − 𝑴𝒆)𝒙€𝒌/𝒙𝒌 +) (𝑰 − 𝑴𝒆)𝒙𝒌. +Hence, 𝑅X% +" = 0 if and only if 𝒙𝒌 +) (𝑴M 𝒌 − 𝑴𝒆)𝒙𝒌=0. Therefore, for any k, 𝐻% = 1 if and +only if 𝑥̄% = 0 and 𝒙𝒌 +) 𝑴M 𝒌𝒙𝒌 = 𝒙𝒌′𝑴𝒆𝒙𝒌 =0 since 𝑁𝑥̄% = 𝒆′𝒙𝒌, which in turn holds if +and only if 𝑥̄% = 0 and 𝑿M𝒌′𝒙𝒌 = 𝟎, implying 𝒙𝒌′𝒙𝒋 = 0 (𝑘 ≠ 𝑗). Thus the result follows. +Example 2.1 An example of 𝑿 satisfying the conditions (1) and (2) in Proposition 2.1 is +𝑿 = {𝒙𝟏, 𝒙𝟐, ⋯ , 𝒙𝑲} with 𝒙𝒊 = 𝛼B𝜹𝒊 (𝛼B > 0), where 𝑬 ≡ (𝜹𝟏, 𝜹𝟐, ⋯ , 𝜹𝑲): 𝑁 × 𝐾 is the +matrix consisting of the first K columns of the Helmert’s orthogonal matrix with + 𝜹𝟏 = 𝒆/√𝑁, 𝒙𝟐 = ( +& +√(∙& , +*& +√(∙&,0, ⋯ ⋯ ,0)′ , + 𝜹𝒌 = ( +$ +%!∙(!($) , +$ +%!∙(!($) , ⋯ , +$ +%!∙(!($) , +((!($) +%!∙(!($) , 0, ⋯ ,0)′ for 𝑘 ∈ {2, ⋯ , 𝑁}. +In fact, this matrix satisfies 𝑥̄% = 0 and 𝑿-𝑿 = 𝑑𝑖𝑎𝑔{𝑎$, 𝑎., ⋯ , 𝑎"} with 𝑎B = 𝛼B +". While, +since it is assumed that model (1.1) includes constant term 𝒙𝟏 = 𝒆, 𝑿) = [𝑰, 𝟎)]: 𝐾 × 𝑁 +cannot be a model for 𝑿 satisfying (1) and (2). +Next, let us study some properties of the I-factor, which will be used in 3. +Proposition 2.2 (1) When 𝐾 ≥ 3, the inefficiency factor is independently measured by + +11 + +(2.4) 𝑧̅% +" ≡ 𝑧̅% +"(𝑝%𝑿) = +0̅" +! +9" +! = +𝒙𝒌 +, 𝑴𝒆𝒙𝒌 +𝒙𝒌 +, (𝑰.𝑴𝒆)𝒙𝒌 = +K" +/.K", +which takes values on [0, ∞) and is increasing in 𝑞% ∈ [0,1), where + 𝑞% = (𝒙𝒌 +) 𝒆)"/[𝒙𝒌 +) 𝒙𝒌 ∙ 𝒆)𝒆] = 𝑐𝑜𝑠"(𝜃%) +is the squared raw correlation of 𝒙𝟏 = 𝒆 and 𝒙𝒌, and 𝑴𝒆 = 𝒆(𝒆)𝒆).𝟏𝒆). +(2) If K=2, then 𝑧̅" +" = 𝑞"/(1 − 𝑞") and 𝑉𝐼𝐹" = 1/(1 − 𝑞"), and hence the two factors +are both a function of 𝑞" and 𝐻% = 1/(1 − 𝑞")". Therefore, they are not separable and +𝐻% = 1 if and only if 𝒙𝒌 +) 𝒆 = 0. +(3) For 𝐾 ≥ 2 , the inefficiency factor 𝑧̅% +" or equivalently 𝑞% = 𝑧̅% +"/(1 + 𝑧̅% +") is a +collinearity measure between 𝒙𝟏 = 𝒆 and 𝒙𝒌. +Proof. Straightforward and omitted. +Thus 𝑧̅% +" is an increasing function of 𝑞%, and the closer the angle between two vectors 𝒙𝒌 +and 𝒆 is to 0, the larger 𝑧̅% +" is. In other words, if and only if 𝒙𝒌 and 𝒆 are orthogonal, 𝑧̅% +" +attains the minimum 0 (Proposition 2.2(3)), implying that 𝑧̅% +" is also measuring the +collinearity of 𝒙𝒌 with 𝒆. Hence, it will be better to choose such variables 𝒙𝒌’s that +𝑐𝑜𝑠"(𝜃%) is close to 0 or equivalently 𝒙𝒌 is less collinear with 𝒆 so that the I-factor is +smaller, provided such a selection of variables is possible. +Note that the inefficiency factor 1 + 𝑧̅% +" is expressed as + +/ +- ∑ +• +0$" +9#"‚ +" +- +$1/ += 1 + 𝑧̅% +" = (1 − 𝑞%)./ = 1 + 𝑞% + 𝑞% +" + ⋯. +2.1 Comparisons of IndSV 𝑉𝑎𝑟r𝛽_%s and IndPSV 𝐼𝑛𝑑𝑃𝑆𝑉% + To compare IndSV and IndPSV, let us make some basic observations. +First, in the case of IndSV 𝑉𝑎𝑟r𝛽_%s that depends on (𝑠0%, 𝑉𝐼𝐹%) in (1.4), it is desirable +for 𝑠0% to be large in order to make the IndSV small. While, in the case of the IndPSV +that depends on ( 𝑧̅% +", 𝑉𝐼𝐹% ) in (2.2), 𝑧̅% +" is the I-factor with 𝑠L% = 1 , where 𝑠L% +" = +∑ +(𝑧$% − 𝑧̅%)"/𝑁 +- +$1/ +. Hence, under 𝑠L% = 1, it is desirable for (𝑥̄%)" to be small as well +as for 𝑠0% to be large for the sake of the IndSV. In other words, it is desired that { 𝑥$% }is +distributed over a broad interval including 0 and its mean is close to 0, even though this +is not controllable since 𝑿∗ is given. While, the case of {𝑥$% > 0 for any n} tends to make +the I-factor large. In addition, if 𝑠0% is also desired to be large for the sake of IndSV, then +the distribution of { 𝑥$% } needs to be distributed more like 𝑁(0, 𝛾) with large 𝛾. + If the standardized variable 𝑤$% = (𝑥$% − 𝑥̅%) 𝑠0% +⁄ + can be used for 𝑥$% in the model, +then 𝑤•% = 0 obtains and the 𝐼𝑛𝑑𝑃𝑆𝑉% = 𝜎"𝑉𝐼𝐹%. This is an important implication for +empirical modelling in treating collinearity via standardization (e.g., Aitken and West, +1996), when the IndPSV is used. In Section 4, it will be shown that the VIF based on + +12 + +{𝑤$%} is the same as the VIF based on { 𝑥$% }. + Second, the XMSP does not aim to minimize 𝐻%′𝑠 or its mean, but aims to control the +two factors of 𝑧̅% +" and 𝑉𝐼𝐹% separately because the strong collinearity of 𝒙% +∗ can seriously +cause confounding effects on the other variables. +Now, let us compare the standard error (SE) of 𝛽_% and the corresponding predictive +standard error (PSE), which are respectively given by +(2.5a) 𝑆𝐸 •𝛽_%(𝑿∗)‚ = 𝜎•(𝒚𝒐, 𝑿∗)(𝑠0%).. +!√𝑉𝐼𝐹% +(2.5b) 𝑃𝑆𝐸 •𝛽_%(𝑿∗)‚ = 𝜎•(𝒚𝒐, 𝑿∗)(1 + 𝑧̅% +") +. +!√𝑉𝐼𝐹% ≡ 𝜎•𝐻% +//", +where 𝜎• is the residual standard error (RSE) under model 𝑿∗ via OLS with 𝒚𝒐, which is +another important measure of model performance with 𝒚𝒐. Each IndSV in (2.5a) can be +smaller than 𝜎" because of (𝑠0%).//". Hence, it is not easy to compare and interpret +those SEs, although the SEs in (2.5a) have been traditionally given as computer outputs +with other statistics such as adjusted CD. In the case of 𝜎•𝐻% +//" in (2.5b), which we call +the PSE of the term {𝑥$%𝛽_%: 𝑛 = 1, ⋯ , 𝑁}, each 𝐻% +//" can be compared with other 𝐻/ +$/.′𝑠. +Hence the individual estimates in (2.2) are comparable in terms of the PSEs together with +RSE 𝜎•. In empirical analysis, it will be better to have the PSE values 𝑃𝑆𝐸(𝛽J𝑘(𝑿∗))′𝑠 as +computer outputs that reveal the contribution of each term {𝑥$%𝛽_%} to 𝒚𝒐 with relative +roles of 𝒙% +∗ 𝛽%′𝑠. + Finally it is remarked that by (1.5), the k-th SE is expressed as +(2.6) 𝑆𝐸 •𝛽_%(𝑿∗)‚ = 𝜎•(𝒚𝒐, 𝑿∗)‖𝒙% +∗‖./(1 + 𝑧̅% +").//"√𝑉𝐼𝐹%. +This expression shows that the k-th SE depends on the three effects; ‖𝒙% +∗‖, 𝑧̅% +" and 𝑉𝐼𝐹%. +In Section 3, it will be pointed out from (2.6) that the SE is controlled by controlling +‖𝒙% +∗ ‖ after a 𝒚𝒐-accommodating class of models with the 𝑧̅% +" and 𝑉𝐼𝐹% controlled is +obtained, where the three effects are functionally independent if 𝐾∗ ≥ 3. +As was stated, the performances of different models 𝑿∗′𝑠 may be compared by the +mean of {𝐻%} as +(2.7) 𝐻• ≡ 𝐻•(𝑿∗) = +/ +?∗./ ∑ +𝐻% +?∗ +%1" += +/ +?∗./ ∑ +(1 + 𝑧̅% +") × 𝑉𝐼𝐹% +?∗ +%1" + +for 𝑿∗: 𝑁 × 𝐾∗ in the IndPSVs, provided 𝑉𝐼𝐹%′𝑠 are controlled in such a way that 𝑉𝐼𝐹% ≤ +𝑑 for any k. This gives a direct comparability of different models { 𝑿∗} in a certain class +of such models as, e.g., those of the same column size J as in 3.3. However, in model +comparison, it will be necessary to take the RSE 𝜎•(𝒚𝒐, 𝑿∗) into consideration, implying +that after a class of IC-controlled models, models in the class may be compared in terms + +13 + +of 𝜎•(𝒚𝒐, 𝑿∗)𝐻•(𝑿∗) for each fixed J . In addition, it will be necessary to use a 𝑦#MSP to +evaluate a whole model performance with (𝒚𝒐, 𝑿 ) and select an optimal model by such +MSPs with AIC, BIC, adjusted CD 𝑅Ÿ", etc. +3 The XMSP for EEM-M + As has been stated, not by a diagnostic checking after a model is estimated, the XMSP +in the EEM-M aims to control (𝐼%, 𝐶%) or equivalently (𝑧̅% +", 𝑉𝐼𝐹%) in (1.3) in such a way +that 𝑧̅% +" ≤ 𝑐 𝑎𝑛𝑑 𝑉𝐼𝐹% ≤ 𝑑 , before 𝒚𝒐 is observed. Then it gives a class 𝐷 of IC- +controlled models for efficiency and stability of OLS estimates. The class is also called a +𝒚𝒐 -accommodating class for 𝑦# MSP. The whole EEM-M process is symbolically +described as 𝐵ℳ([𝑿]) −XMSP —𝑦# MSP (see (1.1a,b) for 𝐵ℳ([𝑿])). It is shown in +Proposition 2.1 that the most efficiently IC-controlled model with 𝐻% = 1 for any 𝑘(≥ +2) is characterized as a model that satisfies +(1) 𝑥̄% = 0 for any 𝑘(≥ 2) and (2) 𝑿-𝑿 = 𝑑𝑖𝑎𝑔{𝑎$, 𝑎., ⋯ , 𝑎"}, +where 𝑎/ = 𝒙𝒊′𝒙𝒊. One may develop a method for comparing models. +In 3.1, the bundle model 𝐵ℳ([𝑿]) in 1.2 is further discussed to develop the XMSP +and in 3.2, we discuss some methods of obtaining the class 𝐷 of IC-controlled models, +where the column size of each model 𝑿∗ is taken into account. In 3.3, for each model 𝑿∗, +using inefficiency-collinearity risk index (ICRI) (see 1.4), a partial ordering is introduced +onto the set [𝑿] to compare models in terms of ICRI and the concept of admissibility of +model is defined by the ICRI. +3.1 Bundle Model and Stochastic Specification. +Recall that the bundle model in (1.1a, b) is given by +(1.1a) 𝐵ℳ([𝑿]) = { 𝒚𝝉 = 𝑿𝝉𝜷𝝉 + 𝒖𝝉 | 𝑿𝝉 ∈ [𝑿], 𝒖𝝉 ∈ [𝒖]} with +(1.1b) [𝒖] = {𝒖𝝉: 𝜏 ∈ Λ, 𝐸(𝒖𝝉) = 𝟎, 𝑉𝑎𝑟(𝒖𝝉) = 𝜎"𝑰}. +This is the set of different models with their own error terms from which we aim to select +a model to use for empirical applications in view of the EEM-M. Note that 𝒚𝒐 is regarded +as generated from a model in the bundle model. This model implies 𝑉𝑎𝑟( 𝒚𝝉) = 𝜎"𝑰 for +any sub-model 𝑿𝝉 in the bundle set, which in turn implies the structure of IndPSV +𝐻(𝑧̅=% +" , 𝑉𝐼𝐹=%) in (2.3) as a function of 𝑿𝝉. This is the information we use to select 𝐷 in +the XMSP and it is symbolically expressed as + 𝜗= ≡ ({𝐻(𝑧̅=% +" , 𝑉𝐼𝐹=%): 𝑘 = 2, ⋯ , 𝐾=}, 𝑉𝑎𝑟(𝒚𝝉) = 𝜎"𝑰), where 𝜏 ∈ Λ, +which is independent of 𝒚𝝉 (including 𝒚𝒐) but dependent on 𝑿𝝉. This structure is in fact +an essential part of empirical modelling connected with the predictive sampling variances +of individual OLS estimators and it enables the XMSP to aim to select a class of IC- + +14 + +controlled models by controlling all the (𝑧̅% +", 𝑉𝐼𝐹%)′𝑠 of individual OLS estimates in each +model without observing 𝒚𝒐. Here we drop the suffix 𝜏 distinguishing models unless it +is necessary and denote a representative sub-model by 𝑿∗. +To develop this XMSP, first note that when model sequence {𝑿𝒊} is increasing in such +a way as 𝑿𝒊 ⊂ 𝑿𝒊N𝟏 ⊂ ⋯ ⊂ 𝑿, the k-th CD 𝑅X% +"(𝑿𝒊) or equivalently 𝑉𝐼𝐹% is increasing +in the column size of each matrix i for fixed k. Hence let +(3.1a) [𝑿] =∪34. +" +[𝑿: 𝐽] with +(3.1b) [𝑿: 𝐽] = M𝑿∗ ∈ [𝑿]|𝑿∗ ∶ 𝑁 × 𝐽, 𝑟𝑎𝑛𝑘(𝑿∗) = 𝐽, 𝒙1 = 𝒆 ∈ 𝑿∗Q. +Note [𝑿] ≡ [𝑿: 𝐾]. To connect 𝑿∗ ∈ [𝑿] with y, let 𝒖∗ ∈ [𝒖] represent a corresponding +error term that has no association with (𝒚, 𝑿) when 𝑿∗ ≠ 𝑿. Then the observed 𝒚𝒐 is +regarded as realized through one of the models in the following set: +(3.2a) 𝐵ℳ([𝑿]) =∪𝐽=2 +𝐾 +𝐵ℳ([𝑿: 𝐽]) with +(3.2b) 𝐵ℳ([𝑿: 𝐽]) = {𝒚∗ = 𝑿∗𝜷∗ + 𝒖∗| 𝑿∗ ∈ [𝑿: 𝐽], 𝒖∗ ∈ [𝒖] }. +The model (1.1) itself belongs to Bℳ([𝑿]). +3.2 𝒚𝒐-Accommodating Class : Controlling (𝐼%, 𝐶%) +In this bundle model, the XMSP is a process of exploring a class of models {𝑿∗} with +𝑧̅% +")𝑠 𝑎𝑛𝑑 𝑉𝐼𝐹%′𝑠 controlled separately, so that all the IndPSVs of each model in the +selected class are controlled. This implies that all the models in the class are candidates +of empirically effective models because an optimal model is selected from them by a +𝑦#MSP. Of course, the optimality depends on the choice of the 𝑦#MSP. In other words, +the XMSP is a self-fulfilling process to make a selected class of models IC-controlled for +𝑦#MSPs. +Now, we formally define the concept of 𝒚𝒐-accommodating set of models. Let 𝐽(𝑿∗) +denote the column size of matrix 𝑿∗, and let 𝑝%𝑿∗ = 𝒙𝒌 +∗ for 𝑘 ≤ 𝐽(𝑿∗) as before. +Definition 3.1 Let (c, d) be prespecified as control parameters. The 𝒚𝒐-accommodating +class of models at level (c, d) is defined to be the class 𝐷23 ≡ 𝐷23([𝑿]) of models, where +(3.3) 𝐷23 = {𝑿∗ ∈ [𝑿]| 𝑧̅% +"(𝑝%𝑿∗) ≤ 𝑐, 𝑉𝐼𝐹%(𝑿∗) ≤ 𝑑, (𝑘 = 2, ⋯ , 𝐽(𝑿∗))} + = 𝐷/ +2 ∩ 𝐷" +3. +Here +(3.3a) 𝐷/ +2 = {𝑿∗ ∈ [𝑿]| 𝑧̅% +"(𝑝%𝑿∗) ≤ 𝑐, (𝑘 = 2, ⋯ , 𝐽(𝑿∗))}, and +(3.3b) 𝐷" +3 = {𝑿∗ ∈ [𝑿]| 𝑉𝐼𝐹%(𝑿∗) ≤ 𝑑, (𝑘 = 2, ⋯ , 𝐽(𝑿∗))}. +Proposition 3.1. (1) If 𝑿∗ ∈ 𝐷23 and 𝑿∗∗ ⊂ 𝑿∗, then 𝑿∗∗ ∈ 𝐷23. +(2) If 𝑿𝟏, 𝑿𝟐 ∈ 𝐷23, then 𝑿𝟏 ∩ 𝑿𝟐 ∈ 𝐷23. + +15 + +This obvious proposition is practically useful to find a model in 𝐷23. If 𝑿∗ is found to +belong to 𝐷23 , so does any sub-model 𝑿∗∗ of 𝑿∗. This will be used in Section 4 to +develop an algorithm of finding models in 𝐷23 with principal component analysis. + It is noted that by (3.3), I-factors 𝑧̅% +"′𝑠 and C-factors 𝑉𝐼𝐹%′𝑠 can be separately controlled +provided 𝐽(𝑿∗) ≥ 3 (see Proposition 2.2). Also note that both 𝑧̅% +" and 𝑅X% +" (or +equivalently 𝑉𝐼𝐹%) are scale-invariant for transforming 𝒙𝒌 +∗ into 𝑎%𝒙𝒌 +∗ for 𝑎% ≠ 0 (𝑘 = +2, ⋯ , 𝐾), implying the independence of their physical units. In addition, as will be shown +in Section 4, 𝑅X% +" is invariant under the transformation from 𝑥$% into (𝑥$% − 𝑥̅%) 𝑠0% +⁄ + for +≥ 2 (see Lemma 4.1). Also, note that by Proposition 2.1 when 𝑅X% +" =0, 𝐼𝑛𝑑𝑃𝑆𝑉% = +𝜎"(1 + 𝑧̅% +"), and that when 𝑧̅% +" = 0, 𝐼𝑛𝑑𝑃𝑆𝑉% = 𝜎"𝑉𝐼𝐹%. +Now, to get a 𝐷23, we need to choose (c, d), but it will be difficult to discuss the choice +in a unified manner. A difficulty lies in the fact that the set of variables in data X is given. +In many areas of social sciences, we have to use the data as it stands. Then, so long as the +systems are not so much volatile, 𝑅X% +")𝑠 will carry some predictability as a whole for +dependent variable. On the other hand, in natural sciences where data X can be designed +to a large extent, then 𝑅X% +")𝑠 will be controlled with smaller d and then 𝑧̅% +"′𝑠 will be more +concerned. In general, it will be reasonable to first find 𝐷/ +2 by deleting variables with 𝑧̅% +"′𝑠 +larger than c and then to find 𝐷" +3 of models with 𝑅X% +")𝑠 ≤ 𝑑: = 1 − 𝑑./ from 𝐷/ +2. This +will save some computation. +Some remarks follow. First, in the literature, concerning collinearity, it is often +suggested that in an MSP via VIF, 𝒙𝒌 +∗ be dropped if 𝑉𝐼𝐹%>10 (or 5) or equivalently +𝑅X% +" >0.9 (or 0.8 resp.) in each estimated model with 𝒚𝒐, though there is no theoretically +solid ground (e.g., O’Brien (2007)) and it is likely to a wrong model since the final model +is dependent on the order of selection. In addition, there is no control on the I-factors. + Second, once 𝐷23 is obtained, it is possible to control IndSVs via (2.6b). Since +𝐼𝑛𝑑𝑃𝑆𝑉%/‖𝒙𝒌 +∗‖ = 𝐼𝑛𝑑𝑆𝑉% and since 𝑧̅% +" and 𝑅X% +" do not depend on ‖𝒙𝒌 +∗‖, we can control +‖𝒙𝒌 +∗‖ by setting its lower bound in the class 𝐷23 of models as 𝐷23P = 𝐷23 ∩ 𝐷Q +P with + 𝐷Q +P = {𝑿∗ ∈ [𝑿] | || 𝑝%𝑿∗|| ≥ 𝑒, (𝑘 = 2, ⋯ , 𝐽(𝑿∗))}. +Since any model in the set 𝐷23satisfies +(3.4) 𝐻T𝑧̅5 +., 𝑉𝐼𝐹5Z ≡ (1 + 𝑧̅5 +.(𝑝5𝑿∗))𝑉𝐼𝐹5(𝑿∗) ≤ (1 + 𝑐)𝑑 ≡ ∆67 +for any k, when model is restricted to the class 𝐷23P, the SE in (2.6) is also controlled as +𝑆𝐸 ^𝛽`5(𝑿∗)a = 𝜎c(𝒚𝒐, 𝑿∗)𝐿T𝑧̅5 +., 𝑉𝐼𝐹5Z with + 𝐿T𝑧̅5 +., 𝑉𝐼𝐹5Z ≡ 𝐻T𝑧̅5 +., 𝑉𝐼𝐹5Z/‖𝒙𝒌 +∗‖ ≤ (1 + 𝑐)𝑑/𝑒, +unless 𝐷23P = ∅. While, since the IndPSVs is bounded above by 𝜎"∆23 from (3.4), all + +16 + +the PSEs in (2.5b) are below or equal to 𝜎•R∗ +S√∆23 where 𝜎•R∗ +S is the RSE 𝜎c(𝒚𝒐, 𝑿∗). +3.3 Finding 𝐷23 and Inefficiency-Collinearity Risk Index (ICRI) of 𝑿∗ +Now let us consider some methods of finding the class 𝐷23. One possible method is to +try all the models via all the possible combinations of variables to get 𝐷23though its +computational process may not be feasible if K is large. Although the two algorithms for +finding 𝐷" +3 in (3.3) can be used, it is better to reduce the total number of variables before +such an algorithm is applied, so that the computational burden becomes as little as +possible. +Empirically it is often the case that the set of variables in X is divided into core part X1 +and such non-core part X2 as dummy variables as in Fox and Monette (1992). And a set +of IC-controlled models is found from the core part X1 and then each variable is added to +the set one by one from X2. +Here, to take a different approach, we define the ICRI to compare different models in +view of the EEM-M and apply it for finding IC-controlled models. In the case of the +largest model X, the ICRI is defined as +(3.5a) 𝑟(𝑿) = (𝑐 𝑐 from X , where the distribution of 𝑧̅% +"′𝑠 is studied +with the graph 𝐺. Renumbering the remaining variables, without loss of generality, let +them be denoted by 𝑿(1) = {𝒙𝟏, 𝒙𝟐, ⋯ , 𝒙𝑲(𝟏)} where 𝑐?3 = (1 + 𝑐𝑀𝜏𝐽)𝑑𝑀𝜏𝐽. Hence averaging this over k’s yields the upper +bound for the averaged IndPSV of the model: +(3.9) 𝜎. 𝐻𝜏𝐽 +lllll = 𝐼𝑛𝑑𝑃𝑆𝑉𝜏𝐽 +lllllllllllll ≤ 𝜎. min {𝐻>?3, (1 + 𝑐)𝑑}, +showing the upper bound for the mean control level of IndPSV for model 𝑿𝝉3 +∗ . + Next, the concept of admissibility is defined on the set [𝑿: 𝐽] with rank J≥ 2. +Definition 3.2 For each J fixed, 𝑿𝝉 +∗r𝑐<=;, 𝑑<=;s is said to better accommodate 𝒚𝒐 than +𝑿𝝉) +∗ r𝑐<=; +) +, 𝑑<=; +) +s if 𝑐<=; ≤ 𝑐<=; +) + and 𝑑<=; ≤ 𝑑<=; +) + hold with one of the inequalities strict. +If no matrix of the same column size accommodates 𝒚𝒐 better than 𝑿𝝉 +∗r𝑐<=;, 𝑑<=;s, then +𝑿𝝉 +∗r𝑐<=;, 𝑑<=;s is said to be admissible in 𝐷; +23 ≡ 𝐷; +23([𝑿]). Hence, the minimal +accommodation set for 𝒚𝒐 is defined by the set of admissible matrices in 𝐷; +23; + 𝐷X3Y; +23 +≡ 𝐷X3Y; +23 +([𝐗]) = N𝑿𝝉 +∗ ∈ 𝐷; +23ª 𝑿𝝉 +∗ is admissible }. +Any model in 𝐷X3Y; +23 + is called an admissible 𝒚𝒐-accommodating model in [𝑿: 𝐽]. + +For J fixed, it will be desirable to have such a minimal set as 𝐷X3Y; +23 + for a possible +model selection and all the admissible models are those in the set 𝐷X3Y +23 +≡ +∪;1" +?∗ N𝐷X3Y; +23 +T, which forms a complete class (see Lehmann and Romano (2005)). +However, it is not necessarily required to have a model in this set. In fact, as IndPSVs are +only required to be uniformly controlled, any model in the 𝒚𝒐-accommodating set (3.3) +will do, and it will be important to select a model for accommodating 𝒚𝒐 in an effective +and balanced way so that I-factor and C-factor are acceptably well controlled. +Furthermore, as was discussed in 3.1, a final model should be selected by a 𝑦#MSP +together with such criteria as AIC, BIC, RSE, CD that use 𝒚𝒐. The XMSP simply gives +a set of models whose two factors are uniformly controlled. +Example 3.1 Let J=2. Then the CD where 𝑥% is regressed on 𝒙𝟏 = 𝒆 is regarded as the +squared inner product 𝑟/% +" = (𝒙𝒌 +) 𝒆)"/[𝒙𝒌 +) 𝒙𝒌 ∙ 𝒆)𝒆] = 𝑐𝑜𝑠"(𝜃%) as the collinearity index +with 𝑉𝐼𝐹% = (1 − 𝑟/% +" )./. By Proposition 2.2, 𝑧̅%" +" and 𝑉𝐼𝐹%" are interdependent, and +𝐻%" = (1 − 𝑟/% +" ).". Here k denotes the k-th matrix 𝑿𝒌 +∗ = (𝒙𝟏, 𝒙𝒌) of N×2. +Proposition 3.3 Let J=2 and let 𝑿% +∗ = (𝒙𝟏, 𝒙𝒌) ∈ [𝑿: 2] with 𝑘 = 2, 3, ⋯ , 𝐾. If 𝑿𝒊 +∗ = +(𝒙/, 𝒙B) ∈ 𝐶" +23 satisfies 𝐻%" = (1 − 𝑟/% +" )." = 𝑚𝑖𝑛%(1 − 𝑟/% +" ).", model 𝑿𝒊 +∗ = (𝒙𝟏, 𝒙𝒊) is +admissible, where 𝑟/B +" = 𝑚𝑖𝑛%𝑟/% +" . + +19 + +Proof. In this case 𝑿B +∗ = (𝒙/, 𝒙B) ∈ 𝐶" +23 is admissible if it satisfies (1 + 𝑧̅B" +" )𝑉𝐼𝐹B" = +𝑚𝑖𝑛%(1 + 𝑧̅%" +" )𝑉𝐼𝐹%" = 𝑚𝑖𝑛%(1 − 𝑟/% +" ).", because no other model is better than this +model. +This is not an example for controlling the two factors separately. As in Proposition +2.2(2), in these simple models the two factors 𝑧̅% +" and 𝑉𝐼𝐹% are not separable and they are +simultaneously controlled by minimizing 𝐻%" = (1 − 𝑟/% +" ).". Hence, it will be better to +have a model with smaller 𝑟/B +" , which makes the two factors simultaneously more +controlled. However, in any regression model, it may be better to select one from 𝐷" +23 by +𝑦#MSP together with a judicious judgement. +4 Collinearity-Controlling Algorithms for MSP +In this Section, we materialize the XMSP by developing two algorithms in order to +derive accommodating classes for 𝑦#MSP. We suggested in Section 3 that a class 𝐷/ +2 in +(3.3a) is first and easily obtained. Hence, here the algorithm of deriving the class 𝐷" +3 of +collinearity-controlled models in (3.3b) is only considered in terms of the coefficient of +determination 𝐶𝐷% ≡ 𝑅X% +", which is 𝐶𝐷% when 𝒙% is regressed on 𝑿n𝒌 = M𝒙𝒋 ∈ 𝑿, 𝑗 ≠ 𝑘Q . +The two algorithms are variable increasing method and variable decreasing method. In +the latter case, principal component analysis (PCA) will be used, where PCA here is +nothing but an orthogonal diagonalization of the correlation matrix of 𝒙Z′𝑠 to select +variables that are strongly correlated with the leading principal components. In the +analysis, it is shown that 𝑅X% +" based on X is equal to 𝑅X% +" based on the 𝑁 × (𝐾 − 1) +matrix 𝒁∗ of standardized variates of 𝒙+ (𝑘 ≥ 2). Note that by Proposition 2.1, if +𝐶𝐷%(𝑿∗) ≤ 𝑑 for any 𝑘, then 𝐶𝐷%(𝑿∗∗) ≤ 𝑑 for any sub-model 𝑿∗∗ of 𝑿∗ and for any k. +4.1 Variable-increasing algorithm. + A variable-increasing algorithm is here developed for controlling strong collinearity in +X in advance to the traditional OLS analysis via 𝑦#MSP. Let 𝐽 = {2,3, ⋯ , 𝐾}, and let +(4.1) 𝐺T𝑝: 𝐴B#$Z = {T𝑖ℎ +𝑝, 𝐴B#$Z: 𝐶𝐷(𝑖ℎ +𝑝, 𝐴B#$) ≤ 𝑑, 𝑖ℎ +𝑝 ∈ 𝐽\𝐴B#$} +denote the set of suffixes 𝑖ℎ ++′𝑠 of x variables in the p-th step after the set 𝐴[./ of x’s +suffixes has been selected in the (p-1)-th step, where 𝐶𝐷(𝑖ℎ ++, 𝐴+($) ≤ 𝑑 denotes the CD +condition for the variable of suffix 𝑖ℎ ++ to satisfy. Here 𝐶𝐷(𝑖ℎ ++, 𝐴+($) is the 𝐶𝐷 -value when +𝒙𝑖ℎ +𝑝 is regressed on the set 𝑋(𝐴[./) of the x variables with suffixes in 𝐴[./. The set 𝐽+ of +available suffices, the set 𝐴+ of selected suffices and the whole picture in each step +respectively move as follows. + +20 + + 𝐽+-$ = 𝐽+ ∖ 𝐺=𝑝 + 1: 𝐴𝑝> with 𝐽$ = {2, ⋯ , 𝐾} and 𝐴$ = {1}. + 𝐴+ ≡ 𝐴+(𝑖. ++) = {𝑖. ++} ∪ 𝐴+($ for 𝑖. ++ ∈ 𝐺=𝑝: 𝐴𝑝−1>, 𝑝 ≥ 2, and + H𝐴$ +𝐽$ I → 𝐺(2: 𝐴1) → H𝐴" +𝐽" I → 𝐺(3: 𝐴2) → H𝐴/ +𝐽/ I → 𝐺(4: 𝐴3) → ⋯ . +Here 𝐴$ ⊂ 𝐴" ⊂ 𝐴/ ⊂ ⋯, 𝐽$ ⊃ 𝐽" ⊃ 𝐽/ ⊃ ⋯, and 𝐺T𝑝: 𝐴B#$Z ⊂ 𝐴𝑝. +It is noted that newly added x variables through the selected suffixes in the p-th step have +𝐶𝐷’s with the variables 𝑋(𝐴[./) that are smaller than or equal to d, securing stability. +Using this notation, we describe our algorithm. +(Step1) Let 𝐴/ = {1} in the first step, implying that 𝒙/ is always included, which +corresponds to the first step. Let 𝐽/ = {2, ⋯ , 𝐾}. +(Step 2) Then the suffixes selected in the second step is given by +(4.2) 𝐺(2: 𝐴/) = {(𝑖ℎ +2: 𝐴/): 𝐶𝐷(𝑖ℎ +2, 𝐴/) ≤ 𝑑, 𝑖ℎ +2 ∈ 𝐽/\𝐴/}. + Here the 𝐶𝐷 of 𝒙% with {𝒙/} is defined by the inner product + 𝐶𝐷5({𝒙$}) = 𝒙5′𝒙$/‖𝒙5‖‖𝒙$‖ , +which is used only for this second step. Let 𝐺(2: 𝐴/) be extensively expressed as +(4.3) 𝐺(2: 𝐴/) = {(𝑖ℎ +2, {1})|𝑖ℎ +2 = 𝑖1 +2, ⋯ , 𝑖𝑙(2) +2 + , 1 < 𝑖1 +2 < ⋯ < 𝑖𝑙(2) +2 +}. +Then each of the pairs {(𝒙B, 𝒙/): 𝑖 = 𝑖1 +2, ⋯ , 𝑖𝑙(2) +2 +} satisfies the 𝐶𝐷 condition and hence +each pair defines 𝐴"(𝑖𝑘 +2) = { 𝑖𝑘 +2, 1} (𝑘 = 1, ⋯ , 𝑙(2)). Note 𝑙(2) ≤ 𝐾 − 1. Let 𝐽" be the +suffix set of the (𝐾 − 1 − 𝑙(2)) remaining variables, since in this step 𝑙(2) suffixes (and +so variables) are deleted from 𝐽/. +Example 4.1 If the suffixes 𝑖$ +" = 3, 𝑖" +" = 5, 𝑖/ +" = 6, 𝑖4 +" = 7, 𝑖5 +" = 9 and 𝑖6 +" = 10 satisfy the CD +condition in 𝐽/ = {2, ⋯ , 𝐾} and the others do not, then the suffix set 𝐽" ={3,5,6,7,9,10} +is carried over to the next choice set, while the suffixes in the set {2,4,8} are deleted +completely. Here 𝑙(2) = 6. +In step 2, variables with the suffixes in the complement set 𝐺(2: 𝐴/)2 do not satisfy the +condition and so those variables are deleted from the model completely so long as 𝒙/ = +𝒆 is included as a part of the model. +(Step 3) The third step concerns the 𝑖G(5) +. + suffix sets for 𝑘 = 1, ⋯ , 𝑖𝑙(𝑘) +2 + given by +(4.4) 𝐺 ^3: 𝐴.T𝑖5 +.Za = {^𝑖ℎ +I, 𝐴.T𝑖5 +.Za : 𝐶𝐷(𝑖ℎ +I, 𝐴.T𝑖5 +.Z) ≤ 𝑑, 𝑖ℎ +I ∈ 𝐽.\𝐴.T𝑖5 +.Z} + ≡ {(𝑖ℎ +3, 𝐴"(𝑖𝑘 +2))| 𝑖ℎ +3 = 𝑖1 +3, ⋯ , 𝑖𝑖(ℎ) +3 +3 , 1 < 𝑖1 +3 < ⋯ < 𝑖𝑙(3) +3 +}, + +21 + + where 𝑙(3) ≤ 𝐾 − 1 − 𝑙(2). In other words, for each choice of 𝐴"(𝑖𝑘 +2) = { 𝑖𝑘 +2, 1} (𝑘 = + 1, ⋯ , 𝑙(2)), the 𝐴Qr𝑖Z +Q: 𝐴"(𝑖% +")s is defined as the three suffixes. + (4.5) 𝐴Qr𝑖Z +Q: 𝐴"(𝑖% +")s = {𝑖Z +Q: 𝐴"(𝑖% +")} = {𝑖Z +Q, 𝑖% +", 1} +For easy exposition, let us stay in Example 4.1 and suppose 𝑖" +" = 3. Then such {𝑖Z +Q = 𝑗} +is chosen so that 𝐶𝐷(𝑗, {3,1}) ≤ 𝑑 when 𝑥Z regressed on {𝑥Q, 𝑥/} for 𝑗 ∈{5,6,7,9,10}. While, +if 𝑖" +" = 5, such {𝑖Z +Q = 𝑗} is chosen so that 𝐶𝐷(𝑗, {5,1}) ≥ 𝑑 when 𝑥Z regressed on {𝑥], 𝑥/} +for 𝑗 ∈{3,6,7,9,10}. Note that 𝐶𝐷(5, {3,1}) and 𝐶𝐷(3, {5,1}) are different. +(Step 4) For each 𝑖% +"𝜖𝐽" ={3,5,6,7,9,10}, there are the 6 suffix sets {𝑖Z +Q: 𝐴"(𝑖% +")} with 𝑖Z +Q +satisfying 𝐶𝐷(𝑖ℎ +3, 𝐴2=𝑖𝑘 +2>) ≤ 𝑑. Then for each of these 6 sets as in (5.6), we find +(4.6) 𝐺 ^4: 𝐴IT𝑖+ +IZa = x^𝑖ℎ +K: 𝐴IT𝑖+ +IZa : 𝐶𝐷 ^𝑖ℎ +K, 𝐴IT𝑖+ +IZa ≤ 𝑑, 𝑖ℎ +K ∈ 𝐽I\𝐴IT𝑖+ +IZy. +This branching process stops when no suffix satisfying the CD condition is found. And +then we will have the set of homogeneous suffix subsets {𝐴3 +∗ } each of which corresponds +to a subset 𝑿𝒅 +∗ of variables satisfying the CD condition within 𝑿𝒅 +∗ . +4.2 Collinearity-Controlled Models via PCA. +(1) First, we show that for fixed 𝑘, the CD 𝑅X% +" ≡ 𝑅X% +"(𝑿) (𝑘 ≥ 2) is expressed in terms +of standardized variables. As is well known, 𝑅X% +" is obtained as its CD by regressing 𝒙% +on ∑ +𝜃Z𝒙Z +𝑲 +Z1/,Z`% +, which is expressed as +(4.7) 𝒙% = 𝜃/ +∗𝒆 + ∑ + 𝜃𝑗±𝑁𝑠𝑥𝑗𝒛Z +𝑲 +Z1",Z`% + +(4.8) 𝒛Z = ( 𝑧$Z) with 𝑧$Z = (𝑥$Z − 𝑥̄Z)/±𝑁𝑠0Z (𝑗 ≥ 2). +where 𝒙𝟏 = 𝒆 and 𝜃/ +∗ = (𝜃/ + ∑ +𝜃Z +Z`% +𝑥̄Z). Hence, as far as the CD is concerned, we can +assume 𝑥̄Z = 0 for any 𝑗 ≥ 2 (𝑗 ≠ 𝑘) without loss of generality. Then since 𝒙Z = +𝒛Z√𝑁𝑠0Z (𝑗 ≠ 𝑘) , and 𝒙5 = 𝑥̄5𝒆 + 𝒛5•𝑁𝑠M5, the whole 𝑿 is expressed as +(4.9) 𝑿 = [𝒆, 𝑿∗] = [𝒆, 𝑶𝒌] + [𝟎, 𝒁∗]𝑑𝑖𝑎𝑔{0, 𝑫}, +where 𝑶𝒌 is 𝑁 × (𝐾 − 1) 𝑧𝑒𝑟𝑜 𝑚𝑎𝑡𝑟𝑖𝑥 except for the k-th column 𝑥̄%𝒆, 𝒁∗ = +(𝒛", ⋯ , 𝒛?): 𝑁 × (𝐾 − 1) with 𝒛% +) 𝒆 = 0 and 𝑫 = 𝑑𝑖𝑎𝑔N±𝑁𝑠0", ⋯ , ±𝑁𝑠0?T. Let +𝑅X% +"(𝒁∗) be the CD of 𝒛% when 𝒛% is regressed on 𝒁M𝒌 +∗, which is 𝒁∗ with the k-th variable +deleted. Note 𝒆′𝒁∗ = 𝟎, implying that the mean of each 𝒛Z is zero. Thus the following +lemma holds. +Lemma 4.1 𝑅X% +"(𝑿) = 𝑅X% +"(𝒁∗) (𝑘 ≥ 2). + +22 + +Proof. To express 𝑅X% +"(𝑿) in terms of 𝒁∗, let 𝑴M 𝑿𝒌 = 𝑿M𝒌(𝑿M𝒌′𝑿M𝒌).𝟏𝑿M𝒌′ with 𝑿M𝒌 = +N𝒙𝒋 ∈ 𝑿, 𝑗 ≠ 𝑘T, 𝒙𝒋 +)𝒆 = 0 as 𝑥̄Z = 0, 𝒙€𝒌 = 𝑴M 𝑿𝒌𝒙𝒌 and 𝑴𝒆 = 𝒆(𝒆)𝒆).𝟏𝒆) with 𝑴M 𝑿𝒌𝒆 = +𝒆. Then +(4.10) 𝑅X% +" = 𝑠0D% +" /𝑠0% +" = +𝒙b𝒌 +, (𝑰.𝑴𝒆)𝒙b𝒌 +𝒙𝒌 +, (𝑰.𝑴𝒆)𝒙𝒌 = +𝒙𝒌 +, (𝑴c 𝑿𝒌.𝑴𝒆)𝒙𝒌 +𝒙𝒌 +, (𝑰.𝑴𝒆)𝒙𝒌 . +Since 𝒙% = 𝑥̄%𝒆 + 𝒛%±𝑁𝑠0% and 𝒛𝒌 +) 𝒛𝒌 = 1, 𝑅X% +" = 𝒛𝒌 +) 𝑴M 𝑿𝒌𝒛𝒌, where r𝑴M 𝑿𝒌 − +𝑴𝒆s𝑥̄%𝒆 = 𝟎 is used. Here, without loss of generality, let k=K for notational convenience. +Also +write + +𝑿M𝑲 = [𝒆, 𝒙𝟐, 𝒙𝟑, ⋯ , 𝒙?./] = [𝒆, 𝑶] + ·𝟎, 𝒁M𝑲 +∗ ¸𝑑𝑖𝑎𝑔{0, 𝑫M𝑲} +, +𝒁M𝑲 +∗ = +[𝒛", ⋯ , 𝒛?./] and 𝑫M𝑲 = 𝑑𝑖𝑎𝑔N√𝑁𝑠0", ⋯ , √𝑁𝑠0?./T. Here 𝑶 is the zero matrix. Then, +(4.11) 𝒛𝑲 +) 𝑴M 𝑿𝑲𝒛𝑲 = r0, 𝒛𝑲 +) 𝒁M𝑲 +∗ 𝑫M𝑲s ¹𝑁 +𝑶′ +𝑶 +𝑫M𝑲𝒁M𝑲 +∗ ′𝒁M𝑲 +∗ 𝑫M𝑲º +.𝟏 +r0, 𝒛𝑲 +) 𝒁M𝑲 +∗ 𝑫M𝑲s +) = 𝒛𝑲 +) 𝑴M 𝒁𝑲𝒛𝑲. +This completes the proof. +An important implication of this lemma is that the collinearity of variables {𝒙𝒌} in 𝑿 +completely corresponds to that of standardized variables {𝒛%} in 𝒁∗ under 𝒙𝟏 = 𝒆 and +𝒆 ∈ 𝑿M𝒌 for any 𝑘 ≥ 2. +(2) Here, we use this lemma to find two sets of variables forming clustering or gathering +around the first and the second principal components. Let 𝒁∗)𝒁∗ = 𝑷)𝜦𝑷 with 𝜦 = +𝑑𝑖𝑎𝑔{ λ", ⋯ , λ?} be the orthogonal decomposition of 𝒁∗)𝒁∗ , where λ" > ⋯ > λ? are the +latent roots with λ" + ⋯ + λ? = 𝐾 − 1 and 𝑷 = [𝒑𝟐, ⋯ , 𝒑𝑲] is a (K-1)×(K-1) +orthogonal matrix. Then setting 𝑭 = 𝒁∗𝑷) and 𝒑Z = (𝑝"Z, ⋯ , 𝑝?Z)′ , then 𝒁∗ = 𝑭𝑷, +𝑭)𝑭 = 𝜦 and 𝑭 = [𝒇", ⋯ , 𝒇?]. Here by the definition of 𝑭, with 𝒇%′𝒇Z = 0 (𝑘 ≠ 𝑗) and +𝒇%′𝒇% = λ%. Then 𝒇Z = 𝒛"𝑝Z" + ⋯ + 𝒛?𝑝Z?, while +(4.12) 𝒛% = 𝒇"𝑝"% + 𝒇Q𝑝Q% + ⋯ + 𝒇?𝑝?% + = 𝒈"√λ"𝑝"% + 𝒈Q±λQ𝑝Q% + ⋯ + 𝒈?√λ?𝑝?%. +where 𝒈Z = 𝒇Z/√λZ is the standardized component with mean 0 and variance 1 as 𝒆′𝒈Z = +0 and 𝒈Z′𝒈Z = 1. In this formulation, 𝒇Z should be the (j-1)-th principal component, but +understanding it, we shall call 𝒇Z the j-th component for 𝑗 ≥ 2 by its original name of the +suffix. Then 𝒈Z′𝒛%/||𝒈Z||‖𝒛%‖ = ±λZ𝑝Z% is the correlation of the k-th variable 𝒛% and +the j-th (standardized) component 𝒈Z, since 𝑒)𝒛% = 0 and 𝒛%′𝒛% = 1. + To describe our decreasing algorithm, let δ8 = λZ/(𝐾 − 1) be the relative contribution +of the j-th principal component to the total variation (𝐾 − 1). The larger δ" is, the more +strongly the 2nd component 𝒈" impacts on the total variations of variables {𝒛%}. This +implies that when δ" is large, those variables {𝒛%} that have larger correlations √λ"𝑝"%′𝑠 + +23 + +with 𝒈" tend to be strongly correlated within the variable. In fact, from (4.12) the +correlation of 𝒛% and 𝒛Z is decomposed as +(4.13a) 𝒛% +) 𝒛Z = λ"𝑝"%𝑝"Z + λQ𝑝Q%𝑝QZ + ⋯ + λ?𝑝?%𝑝?Z, +(4.13b) 1= 𝒛% +) 𝒛% = λ"𝑝"% +" + λQ𝑝Q% +" + ⋯ + λ?𝑝?% +" +and so if |0λ2𝑝2𝑘| is large for 𝑖 = 𝑘, 𝑗, the correlation of 𝒛% and 𝒛Z is large in absolute +value due to the common component 𝒈" and they are less correlated with those variables +that have small correlation with 𝒈". Hence let +(4.14) 𝐺"XL = {𝒛%: 𝑁 × 1| |0λ2𝑝2𝑘| ≥ 𝑎} +be the class of the variables with |0λ2𝑝2𝑘| ≥ 𝑎, where 𝑎 is a pre-selected number in e.g., +(0.9,1). In this case the squared correlations of 𝒛% in 𝐺"XLand 𝒈" are all larger 0.8. Then +the class 𝐺"XL is a class of variables each of which is very likely to be strongly collinear +with the rest of variables in the class. Then no pair of variables in the class will be unable +to be used in a same model in order to avoid strong collinearity. + Similarly. let the class of the variables with |0λ3𝑝3𝑘| ≥ 𝑎 be denoted by + 𝐺YXL = {𝒛%: 𝑁 × 1| |0λ𝑚𝑝𝑚𝑘| ≥ 𝑎} +to which the same argument is applied. It is noted that 𝐺ZXL ∩ 𝐺YXL =Ø for 𝑗 ≠ +𝑚 because of (4.13b) . By Lemma 4.1, put 𝐺BX0 = {𝒙%: 𝑁 × 1|𝒛% ∈ 𝐺BXL} and let +#(𝐺BX0) = ℎBX denote the number of variables in the class 𝐺BX0 that satisfies ℎBX ≥ 2. +Then we assume that there are such M classes and by renumbering the suffixes, without +loss of generality, let them denote 𝑖 = 2, ⋯ , 𝑀 and ℎ"X + ⋯ + ℎ 𝑑, then delete 𝒙? and replace 𝑿(0) by 𝑿M? +(j) = 𝑿M? . +(Step 2) Denote the set of remaining variables by 𝑿(1)=𝑿M? +(j)={𝑥/, 𝒙", ⋯ , 𝒙?./} via + +24 + +renumbering and let Ξj +(/) = {𝐶𝐷% +(/): 𝑘 = 2, ⋯ , 𝐾 − 1}. Suppose that 𝐶𝐷?./ +(/) of 𝒙?./ +(/) ≠ +𝒙/ = 𝒆 is the largest CD in Ξj +(/)and 𝐶𝐷𝐹?./ +(/) = maxB 𝐶𝐷B +(/) > 𝑑. Then 𝑿(2) = 𝑿M%./ +(/) = +{𝒙𝟏, 𝒙𝟐, ⋯ , 𝒙𝑲.𝟐} is the set of K-2 variables after 𝒙?./ +(/) is deleted from 𝑿(1)=𝑿M? +(j). +(Step 3) Repeating this variable-reducing process, we stop it at the p-th step when there +is no CD such that 𝐶𝐹B +([) > 𝑑 . Then 𝑿(𝑝)=𝑿_9($-+ +(+($) ={𝒙/, 𝒙", ⋯ , 𝒙?.[} by renumbering is +a final set for which the OLS analysis is applied without any serious effect of collinearity. + By Proposition 2.1, any sub-model from this set satisfies 𝑚𝑎𝑥%𝐶𝐷% ≤ 𝑑. +5 Conclusion + By the concept of EEM-M, we developed a new integrated process of bundle model – +XMSP - 𝑦#MSP to obtain an empirical effective model for given data (𝒚𝒐, 𝑿). The XMSP +we proposed selects a class of models with inefficiency-controlled and collinearity- +controlled OLS estimates in each model without using 𝒚𝒐. The property of the estimates +corresponds to inefficiency measure and collinearity measure associated with individual +predictive sampling variances. Using these two measures of individual estimates in each +estimated model, we defined the IC Risk Index, by which models are made to be +compared in the XMSP. Finally, to materialize our conceptual and analytical results, we +proposed two algorithms to control collinearity. + +6 Acknowledgement +Kariya and Hayashi’ s portion of this work was supported by JSPS KAKENHI Grant Number +21K01431. Kurata’s portion is partially supported by JSPS KAKENHI Grant +Number 19K11853. This paper was presented in the 2022 Meeting of the Japan +Statistical Society and some useful questions received from Prof. N. Hoshino are +reflected here. + +References +Aitken, L.S. and West, S.G. (1996), Multiple Regression: Testing and Interpreting Interaction, +Sage Publication. +Draper, N. R. and Smith, H. (1998), Applied Regression Analysis, Third Edition, John Wiley. +Fox, J. and Monette, G. (1992), “Generalized Collinearity Diagnostics”, Journal of the Statistical +Association, 87:417, 178-183. +Hoerl, A. and Kennard, R.W. (1988), “Ridge Regression”, in Encyclopedia of Statistical Sciences, +vol. 8, pp. 129–136, John Wiley & Sons New York. +Jolliffe, I.T. (1986), Principal Component Analysis, Springer-Verlag, New York. +Kariya, T. and Kurata, H. (2003), Generalized Least Squares, John Wiley, London. + +25 + +Kubokawa, T. and Srivastava, M.S. (2004), “Improved Empirical Bayes Ridge Regression +Estimators under Multicollinearity”, Communications in Statistics Theory and Methods, 33, +No. 8, 1943–1973. +Lavery, R.M., Acharya, P., Sivo, A.S. and Xu, L. (2019), “Number of predictors and +multicollinearity: What are their effects on error and bias in regression?”, Communications in +statistics—Simulation and Computation, 48, No1, 27-38. +Lehmann, E.L. and Romano, J.R. (2005), Testing Statistical Hypotheses, Springer, NY. +O’Brien, R.M. (2007), “A Caution Regarding Rules of Thumb for Variance Inflation Factor”, +Quality and Quantity, 41, 673–690. +Saleh, A. K. Md. Ehsanes (2006), Theory of Preliminary Test and Stein-Type Estimation With +Applications, John Wiley & Sons. +Stewart, G.W. (1987), “Collinearity and Least Squares Regression”, Statistical Science, 2, No.1, +68-100. +Tsao, M. (2019), “Estimable Group Effects for Strongly Correlated Variables in Linear Models”, +Journal of Statistical Planning and Inference, 198, 29-42. +Zou, H. and Hastie, T. (2005), “Regularization and Variable Selection via the Elastic Net”, Journal +of Royal Statistical Society Series B, Statistical Methodology, 67, No. 2, 301–320. (Addendum, +no. 5, 768). + diff --git a/ndE1T4oBgHgl3EQfOgNK/content/tmp_files/load_file.txt b/ndE1T4oBgHgl3EQfOgNK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d8d8f08e33f83258c5a00f1c8d9a7928c057cd5c --- /dev/null +++ b/ndE1T4oBgHgl3EQfOgNK/content/tmp_files/load_file.txt @@ -0,0 +1,907 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf,len=906 +page_content='1 A Modelling Framework for Regression with Collinearity Takeaki KARIYA１, Nagoya University of Commerce and Business, Graduate School of Management, Nisshin, Aichi, Japan 470-0193 The corresponding author: thekariya70@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='com Hiroshi KURATA2 University of Tokyo, Graduate School of Arts and Sciences, Meguroku, Tokyo, Japan 153-8902 kurata@waka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='jp Takaki HAYASHI3 Keio University, Graduate School of Business Administration, Yokohama, Kanagawa, Japan 223-8526 hayashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='kbs@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='com Abstract This study addresses a fundamental, yet overlooked, gap between standard theory and empirical modelling practices in the OLS regression model 𝒚 = 𝑿𝜷 + 𝒖 with collinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In fact, while an estimated model in practice is desired to have stability and efficiency in its “individual OLS estimates”, 𝒚 itself has no capacity to identify and control the collinearity in 𝑿 and hence no theory including model selection process (MSP) would fill this gap unless 𝑿 is controlled in view of sampling theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In this paper, first introducing a new concept of “empirically effective modelling“ (EEM), we propose our EEM methodology (EEM-M) as an integrated process of two MSPs with data (𝒚𝒐, 𝑿) given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The first MSP uses 𝑿 only, called the XMSP, and pre-selects a class 𝐷 of models with individually inefficiency-controlled and collinearity-controlled OLS estimates, where the corresponding two controlling variables are chosen from predictive standard error of each estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Next, defining an inefficiency-collinearity risk index for each model, a partial ordering is introduced onto the set of models to compare without using 𝒚𝒐, where the better-ness and admissibility of models are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The second MSP is a commonly used MSP that uses (𝒚𝒐, 𝑿), and evaluates total model performance as a whole by such AIC, BIC, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' to select an optimal model from 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Third, to materialize the XMSP, two algorithms are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Key words: OLS, model selection process, collinearity effect, empirically modelling, t- test AMS Classification: 62J05, 62J20 2 1 Introduction Within the traditional ordinary least squares (OLS) framework in the linear regression model with collinearity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1) 𝒚 = 𝑿𝜷 + 𝒖 with 𝐸(𝒖) = 𝟎 and 𝑉𝑎𝑟(𝒖) = 𝜎"𝑰, this paper first proposes a new concept of “Empirically Effective Modelling” (EEM) together with the EEM methodology (EEM-M) for finding an empirically effective model from an observation (𝒚𝒐, 𝑿) for model in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The EEM-M is based on the EEM concept and forms a new integrated model selection process (MSP) consisting of bundle model concept, two different MSPs, new model comparison methodology and algorithms for selecting inefficiency-controlled and collinearity-controlled models, where the two MSPs are the XMSP with using X only and 𝑦#MSP with using (𝒚𝒐, 𝑿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' As notation, let 𝑿 = (𝑥$%) = (𝒙𝟏, 𝒙𝟐, ⋯ , 𝒙𝑲) with 𝒙𝟏 = (1, ⋯ ,1)) ≡ 𝒆 be an 𝑁 × 𝐾(𝑁 > 𝐾) explanatory matrix of rank K, and 𝑿 is assumed to contain all possible variables for MSP and to be used as they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' We call each vector 𝒙𝒌 a “variable”, and X a “model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' And X is interchangeably used as the set consisting of K vectors {𝒙𝒌} as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Similarly, a sub-set 𝑿∗ of 𝑿 is called a sub-model or simply a model formed as a matrix and it is assumed that 𝑿∗ always includes a constant term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', 𝒙𝟏 = 𝒆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Our concept of collinearity is confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Since models always include 𝒙𝟏 = 𝒆, so that the coefficient of determination (CD) in regression can be used as a measure of collinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Let 𝑿M𝒌 = N𝒙𝒋 ∈ 𝑋, 𝑗 ≠ 𝑘T in 𝑅- be the set X with the k-th variable deleted, where 𝒙𝟏 = 𝒆 ∈ 𝑿M𝒌 and 𝑘 = 2, ⋯ , 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' We also use 𝑿M𝒌 as the 𝑁 × (𝐾 − 1) matrix formed by its columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Let 𝑅X% " is the CD when 𝒙𝒌 is regressed on 𝑿M𝒌 in 𝑅-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In our terminology, each 𝒙𝒌 (𝑘 ≥ 2) is said to be collinear with 𝑿M𝒌 unless 𝑅X% " = 0 or 𝑅X% " = 1, and it is said to be perfectly collinear with 𝑿M𝒌 if 𝑅X% " = 1, and zero-collinear with 𝑿M𝒌 if 𝑅X% " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' When 𝑅X% " is close to 1, 𝒙𝒌 is said to be strongly collinear with 𝑿M𝒌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' By assuming rank(X)=K, the case of 𝑅X% " = 1 is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In this definition, collinear relations always exist among the variables in each model unless 𝑅X% " = 0 for any 𝑘 ≥ 2, and model performance comes with the collinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' What to do in modelling is not to avoid it but to exclude each variable 𝒙𝒌 that has “strong” collinearity with 𝑿M𝒌 in each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In the EEM-M, the XMSP will screen such variables out and select models with 𝑅X% " ≤ 𝑑 for any 𝑘 ≥ 2, where 𝑑 is a control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Concerning collinearity, as its vast literature shows, 𝒚𝒐 itself does not have a capacity to identify the collinearity structure of 𝑿 via a 𝑦#MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In fact, there will be no literature on such a 𝑦#MSP that mitigates the ill-effect within the OLS framework even if 𝒚𝒐 is really generated from it, though a lot of hybrid remedies such as shrinkage-type estimation have been provided outside of the OLS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Besides, under strong 3 collinearity, 𝒚𝒐 loses its own capacity to evaluate model performances by a 𝑦#MSP, because OLSEs and maximum likelihood estimates suffer from the ill-effect, which invalidates such a 𝑦# MSP as the ones with Akaike’s information criterion (AIC), Bayesian information criterion (BIC), adjusted coefficient of determination (CD), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In our terminology, MSP is defined as a procedure to select a model 𝑿∗ from X that is estimated by the OLS method as 𝜷]∗ = 𝜷]∗(𝑿∗, 𝒚), where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2) 𝜷] ≡ 𝜷](𝑿, 𝒚) = (𝛽_%) = (𝑿)𝑿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='/𝑿)𝒚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' By the OLSE we mean either OLS estimator 𝜷]∗ or OLS estimate 𝜷]∗𝒐 = 𝜷](𝑿∗, 𝒚𝒐) for given (𝒚𝒐, 𝑿∗) along its context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here it is noted that collinearity does not affect the optimality of 𝜷] in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2) as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In fact, if the initial model in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1) is supposedly “true” with rank(X) =K, 𝜷] is the best linear unbiased estimator in the 𝐾 × 𝐾 nonnegative definite ordering, in which no shrinkage-type estimator will beat the OLS estimator in risk matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This optimality holds no matter how strong the collinearity in X may be, implying no collinearity effect on this basic optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' However, strong collinearity seriously affects some individual OLSEs so that a selected model will be sensitive to small changes of predictive variables and lose stability or trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Our results do not much depend on other authors’ work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence, in the sequel, we summarize our work in some details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 Empirically Effective Modelling Methodology (EEM-M) From a practical viewpoint, we first define a new concept of “Empirically Effective Modelling (EEM)” in the OLS framework, where it leads to an empirically effective model from an observation (𝒚𝒐, 𝑿) in its own sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here we let 𝑿 also represents all the cases of sub-model 𝑿∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 A model estimated by the OLS with data (𝒚𝒐, 𝑿) is defined to be empirically effective if it is judged to hold the two properties [1] and [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' [1] Efficiency and stability of each “individual” OLS estimate in the estimated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' [2] Goodness or optimality in total model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' To make this EEM concept implemented, we develop the EEM-M with two MSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The first MSP is the XMSP applied to data 𝑿 without using 𝒚𝒐 for [1] in order to pre- select a class 𝐷 of inefficiency-controlled and collinearity-controlled models, or shortly “IC-controlled models”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In this paper, the property [1] is measured by the pair (𝐼%, 𝐶%) for the k-th OLS estimate 𝛽_%, which is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) 𝐼% ≡ 𝑧̅% " = 𝑥̅% "/𝑠% " ≥ 0 and 𝐶% is 𝑅X% " or equivalently 𝑉𝐼𝐹% = 1/(1 − 𝑅X% ") where mean 𝑥̅% of 𝒙𝒌 = {𝑥%$} and 𝑠0% " = ∑ (𝑥$% − 𝑥̄%)" $1/ /𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' These are taken from 4 the individual predictive sampling variance (IndPSV) of the k-th OLS “estimator” 𝛽_% (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5) in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' For the OLS “estimate” 𝛽_% given, 𝐼% will be called the inefficient factor (I- factor) of 𝛽_%, while 𝐶% ≥ 0 is called the collinearity factor (C-factor) of 𝛽_% (𝑘 ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' It is noted that values of (𝐼%, 𝐶%) are determined only by 𝑿 as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' By this fact, the XMSP aims to control the two objective variables (𝐼%(𝑝%𝑿), 𝑅X% "(𝑿)) with respect to 𝑿 to get a class 𝐷 of “ IC-controlled models” with control level (c, d) before 𝒚𝒐 is observed, where 𝑝%𝑿 = 𝒙𝒌 and (c,d) is prespecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The VIF in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) is the well-known variance inflation factor (VIF) (Jolliffe, 1986 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In general, in empirical analysis, it will be desirable to compute the values of I-factor and C-factor of each estimate 𝛽_% as such performance statistics of each estimate as standard errors (SD) and t-value, because these factors provide the information on how much of the property [1] is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In this paper, the XMSP provides two algorithms to pre-select a class 𝐷23 of IC-controlled models with 𝐼% ≤ 𝑐 and 𝑅X% " ≤ 𝑑 for any 𝑘 ≥ 2 in it’s OLS estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Such a class 𝐷23 (or simply 𝐷) will be called an IC-controlled class or “ 𝒚𝒐-accommodating” class for 𝑦#MSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The second MSP is a 𝑦#MSP applied to data (𝒚𝒐, 𝑿) for [2] in order to select an optimal model from the class 𝐷23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Since there are many useful 𝑦#MSPs available, any reasonable 𝑦#MSP in the EEM-M can be used for finding an optimal model by its own criterion so long as it independently evaluates the total performance of each IC-controlled model in 𝐷23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In this sense, we will take it for granted that an optimal model can be obtained by such a 𝑦#MSP, and so the optimality of an IC-controlled model depends on the choice of 𝑦#MSP, implying that the optimal model is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Examples of 𝑦#MSP are those with AIC, BIC, adjusted CD, or hybrids, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' However, we will not include the 𝑦#MSPs that have a process of pre-testing, model selection and estimation in selecting models within one sample "𝒚𝒐" (Saleh, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In view of sampling theory, these MSPs intrinsically entail a nonlinear and conditional structure in a selected model and so it is out of the OLS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Besides, those MSPs mostly aim to select significant variables via testing and do not seek an optimal model as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Thus, the XMSP fills the overlooked gap between the problem that an estimated model needs to have the property [1] for individual OLS estimates and the problem that the traditional 𝑦#MSP cannot identify and control the ill-effect of collinearity on individual OLS estimates, because 𝒚𝒐 itself have no capacity to do it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In fact, the XMSP redeems the weak point in selecting a class 𝐷 of models with property [1] before 𝒚𝒐 is observed, and then a 𝑦#MSP comes into own capacity in 𝐷 to select an optimal model with property [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Therefore, the EEM-M will give us an empirically effective model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Before some more details are described below, we need to add some more components 5 to the EEM-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The whole picture of the EEM-M is symbolically described as an integrated process of 𝐵ℳ([𝑿]) −XMSP —𝑦# MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here 𝐵ℳ([𝑿]) denotes a bundle model replacing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1) and it makes the two MSP concepts consistent with the EEM (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In addition, the XMSP includes a framework of comparing models in terms of (𝐼%, 𝑅X% ").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In fact, the maximums 𝑚𝑎𝑥%4"𝐼% and 𝑚𝑎𝑥%4"𝐶% in a given model 𝑿∗ are respectively defined to be the inefficiency risk (I-risk) and the collinearity risk (C-risk) of the model 𝑿∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The pair of the maximums will be called inefficiency-collinearity risk index (ICRI) (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The ICRI gives a decision theoretic framework for comparing models because it introduces a partial ordering onto the set of sub-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In particular, in comparing models, they provide the concept of better-ness and admissibility of a model in terms of the ICRI (See 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' As a part of the EEM-M, the two algorithms for the XMSP are given for selecting a 𝐷 of IC-controlled models (See 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2 Bundle Model 𝐵ℳ([𝑿]) As a part of the EEM-M, we replace the traditional model concept expressed in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1) by our bundle model concept for the EEM-M with the XMSP and 𝑦#MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This is because the EEM-M starts with observed data (𝒚𝒐, 𝑿) and pursues the effectiveness of empirical model in the sense of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 with some evaluation criteria derived in the OLS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' As an example, let us consider the problem of selecting one of the models: (1) 𝒚 = 𝑿∗𝜷∗ + 𝑿∗∗𝜷∗∗ + 𝒖 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1) with 𝑿 = (𝑿∗, 𝑿∗∗) (2) 𝒚 = 𝑿∗𝜷∗ + 𝒖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Such a problem is often treated as a binary decision problem where the F-testing scheme is used within the so-called frequentist’s framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This approach leads us to the MSP with a procedure of preliminary test, model selection, and estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' As has been stated, we exclude this MSP because it includes an internal inconsistency so long as the models in the procedure are estimated by the OLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' While, in the EEM-M, 𝒚 is realized as 𝒚𝒐, implying that it has to be generated by either (1) or (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence the error terms in (1) and (2) should be different because the two models cannot generate 𝒚𝒐 under the same 𝒖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Taking this view into the EEM-M framework, we use the bundle model defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1a) 𝐵ℳ([𝑿]) = { 𝒚𝝉 = 𝑿𝝉𝜷𝝉 + 𝒖𝝉 | 𝑿𝝉 ∈ [𝑿], 𝒖𝝉 ∈ [𝒖]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here [𝑿] is the set of all the sub-models {𝑿𝝉: 𝜏 ∈ Λ} with 𝒙𝝉/ = 𝒆 and Λ = {1, 2, ⋯ , 2"#$ − 1}, where 𝜏 is a parameter that distinguishes models, and [𝒖] is the set of error terms corresponding to the set [𝑿] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1b) [𝒖] = {𝒖𝝉: 𝜏 ∈ Λ, 𝐸(𝒖𝝉) = 𝟎, 𝑉𝑎𝑟(𝒖𝝉) = 𝜎"𝑰}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then 𝒚𝒐 is regarded as realized from one of the sub-models in 𝐵ℳ([𝑿]), not from a “true model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' It is noted that no specific stochastic structure is specified among 𝒖𝝉′𝑠 except for 6 its own two moments as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This is because any 𝑦#MSP in the EEM-M is assumed to be able to evaluate the total performance of each individual model in 𝐵ℳ([𝑿]) with its own criterion when (𝒚𝒐, 𝑿) is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In the EEM-M, the XMSP will replace [𝑿] in 𝐵ℳ([𝑿]) by a class 𝐷 of IC-controlled models, which makes 𝑦#MSP better accommodated or equivalently more effectively functioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This can be done in advance before 𝒚𝒐 is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The above argument shows that in our analytical framework, there is no concept of “true” model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' After all, a finally selected empirically effective model via a 𝑦#MSP will have to be regarded as the model having generated 𝒚𝒐 in EEM-M or even in any empirical analysis, so long as 𝒚𝒐 is regressed on the final model and it is used in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3 Individual Predictive Sampling Variance as for [1] To describe the two factors in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) for [1] in a sub-model 𝒚∗ = 𝑿∗𝜷∗ + 𝒖∗, let us first consider the individual sampling variance (IndSV) 𝑉𝑎𝑟(𝛽_% ∗) of each OLSE 𝛽_% ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then, as is well known, it is decomposed as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='4) 𝑉𝑎𝑟r𝛽_% ∗s = 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 778" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' = 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 9#" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' × 𝑉𝐼𝐹% with 𝐸𝐸𝐹% " = 𝑁𝑠𝑥𝑘 2 (1 − 𝑅M𝑘 2), where 𝑘 = 2, ⋯ , 𝐾∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here 𝐸𝐸𝐹% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='" is the (k,k) element of (𝑿∗)𝑿∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='/ and 𝐸𝐸𝐹% is called empirically effective factor (EEF) that controls 𝑉𝑎𝑟r𝛽_% ∗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' It is desirable for 𝐸𝐸𝐹% to be larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Also, 𝑉𝑎𝑟r𝛽_% ∗s in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='4) is linearly affected by 𝑉𝐼𝐹% in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3), which is used as one of the equivalent alternatives for the C-factor in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Clearly each 𝐸𝐸𝐹% " consists of the two components: 1) the variate-own effect 𝑁𝑠0% " of predictive variable 𝒙𝒌 ∗ and 2) the collinearity effect 𝑉𝐼𝐹%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In this case, for given 𝑉𝐼𝐹%, the larger the 𝑁𝑠0% " is, the smaller 𝑉𝑎𝑟r𝛽_% ∗s is, and the more efficient the k-th OLSE is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This is inappropriate in comparing the standard errors (SEs) √𝑉𝑎𝑟 u r𝛽_% ∗s′𝑠 of individual OLSEs in an estimated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Besides, the physical units of 𝒙% ∗ ’s in measurement are different in general and so IndSVs in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) are not comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' To overcome the incomparability of IndSVs, we pay attention to individual terms 𝑥$%𝛽_% ∗)𝑠 in each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Since these terms have a common physical unit with 𝑦$, we adopt the predictive sampling variance (IndPSV) of each 𝛽_% ∗ that is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5) 𝐼𝑛𝑑𝑃𝑆𝑉% ≡ ∑ 𝑉𝑎𝑟r𝑥$%𝛽_% ∗s $1/ = 𝜎"(1 + 𝑧̅% ")𝑉𝐼𝐹% ≡ 𝜎"𝐻(𝑧̅% ", 𝑉𝐼𝐹%), where 𝑧$% = 𝑥$% 𝑠0% ⁄ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then it is clear that the two variables 𝑧̅% " and 𝑉𝐼𝐹% in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5) are measurement-unit-free since they are scale-invariant (see 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Note that 𝜎 carries the same physical unit with 𝑦$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' And the 𝐼𝑛𝑑𝑃𝑆𝑉% of the k-th term {𝑥$%𝛽_% ∗: 𝑛 = 1, ⋯ , 𝑁} consists of the two effects: 7 3) the variate-own inefficiency effect ||𝒙𝒌 ∗||"/ 𝑁𝑠0% " = 1 + 𝑧̅% " and 4) the collinearity effect 𝑉𝐼𝐹%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In the expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5), it is desirable for both 𝑧̅% " ≥ 0 and 𝑉𝐼𝐹% ≥ 1 to be smaller so that 𝐼𝑛𝑑𝑃𝑆𝑉% becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5), for the k-th OLS estimate in model 𝑿∗, 𝐼% ≡ 𝑧̅% " is the I-factor to be controlled as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3), while as one of the two equivalents in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3), the 𝑉𝐼𝐹% is called the C-factor 𝐶% to be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In fact, 𝑉𝐼𝐹% ≤ 𝑑 is equivalent to 𝑅X% " ≤ 𝑑: = 1 − 1/𝑑, and so 𝑉𝐼𝐹% and 𝑅X% " are interchangeably used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In Section 2, for reference, the standard error (SE) of each estimate based on IndSV is compared to the SE based on Ind PSV after model is estimated with 𝒚𝒐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Also, we give a necessary and sufficient condition for a model 𝑿∗ to attain the lower bound for the IndPSVs of all the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='4 𝒚𝒐-Accommodating Class, IC Risk Index and Admissibility of Model In Section 3, we will formulate a decision theoretic framework for comparing models by the ICRI of 𝑿∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Since the degree of collinearity of model 𝑿∗ is not greater than that of 𝑿∗∗ if 𝑿∗ ⊂ 𝑿∗∗, it is necessary to take the column size into account in comparing models by 𝑉𝐼𝐹%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence, letting 𝐽(𝑿∗) be the column size of 𝑿∗, the distinguishability of models {𝑿∗} with 𝐽(𝑿∗) = 𝐽 is made only through the set of I-factors and C-factors of each OLS estimate in 𝑿∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='6) {(𝑧̅% "(𝑝%𝑿∗), 𝑉𝐼𝐹%(𝑿∗))|𝑘 = 2, ⋯ , 𝐽}, where 𝑝%𝑿∗ = 𝒙𝒌 ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' And, let 𝐷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 23 denote the class of IC-controlled models of column size J with control level (c, d), where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='7a) 𝐷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 23 = {𝑿∗ ∈ [𝑿: 𝐽]| 𝑧̅% "(𝑝%𝑿∗) ≤ 𝑐, 𝑉𝐼𝐹%(𝑿∗) ≤ 𝑑, (𝑘 = 2, ⋯ , 𝐽)}, where [𝑿: 𝐽] denotes the set of models whose column sizes are J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Also let (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='7b) 𝐷23 =∪𝐽=2 𝐾 𝐷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 23 be the set of all the IC-controlled models with control level (c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' To materialize the XMSP to find 𝐷23, two algorithms are proposed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In this paper, the problem of how to choose (c, d) is left open (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Also in Section 3, a partial ordering on models with column size J is introduced based on the ICRI of model 𝑿𝝉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ∗ , which is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='8a) 𝑟r𝑿𝝉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ∗ s = r𝑐<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', 𝑑<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='s for each 𝑿𝝉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ∗ , and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='8b) 𝑐<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' = 𝑚𝑎𝑥%𝑧̅% "(𝑝%𝑿𝝉𝑱 ∗ ) and 𝑑<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' = 𝑚𝑎𝑥%𝑉𝐼𝐹%(𝑿𝝉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ∗ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The model 𝑿𝝉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ∗ with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='8b) is also denoted as 𝑿𝝉∗r𝑐<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', 𝑑<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then for each J fixed, 𝑿𝝉∗r𝑐<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', 𝑑<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='s is said to better accommodate 𝒚𝒐 than 𝑿𝝉) ∗ r𝑐<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ) , 𝑑<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ) s if 𝑐<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ≤ 𝑐<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ) and 𝑑<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ≤ 𝑑<=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ) hold with one of the inequalities strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Importantly, this can be 8 determined before 𝒚𝒐 is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' From this set-up, the concept of admissibility naturally follows (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 3), though the characterization of admissible 𝒚𝒐-accommodating class of models is made only in the case of J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5 Two Algorithms for materializing the XMSP In Section 4, to make the XMSP practically feasible, we develop two computational algorithms: variable-increasing and variable-reducing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In the latter case, principal component analysis is used, and it is shown that 𝑅X% "(X) = 𝑅X% "(Z) for the matrix Z of standardized variates of 𝒙+ (𝑘 ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In particular, we focus on the algorithm to make models satisfy the condition 𝑅"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' " ≤ 𝑑𝑅 ≡ 1 − 1/𝑑 for any k, since it is easy to find models that satisfy 𝑧̅% " ≤ 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then a set {𝑿∗} of IC-controlled models is obtained by combining them as the intersection of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='6 A brief Literature Review on Collinearity in Model Selection Research history on collinearity in regression is very long and has been still accumulating a vast amount of literature, though no clear-cut solution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In our OLS context, a recent example is Tsao (2019) proposing estimating a linear combination of regression parameters in a strongly correlated model X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' While, because they are not in the OLS framework, we do not treat such hybrid methods and procedures of the GLS-type (Kariya and Kurata, 2003) , LASSO-type (Zou and Hastie, 2005), ridge-type (Hoerl and Kennard, 1988), Stein-type (Kubokawa and Srivastava, 2004), principal component-type (Jolliffe, 1986), and standardization-type (Aitken and West, 1996) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' These sophisticated methods basically aim to secure the stability of estimates in the context of collinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Stewart (1987) mathematically clarified the algebraic structure of collinearity in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Fox and Monette (1991) generalized the concept of VIF to the case where 𝑿 is divided into three categories of variables, including dummy variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Lavery, Acharya, Sivo and Xu (2019) studied on the relation between the number of variables and collinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Some 𝑦#MSPs are associated with a pre-testing, variable selection and estimation procedure and often used as those of stepwise forward, backward or hybrid methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In their textbook, Draper and Smith (1998) well describe them with various empirical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' But these MSPs do not lead to an optimal model as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In addition, the ill-effect of collinearity may let us select a wrong model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 2 Sampling Variance and Predictive Sampling Variance of OLSE In this section, first, using the individual predictive sampling variance (IndPSV) of each individual OLS estimate in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5), we study some important relations between I-C 9 factors (𝐼%, 𝐶%) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) and the structure of model 𝑿∗that carries ({(𝐼%, 𝐶%): 𝑘 = 2, ⋯ , 𝐾∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2, we compare the traditional standard errors (SDs) of individual OLS estimates based on the IndSV and the corresponding predictive standard errors (PSDs) based on the IndPSV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In the sequel, we let X represent all the cases of 𝑿∗ ∈ [𝑿].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The effectiveness of the XMSP relies on the fact that the predictive sampling variance (IndPSV) of each term {𝑥$%𝛽_%} = {𝑥$%𝛽_%: 𝑛 = 1, ⋯ , 𝑁} with 𝛽_% being the OLS “estimator” is expressed as 𝜎"𝐻(𝑧̅% ", 𝑉𝐼𝐹%) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5), which is a function of I-factor 𝑧̅% " and C-factor 𝑉𝐼𝐹%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Since the 𝐻 function depends only on 𝑿, an “estimated” model is made IC-controlled by controlling the I-factor and the C-factor separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence, the IndPSVs as a sampling property of {𝑥$%𝛽_%} is connected with the effectiveness of the OLS estimates in the estimated model before 𝒚𝒐 is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 Relations between (𝐼%, 𝐶%) and the Structure of 𝑿 More specifically, let us consider an estimated predictive model: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1) 𝑦•$ = 𝛽_/ + 𝑥$"𝛽_" + ⋯ + 𝑥$?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='𝛽_?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', which is the n-th element 𝑦•$ of 𝒚€ = ∑ 𝒙𝒌𝛽_% ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' %1/ = 𝑿𝜷].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This is also viewed as pre- sampled version as 𝑦•$ ≡ 𝑦•$(𝒚) for given X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then, for {𝑥$%} given, the sampling distribution of 𝑦•$(𝒚) is dependent on the whole covariance matrix 𝑿𝑉𝑎𝑟r𝜷]s𝑿′, but in view of the EEM, what matters in the post-sampled predictive model with 𝒚𝒐 given is the efficiency and stability of individual terms {𝑥$%𝛽_%}′𝑠 by Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' By the above observation, it suffices to control the inefficiency factor (I-factor) and the collinearity factor (C-factor) of each term, since it controls 𝑉𝑎𝑟(𝑥$%𝛽_%) for each 𝑥$% whether or not 𝒚𝒐 is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' If the IndPSV is small, the contribution of 𝑥$%𝛽_% to 𝑦•$ will be stable and 𝑥$%𝛽_% is likely to be realized in a neighborhood of its mean 𝑥$%𝛽% for fixed 𝑥$%, though the variation is that of 𝛽_%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Since 𝑥$% varies over {𝑥$%}, summing 𝑉𝑎𝑟(𝑥$%𝛽_%) up over 𝑛 = 1, ⋯ , 𝑁 yields the IndPSV as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2) 𝐼𝑛𝑑𝑃𝑆𝑉% ≡ ∑ 𝑉𝑎𝑟(𝑥$%𝛽_%) $1/ = 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ∑ 𝑥𝑛𝑘2 𝑁 𝑛=1 778" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ≡ 𝜎"𝐻%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here letting 𝑧$% = 𝑥$% 𝑠0% ⁄ and using ||𝒙𝒌||" = ∑ 𝑥$% " $1/ , ||𝒙𝒌||"/𝑁𝑠0% " = / ∑ 0$" 9#"‚ " $1/ = 1 + 𝑧̅% ", and hence each IndPSV is determined by the measurement-unit-free variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) 𝐻% = 𝐻(𝑧̅% ", 𝑉𝐼𝐹%) ≡ (1 + 𝑧̅% ") × 𝑉𝐼𝐹% (≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This 𝐻% is clearly free from the physical unit of 𝑥$% so that its size is comparable with those of the others, and the physical unit of y is carried over to its population standard deviation 𝜎 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3), the IndPSV is separated into the inefficiency factor 𝑧̅% " and 10 the collinearity factor 𝑉𝐼𝐹%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3), it is easy to observe the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' (a) 𝑧̅% "=0 if and only if 𝑥̄% = 0, and 𝑉𝐼𝐹% =1 if and only if 𝑅X% " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' (b) 𝐻% = 1 if and if 𝑥̄% = 0 and 𝑅X% " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' When 𝑥̄% = 0, then 𝐻% = 𝑉𝐼𝐹%, and so controlling 𝑉𝐼𝐹% is equivalent to controlling the IndPSV in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' While, when 𝑉𝐼𝐹%=1, then 𝐻% = 1 + 𝑧̅% " so that it is desirable for 𝑧̅% " to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' When 𝐻%(≥ 1) is small, the term {𝑥$%𝛽_%} in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1) is stabilized in the sense of 𝐼𝑛𝑑𝑃𝑆𝑉% and the collinearity factor is not large whether or not y is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The next proposition gives the condition for the attainment of the lower bound of 𝐻%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3), for given 𝑘(≥ 2), 𝐻% = 1 if and only if 𝑥̄% = 0 and 𝒙𝒊′𝒙𝒌 = 0 for any 𝑖 ≠ 𝑘 (𝑖 ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence, 𝐻% = 1 for any 𝑘(≥ 2) if and only if (1) 𝑥̄% = 0 for any 𝑘(≥ 2) and (2) 𝑿-𝑿 = 𝑑𝑖𝑎𝑔{𝑎$, 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', ⋯ , 𝑎"}, where 𝑎/ = 𝑁 and 𝑎B = 𝒙𝒊′𝒙𝒊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' If model satisfies (2) only, 𝐻% = 1 + 𝑧̅% " only denotes the effect of inefficiency value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here 𝑑𝑖𝑎𝑔{𝑎$, ⋯ , 𝑎"} denotes diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' By the definition of 𝑅X% ", letting 𝑴M 𝒌 = 𝑿M𝒌(𝑿M𝒌′𝑿M𝒌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='𝟏𝑿M𝒌′ with 𝑿M𝒌 = N𝒙𝒋 ∈ 𝑿, 𝑗 ≠ 𝑘T, 𝒙€𝒌 = 𝑴M 𝒌𝒙𝒌 and 𝑴𝒆 = 𝒆(𝒆)𝒆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='𝟏𝒆) with 𝑴M 𝒌𝒆 = 𝒆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then 𝑅X% " = 𝑠0D% " /𝑠0% " = 𝒙€𝒌 ) (𝑰 − 𝑴𝒆)𝒙€𝒌/𝒙𝒌 ) (𝑰 − 𝑴𝒆)𝒙𝒌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence, 𝑅X% " = 0 if and only if 𝒙𝒌 ) (𝑴M 𝒌 − 𝑴𝒆)𝒙𝒌=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Therefore, for any k, 𝐻% = 1 if and only if 𝑥̄% = 0 and 𝒙𝒌 ) 𝑴M 𝒌𝒙𝒌 = 𝒙𝒌′𝑴𝒆𝒙𝒌 =0 since 𝑁𝑥̄% = 𝒆′𝒙𝒌, which in turn holds if and only if 𝑥̄% = 0 and 𝑿M𝒌′𝒙𝒌 = 𝟎, implying 𝒙𝒌′𝒙𝒋 = 0 (𝑘 ≠ 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Thus the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 An example of 𝑿 satisfying the conditions (1) and (2) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 is 𝑿 = {𝒙𝟏, 𝒙𝟐, ⋯ , 𝒙𝑲} with 𝒙𝒊 = 𝛼B𝜹𝒊 (𝛼B > 0), where 𝑬 ≡ (𝜹𝟏, 𝜹𝟐, ⋯ , 𝜹𝑲): 𝑁 × 𝐾 is the matrix consisting of the first K columns of the Helmert’s orthogonal matrix with 𝜹𝟏 = 𝒆/√𝑁, 𝒙𝟐 = ( & √(∙& , & √(∙&,0, ⋯ ⋯ ,0)′ , 𝜹𝒌 = ( $ %!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='∙(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ($) , $ %!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='∙(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ($) , ⋯ , $ %!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='∙(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ($) , ((!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ($) %!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='∙(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ($) , 0, ⋯ ,0)′ for 𝑘 ∈ {2, ⋯ , 𝑁}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In fact, this matrix satisfies 𝑥̄% = 0 and 𝑿-𝑿 = 𝑑𝑖𝑎𝑔{𝑎$, 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', ⋯ , 𝑎"} with 𝑎B = 𝛼B ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' While, since it is assumed that model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1) includes constant term 𝒙𝟏 = 𝒆, 𝑿) = [𝑰, 𝟎)]: 𝐾 × 𝑁 cannot be a model for 𝑿 satisfying (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Next, let us study some properties of the I-factor, which will be used in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2 (1) When 𝐾 ≥ 3, the inefficiency factor is independently measured by 11 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='4) 𝑧̅% " ≡ 𝑧̅% "(𝑝%𝑿) = 0̅" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 9" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' = 𝒙𝒌 , 𝑴𝒆𝒙𝒌 𝒙𝒌 , (𝑰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='𝑴𝒆)𝒙𝒌 = K" /.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='K", which takes values on [0, ∞) and is increasing in 𝑞% ∈ [0,1), where 𝑞% = (𝒙𝒌 ) 𝒆)"/[𝒙𝒌 ) 𝒙𝒌 ∙ 𝒆)𝒆] = 𝑐𝑜𝑠"(𝜃%) is the squared raw correlation of 𝒙𝟏 = 𝒆 and 𝒙𝒌, and 𝑴𝒆 = 𝒆(𝒆)𝒆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='𝟏𝒆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' (2) If K=2, then 𝑧̅" " = 𝑞"/(1 − 𝑞") and 𝑉𝐼𝐹" = 1/(1 − 𝑞"), and hence the two factors are both a function of 𝑞" and 𝐻% = 1/(1 − 𝑞")".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Therefore, they are not separable and 𝐻% = 1 if and only if 𝒙𝒌 ) 𝒆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' (3) For 𝐾 ≥ 2 , the inefficiency factor 𝑧̅% " or equivalently 𝑞% = 𝑧̅% "/(1 + 𝑧̅% ") is a collinearity measure between 𝒙𝟏 = 𝒆 and 𝒙𝒌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Straightforward and omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Thus 𝑧̅% " is an increasing function of 𝑞%, and the closer the angle between two vectors 𝒙𝒌 and 𝒆 is to 0, the larger 𝑧̅% " is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In other words, if and only if 𝒙𝒌 and 𝒆 are orthogonal, 𝑧̅% " attains the minimum 0 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2(3)), implying that 𝑧̅% " is also measuring the collinearity of 𝒙𝒌 with 𝒆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence, it will be better to choose such variables 𝒙𝒌’s that 𝑐𝑜𝑠"(𝜃%) is close to 0 or equivalently 𝒙𝒌 is less collinear with 𝒆 so that the I-factor is smaller, provided such a selection of variables is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Note that the inefficiency factor 1 + 𝑧̅% " is expressed as / ∑ 0$" 9#"‚ " $1/ = 1 + 𝑧̅% " = (1 − 𝑞%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='/ = 1 + 𝑞% + 𝑞% " + ⋯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 Comparisons of IndSV 𝑉𝑎𝑟r𝛽_%s and IndPSV 𝐼𝑛𝑑𝑃𝑆𝑉% To compare IndSV and IndPSV, let us make some basic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' First, in the case of IndSV 𝑉𝑎𝑟r𝛽_%s that depends on (𝑠0%, 𝑉𝐼𝐹%) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='4), it is desirable for 𝑠0% to be large in order to make the IndSV small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' While, in the case of the IndPSV that depends on ( 𝑧̅% ", 𝑉𝐼𝐹% ) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2), 𝑧̅% " is the I-factor with 𝑠L% = 1 , where 𝑠L% " = ∑ (𝑧$% − 𝑧̅%)"/𝑁 $1/ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence, under 𝑠L% = 1, it is desirable for (𝑥̄%)" to be small as well as for 𝑠0% to be large for the sake of the IndSV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In other words, it is desired that { 𝑥$% }is distributed over a broad interval including 0 and its mean is close to 0, even though this is not controllable since 𝑿∗ is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' While, the case of {𝑥$% > 0 for any n} tends to make the I-factor large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In addition, if 𝑠0% is also desired to be large for the sake of IndSV, then the distribution of { 𝑥$% } needs to be distributed more like 𝑁(0, 𝛾) with large 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' If the standardized variable 𝑤$% = (𝑥$% − 𝑥̅%) 𝑠0% ⁄ can be used for 𝑥$% in the model, then 𝑤•% = 0 obtains and the 𝐼𝑛𝑑𝑃𝑆𝑉% = 𝜎"𝑉𝐼𝐹%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This is an important implication for empirical modelling in treating collinearity via standardization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', Aitken and West, 1996), when the IndPSV is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In Section 4, it will be shown that the VIF based on 12 {𝑤$%} is the same as the VIF based on { 𝑥$% }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Second, the XMSP does not aim to minimize 𝐻%′𝑠 or its mean, but aims to control the two factors of 𝑧̅% " and 𝑉𝐼𝐹% separately because the strong collinearity of 𝒙% ∗ can seriously cause confounding effects on the other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Now, let us compare the standard error (SE) of 𝛽_% and the corresponding predictive standard error (PSE), which are respectively given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5a) 𝑆𝐸 •𝛽_%(𝑿∗)‚ = 𝜎•(𝒚𝒐, 𝑿∗)(𝑠0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='. !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='√𝑉𝐼𝐹% (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5b) 𝑃𝑆𝐸 •𝛽_%(𝑿∗)‚ = 𝜎•(𝒚𝒐, 𝑿∗)(1 + 𝑧̅% ") .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='√𝑉𝐼𝐹% ≡ 𝜎•𝐻% //", where 𝜎• is the residual standard error (RSE) under model 𝑿∗ via OLS with 𝒚𝒐, which is another important measure of model performance with 𝒚𝒐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Each IndSV in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5a) can be smaller than 𝜎" because of (𝑠0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='//".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence, it is not easy to compare and interpret those SEs, although the SEs in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5a) have been traditionally given as computer outputs with other statistics such as adjusted CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In the case of 𝜎•𝐻% //" in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5b), which we call the PSE of the term {𝑥$%𝛽_%: 𝑛 = 1, ⋯ , 𝑁}, each 𝐻% //" can be compared with other 𝐻/ $/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='′𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence the individual estimates in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2) are comparable in terms of the PSEs together with RSE 𝜎•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In empirical analysis, it will be better to have the PSE values 𝑃𝑆𝐸(𝛽J𝑘(𝑿∗))′𝑠 as computer outputs that reveal the contribution of each term {𝑥$%𝛽_%} to 𝒚𝒐 with relative roles of 𝒙% ∗ 𝛽%′𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Finally it is remarked that by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5), the k-th SE is expressed as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='6) 𝑆𝐸 •𝛽_%(𝑿∗)‚ = 𝜎•(𝒚𝒐, 𝑿∗)‖𝒙% ∗‖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='/(1 + 𝑧̅% ").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='//"√𝑉𝐼𝐹%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This expression shows that the k-th SE depends on the three effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' ‖𝒙% ∗‖, 𝑧̅% " and 𝑉𝐼𝐹%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In Section 3, it will be pointed out from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='6) that the SE is controlled by controlling ‖𝒙% ∗ ‖ after a 𝒚𝒐-accommodating class of models with the 𝑧̅% " and 𝑉𝐼𝐹% controlled is obtained, where the three effects are functionally independent if 𝐾∗ ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' As was stated, the performances of different models 𝑿∗′𝑠 may be compared by the mean of {𝐻%} as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='7) 𝐻• ≡ 𝐻•(𝑿∗) = / ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='/ ∑ 𝐻% ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='∗ %1" = / ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='/ ∑ (1 + 𝑧̅% ") × 𝑉𝐼𝐹% ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='∗ %1" for 𝑿∗: 𝑁 × 𝐾∗ in the IndPSVs, provided 𝑉𝐼𝐹%′𝑠 are controlled in such a way that 𝑉𝐼𝐹% ≤ 𝑑 for any k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This gives a direct comparability of different models { 𝑿∗} in a certain class of such models as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', those of the same column size J as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' However, in model comparison, it will be necessary to take the RSE 𝜎•(𝒚𝒐, 𝑿∗) into consideration, implying that after a class of IC-controlled models, models in the class may be compared in terms 13 of 𝜎•(𝒚𝒐, 𝑿∗)𝐻•(𝑿∗) for each fixed J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In addition, it will be necessary to use a 𝑦#MSP to evaluate a whole model performance with (𝒚𝒐, 𝑿 ) and select an optimal model by such MSPs with AIC, BIC, adjusted CD 𝑅Ÿ", etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 3 The XMSP for EEM-M As has been stated, not by a diagnostic checking after a model is estimated, the XMSP in the EEM-M aims to control (𝐼%, 𝐶%) or equivalently (𝑧̅% ", 𝑉𝐼𝐹%) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) in such a way that 𝑧̅% " ≤ 𝑐 𝑎𝑛𝑑 𝑉𝐼𝐹% ≤ 𝑑 , before 𝒚𝒐 is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then it gives a class 𝐷 of IC- controlled models for efficiency and stability of OLS estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The class is also called a 𝒚𝒐 -accommodating class for 𝑦# MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The whole EEM-M process is symbolically described as 𝐵ℳ([𝑿]) −XMSP —𝑦# MSP (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1a,b) for 𝐵ℳ([𝑿])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' It is shown in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 that the most efficiently IC-controlled model with 𝐻% = 1 for any 𝑘(≥ 2) is characterized as a model that satisfies (1) 𝑥̄% = 0 for any 𝑘(≥ 2) and (2) 𝑿-𝑿 = 𝑑𝑖𝑎𝑔{𝑎$, 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', ⋯ , 𝑎"}, where 𝑎/ = 𝒙𝒊′𝒙𝒊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' One may develop a method for comparing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1, the bundle model 𝐵ℳ([𝑿]) in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2 is further discussed to develop the XMSP and in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2, we discuss some methods of obtaining the class 𝐷 of IC-controlled models, where the column size of each model 𝑿∗ is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3, for each model 𝑿∗, using inefficiency-collinearity risk index (ICRI) (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='4), a partial ordering is introduced onto the set [𝑿] to compare models in terms of ICRI and the concept of admissibility of model is defined by the ICRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 Bundle Model and Stochastic Specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Recall that the bundle model in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1a, b) is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1a) 𝐵ℳ([𝑿]) = { 𝒚𝝉 = 𝑿𝝉𝜷𝝉 + 𝒖𝝉 | 𝑿𝝉 ∈ [𝑿], 𝒖𝝉 ∈ [𝒖]} with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1b) [𝒖] = {𝒖𝝉: 𝜏 ∈ Λ, 𝐸(𝒖𝝉) = 𝟎, 𝑉𝑎𝑟(𝒖𝝉) = 𝜎"𝑰}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This is the set of different models with their own error terms from which we aim to select a model to use for empirical applications in view of the EEM-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Note that 𝒚𝒐 is regarded as generated from a model in the bundle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This model implies 𝑉𝑎𝑟( 𝒚𝝉) = 𝜎"𝑰 for any sub-model 𝑿𝝉 in the bundle set, which in turn implies the structure of IndPSV 𝐻(𝑧̅=% " , 𝑉𝐼𝐹=%) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) as a function of 𝑿𝝉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This is the information we use to select 𝐷 in the XMSP and it is symbolically expressed as 𝜗= ≡ ({𝐻(𝑧̅=% " , 𝑉𝐼𝐹=%): 𝑘 = 2, ⋯ , 𝐾=}, 𝑉𝑎𝑟(𝒚𝝉) = 𝜎"𝑰), where 𝜏 ∈ Λ, which is independent of 𝒚𝝉 (including 𝒚𝒐) but dependent on 𝑿𝝉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This structure is in fact an essential part of empirical modelling connected with the predictive sampling variances of individual OLS estimators and it enables the XMSP to aim to select a class of IC- 14 controlled models by controlling all the (𝑧̅% ", 𝑉𝐼𝐹%)′𝑠 of individual OLS estimates in each model without observing 𝒚𝒐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here we drop the suffix 𝜏 distinguishing models unless it is necessary and denote a representative sub-model by 𝑿∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' To develop this XMSP, first note that when model sequence {𝑿𝒊} is increasing in such a way as 𝑿𝒊 ⊂ 𝑿𝒊N𝟏 ⊂ ⋯ ⊂ 𝑿, the k-th CD 𝑅X% "(𝑿𝒊) or equivalently 𝑉𝐼𝐹% is increasing in the column size of each matrix i for fixed k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Hence let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1a) [𝑿] =∪34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' " [𝑿: 𝐽] with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1b) [𝑿: 𝐽] = M𝑿∗ ∈ [𝑿]|𝑿∗ ∶ 𝑁 × 𝐽, 𝑟𝑎𝑛𝑘(𝑿∗) = 𝐽, 𝒙1 = 𝒆 ∈ 𝑿∗Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Note [𝑿] ≡ [𝑿: 𝐾].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' To connect 𝑿∗ ∈ [𝑿] with y, let 𝒖∗ ∈ [𝒖] represent a corresponding error term that has no association with (𝒚, 𝑿) when 𝑿∗ ≠ 𝑿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then the observed 𝒚𝒐 is regarded as realized through one of the models in the following set: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2a) 𝐵ℳ([𝑿]) =∪𝐽=2 𝐾 𝐵ℳ([𝑿: 𝐽]) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2b) 𝐵ℳ([𝑿: 𝐽]) = {𝒚∗ = 𝑿∗𝜷∗ + 𝒖∗| 𝑿∗ ∈ [𝑿: 𝐽], 𝒖∗ ∈ [𝒖] }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1) itself belongs to Bℳ([𝑿]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2 𝒚𝒐-Accommodating Class : Controlling (𝐼%, 𝐶%) In this bundle model, the XMSP is a process of exploring a class of models {𝑿∗} with 𝑧̅% ")𝑠 𝑎𝑛𝑑 𝑉𝐼𝐹%′𝑠 controlled separately, so that all the IndPSVs of each model in the selected class are controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This implies that all the models in the class are candidates of empirically effective models because an optimal model is selected from them by a 𝑦#MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Of course, the optimality depends on the choice of the 𝑦#MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In other words, the XMSP is a self-fulfilling process to make a selected class of models IC-controlled for 𝑦#MSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Now, we formally define the concept of 𝒚𝒐-accommodating set of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Let 𝐽(𝑿∗) denote the column size of matrix 𝑿∗, and let 𝑝%𝑿∗ = 𝒙𝒌 ∗ for 𝑘 ≤ 𝐽(𝑿∗) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 Let (c, d) be prespecified as control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' The 𝒚𝒐-accommodating class of models at level (c, d) is defined to be the class 𝐷23 ≡ 𝐷23([𝑿]) of models, where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) 𝐷23 = {𝑿∗ ∈ [𝑿]| 𝑧̅% "(𝑝%𝑿∗) ≤ 𝑐, 𝑉𝐼𝐹%(𝑿∗) ≤ 𝑑, (𝑘 = 2, ⋯ , 𝐽(𝑿∗))} = 𝐷/ 2 ∩ 𝐷" 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3a) 𝐷/ 2 = {𝑿∗ ∈ [𝑿]| 𝑧̅% "(𝑝%𝑿∗) ≤ 𝑐, (𝑘 = 2, ⋯ , 𝐽(𝑿∗))}, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3b) 𝐷" 3 = {𝑿∗ ∈ [𝑿]| 𝑉𝐼𝐹%(𝑿∗) ≤ 𝑑, (𝑘 = 2, ⋯ , 𝐽(𝑿∗))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' (1) If 𝑿∗ ∈ 𝐷23 and 𝑿∗∗ ⊂ 𝑿∗, then 𝑿∗∗ ∈ 𝐷23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' (2) If 𝑿𝟏, 𝑿𝟐 ∈ 𝐷23, then 𝑿𝟏 ∩ 𝑿𝟐 ∈ 𝐷23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 15 This obvious proposition is practically useful to find a model in 𝐷23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' If 𝑿∗ is found to belong to 𝐷23 , so does any sub-model 𝑿∗∗ of 𝑿∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This will be used in Section 4 to develop an algorithm of finding models in 𝐷23 with principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' It is noted that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3), I-factors 𝑧̅% "′𝑠 and C-factors 𝑉𝐼𝐹%′𝑠 can be separately controlled provided 𝐽(𝑿∗) ≥ 3 (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Also note that both 𝑧̅% " and 𝑅X% " (or equivalently 𝑉𝐼𝐹%) are scale-invariant for transforming 𝒙𝒌 ∗ into 𝑎%𝒙𝒌 ∗ for 𝑎% ≠ 0 (𝑘 = 2, ⋯ , 𝐾), implying the independence of their physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In addition, as will be shown in Section 4, 𝑅X% " is invariant under the transformation from 𝑥$% into (𝑥$% − 𝑥̅%) 𝑠0% ⁄ for ≥ 2 (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Also, note that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='1 when 𝑅X% " =0, 𝐼𝑛𝑑𝑃𝑆𝑉% = 𝜎"(1 + 𝑧̅% "), and that when 𝑧̅% " = 0, 𝐼𝑛𝑑𝑃𝑆𝑉% = 𝜎"𝑉𝐼𝐹%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Now, to get a 𝐷23, we need to choose (c, d), but it will be difficult to discuss the choice in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' A difficulty lies in the fact that the set of variables in data X is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In many areas of social sciences, we have to use the data as it stands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Then, so long as the systems are not so much volatile, 𝑅X% ")𝑠 will carry some predictability as a whole for dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' On the other hand, in natural sciences where data X can be designed to a large extent, then 𝑅X% ")𝑠 will be controlled with smaller d and then 𝑧̅% "′𝑠 will be more concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In general, it will be reasonable to first find 𝐷/ 2 by deleting variables with 𝑧̅% "′𝑠 larger than c and then to find 𝐷" 3 of models with 𝑅X% ")𝑠 ≤ 𝑑: = 1 − 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='/ from 𝐷/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' This will save some computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Some remarks follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' First, in the literature, concerning collinearity, it is often suggested that in an MSP via VIF, 𝒙𝒌 ∗ be dropped if 𝑉𝐼𝐹%>10 (or 5) or equivalently 𝑅X% " >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='9 (or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='8 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=') in each estimated model with 𝒚𝒐, though there is no theoretically solid ground (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', O’Brien (2007)) and it is likely to a wrong model since the final model is dependent on the order of selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In addition, there is no control on the I-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Second, once 𝐷23 is obtained, it is possible to control IndSVs via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Since 𝐼𝑛𝑑𝑃𝑆𝑉%/‖𝒙𝒌 ∗‖ = 𝐼𝑛𝑑𝑆𝑉% and since 𝑧̅% " and 𝑅X% " do not depend on ‖𝒙𝒌 ∗‖, we can control ‖𝒙𝒌 ∗‖ by setting its lower bound in the class 𝐷23 of models as 𝐷23P = 𝐷23 ∩ 𝐷Q P with 𝐷Q P = {𝑿∗ ∈ [𝑿] | || 𝑝%𝑿∗|| ≥ 𝑒, (𝑘 = 2, ⋯ , 𝐽(𝑿∗))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Since any model in the set 𝐷23satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='4) 𝐻T𝑧̅5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', 𝑉𝐼𝐹5Z ≡ (1 + 𝑧̅5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' (𝑝5𝑿∗))𝑉𝐼𝐹5(𝑿∗) ≤ (1 + 𝑐)𝑑 ≡ ∆67 for any k, when model is restricted to the class 𝐷23P, the SE in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='6) is also controlled as 𝑆𝐸 ^𝛽`5(𝑿∗)a = 𝜎c(𝒚𝒐, 𝑿∗)𝐿T𝑧̅5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', 𝑉𝐼𝐹5Z with 𝐿T𝑧̅5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', 𝑉𝐼𝐹5Z ≡ 𝐻T𝑧̅5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=', 𝑉𝐼𝐹5Z/‖𝒙𝒌 ∗‖ ≤ (1 + 𝑐)𝑑/𝑒, unless 𝐷23P = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' While, since the IndPSVs is bounded above by 𝜎"∆23 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='4), all 16 the PSEs in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5b) are below or equal to 𝜎•R∗ S√∆23 where 𝜎•R∗ S is the RSE 𝜎c(𝒚𝒐, 𝑿∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3 Finding 𝐷23 and Inefficiency-Collinearity Risk Index (ICRI) of 𝑿∗ Now let us consider some methods of finding the class 𝐷23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' One possible method is to try all the models via all the possible combinations of variables to get 𝐷23though its computational process may not be feasible if K is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Although the two algorithms for finding 𝐷" 3 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='3) can be used, it is better to reduce the total number of variables before such an algorithm is applied, so that the computational burden becomes as little as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Empirically it is often the case that the set of variables in X is divided into core part X1 and such non-core part X2 as dummy variables as in Fox and Monette (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' And a set of IC-controlled models is found from the core part X1 and then each variable is added to the set one by one from X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' Here, to take a different approach, we define the ICRI to compare different models in view of the EEM-M and apply it for finding IC-controlled models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content=' In the case of the largest model X, the ICRI is defined as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE1T4oBgHgl3EQfOgNK/content/2301.03015v1.pdf'} +page_content='5a) 𝑟(𝑿) = (𝑐 density2. +.output Answer +No +5 +grandson +1. find(coke) +2. pick(coke) +3. find(trash) +4. put(coke) +5. find(sponge) +6. pick(sponge) +7. find(user) +8. put(sponge) +Query +Reasoning Chain +Answer +Goal in PDDL: +(:goal + ; I need to do both things + (and + ; first throw away the coke + (at coke trash) + ; then bring the user a sponge as a cleaning tool + (at sponge user) + ) +) +# 1. How is [Vincent] related to [Dorothy]? (independent, support: " +[Dorothy]'s son [Vincent] showed up shortly after.") +relation(Vincent, Dorothy) = son +# 2. How is [Dorothy] related to [Gabrielle]? (independent, support: +"[Gabrielle] drove her daughter [Dorothy] to the hospital.") +relation(Dorothy, Gabrielle) = daughter +# 3. Final answer: How is [Vincent] related to [Gabrielle]? (depends +on 1, 2) +relation(Vincent, Gabrielle) = relation(Vincent, Dorothy) @ + + + + relation(Dorothy, Gabrielle) +Deterministic +Solver +Translator +(LM) +Figure 3. Examples from each task (Math Word Problems, Multi-hop QA, Planning, Logical Inference) showing our 2-stage Translation +and Problem Solving pipeline. +to-Most prompting (Zhou et al., 2022) uses an LM to first +reduce the question to subquestions and then sequentially +answer them. However, the reasoning chain is entirely in +NL, so there is still no faithfulness guarantee that the answer +follows from it. +We note that our work is concurrent with Chen et al. (2022) +and Gao et al. (2022), both generating the reasoning chain +in Python code and calling a Python interpreter to derive the +answer. While we do not compare with them empirically +since they are not yet published, we do want to highlight +the following differences: (a) We demonstrate the general- +izability of our approach to multiple symbolic languages +beyond Python and multiple domains beyond arithmetic +reasoning and simple symbolic reasoning. (b) In particular, +we innovatively recast a diverse set of realistic tasks (Plan- +ning, Multi-hop QA, and Logical Inference) into a symbolic +representation, which allows us to tackle them with a single +framework. (c) Our reasoning chain interleaves NL and +SL in a structured fashion, which allows the user to better +understand and potentially interact with the model. +3. Method +Our method, Faithful CoT, is a 2-stage pipeline, as seen +in Figure 2. Like previous CoT-style work, our prompt +consists of (Q, C, A) triples. Notable differences lie in +our unique interleaving of NL (natural language) and SL +(symbolic language) in C, as well as the way we derive the +final answer A. +In the Translation stage, given a complex query Q in NL, +we prompt an LM to translate it into a reasoning chain C, +which interleaves NL and a task-specific SL (e.g., Python, +Datalog, or PDDL).3 In the Problem Solving stage, we call +a deterministic external solver, e.g., a Python interpreter, +a Datalog executor, or PDDL planner, depending on the +task, to obtain the answer A from the reasoning chain C. +As shown in Figure 3, we define CNL to be the NL com- +ponent (black) and CSL to be the SL component (blue) in +C. Though we separate the two components notationally, +they are interleaved in the generation. Using this approach, +C is guaranteed to be a faithful model explanation, since +our final A is the result of deterministically executing CSL. +Moreover, CNL allows the user to better understand the +3Our prompts can be found in the Supplementary Materials. + +Faithful Chain-of-Thought Reasoning +reasoning process.4 +We apply this method to 4 types of complex reasoning +tasks: MWP, Multi-hop QA, Planning, and Logical Infer- +ence. Next, we will illustrate how our method works for +each of them, with examples from Figure 3. +3.1. Math Word Problems (MWP) +Given a grade-school math question Q written in NL (“If +there are 3 cars in the parking lot and 2 more cars arrive, +how many cars are in the parking lot?”, shown in green in +Figure 3), we want to obtain A as a real-valued number (5). +In the Translation stage, we prompt the LM to take in Q and +generate a reasoning chain C, which interleaves CNL and +CSL. Specifically, the CNL component consists of three +types of information: +(a) Subquestions: Q is broken down into multiple smaller- +scale subquestions, e.g., “1. how many cars are there in the +beginning?”, “2. how many cars arrive?”, and “3. how many +cars are in the parking lot?”. +(b) Dependency Graph: Each subquestion can either be +answered directly via context (subquestions 1 and 2 are +“independent”) or rely on answers to previous subquestions +(subquestion 3 “depends on 1 and 2”). +(c) Rationales: Each subquestion is accompanied with ra- +tionale(s) to support the answer (the “support” field). The +rationales can be either a subset of the original context (“2 +more cars arrive”) or any external knowledge (“there are 7 +days in a week”) relevant to the subquestion. +Each subquestion and its corresponding dependencies and +rationales inform the subsequent generation of CSL. In our +example in Figure 3, CSL consists of Python code generated +to answer each subquestion in CNL. During the Problem +Solving stage, we execute CSL using our solver, a Python +interpreter, to derive A (5 cars in the end). +3.2. Multi-hop QA +Given a complex question Q that involves multiple steps of +reasoning (e.g., “Would a pear sink in water?”, shown in red +in Figure 3), we want to obtain the answer A as a Boolean +value or string value variable. Similar to our MWP task +formulation, C interleaves CNL (NL comments), and CSL +(symbolic program). Depending on the nature of the task, +the format of the reasoning chain C is slightly different: for +some datasets, the LM first generates all subquestions and +their answers in NL, and then represents these answers as +SL to derive A (see Figure 3); for others, the LM interleaves +the NL subquestions and the SL program, similar to the +case of MWP (see Table 11 and Table 12 for examples). +4While no constraints are enforced between CNL and CSL in +our main experiments, we analyze this in Section C.3. +In terms of SL, we use both Python and Datalog, also de- +pending on the dataset. As Multi-hop QA problems involve +multi-step reasoning to solve, CSL often utilizes Boolean +algebra and string comparisons (in Python) along with rela- +tion definitions and logic programming (in Datalog). We use +their corresponding interpreter as our deterministic solver +to execute CSL and obtain A. +In the example from Figure 3, the LM first generates the sub- +questions, “1. What is the density of a pear?” and “2. What +is the density of water?”, which are individually answered +in NL. The answers (“Water has a density of 1g/cm3”) are +converted to Datalog statements (Has_density(“water”, +1)), which are then combined to formalize the truth condi- +tion of the final answer. Finally, we execute the Datalog +program to determine that a pear would not sink in water. +3.3. Planning +In a user-robot interaction scenario, given a household task +query Q from a user, we want to come up with a plan of +actions A that the robot should take in order to accomplish +the task. For example, in Figure 3, given user query “I +spilled my coke on the table, could you throw it away and +bring something to clean with?”, a possible plan can be +“find(coke), pick(coke), find(trash), put(coke) ...”. In the +Translation stage, an LM translates Q into C, consisting of +CNL (which breaks down Q into subtasks) and CSL (which +represents the subtasks as a symbolic goal in PDDL5 — a +language to define and solve classical planning problems). +Figure 3 shows this translation, with CSL in blue and CNL +in black. Finally, we call a PDDL Planner as the determin- +istic solver to obtain A, a plan to accomplish the goal CSL +under the predefined scenario. +3.4. Logical Inference +Given a logical inference problem Q written in NL, we +want to obtain A as a string-valued variable. For exam- +ple, the CLUTRR (Sinha et al., 2019) dataset involves in- +ferring the family relationship (e.g., “grandson”) between +two people from a short story (e.g., “[Gabrielle] drove her +daughter [Dorothy] to the hospital. [Dorothy]’s son [Vin- +cent] showed up shortly after. How is [Vincent] related to +[Gabrielle]?”, shown in yellow in Figure 3). During the +Translation stage, we prompt the LM to generate C, consist- +ing of CNL and CSL. Similar to previous tasks, CNL breaks +down Q into subquestions (“How is [Vincent] related to +[Dorothy]” and “How is [Dorothy] related to [Gabrielle]”), +as well as provide input extracts as rationales to support +the answer (“[Dorothy]’s son [Vincent] showed up shortly +after”, etc.). Each subquestion in CNL is answered in CSL +via a logical expression representing the relation between +5https://en.wikipedia.org/wiki/Planning_Domain_ +Definition_Language. A goal is a special construct in PDDL. + +Faithful Chain-of-Thought Reasoning +Q: There are 15 trees in the grove. Grove +workers will plant trees in the grove today. +After they are done, there will be 21 trees. +How many trees did the grove workers +plant today? +A: We start with 15 trees. Later we have +21 trees. The difference must be the +number of trees they planted. So, they +must have planted 21 - 15 = 6 trees. The +answer is 6. +…… (7 more examples) +Q: Claire makes a 3 egg omelet every +morning for breakfast. How many dozens +of eggs will she eat in 4 weeks? +A: Claire makes a 3 egg omelet every +morning. In one week she will eat 3 * 7 = +21 eggs. In 4 weeks she will eat 4 * 21 = +84 eggs. The answer is 84. +Input +Model Output +Standard prompting +Chain of Thought (COT) prompting +(Wei et al., 2022) +Faithful COT prompting +(our method) +Q: There are 15 trees in the grove. Grove +workers will plant trees in the grove today. +After they are done, there will be 21 trees. +How many trees did the grove workers +plant today? +A: The answer is 6. +…… (7 more examples) +Q: Claire makes a 3 egg omelet every +morning for breakfast. How many dozens +of eggs will she eat in 4 weeks? +A: The answer is 12. +Input +Model Output +# Q: There are 15 trees in the grove. Grove workers will plant trees in the grove today. After they +are done, there will be 21 trees. How many trees did the grove workers plant today? +# To answer this question, we write a Python program to answer the following subquestions: +# 1. How many trees are there in the beginning? (independent, support: ["There are 15 trees"]) +trees_begin = 15 +# 2. How many trees are there in the end? (independent, support: ["there will be 21 trees"]) +trees_end = 21 +# 3. How many trees did the grove workers plant today? (depends on 1 and 2, support: []) +trees_today = trees_end - trees_begin +# 4. Final Answer: How many trees did the grove workers plant today? (depends on 3, support: []) +answer = trees_today +…… (7 more examples) +# Q: Claire makes a 3 egg omelet every morning for breakfast. How many dozens of eggs will +she eat in 4 weeks? +# To answer this question, write a Python program to answer the following subquestions: +# 1. How many eggs are in one dozen? (independent, support: ["External knowledge: there are +12 eggs in a dozen"]) +eggs_in_dozen = 12 +# 2. How many eggs are in one omelet? (independent, support: ["Claire makes a 3 egg omelet +every morning"]) +eggs_in_omelet = 3 +# 3. How many omelets does Claire make in one week? (independent, support: ["External +knowledge: there are 7 days in a week"]) +omelets_per_week = 7 +# 4. How many eggs does Claire eat in one week? (depends on 2 and 3, support: []) +eggs_per_week = eggs_in_omelet * omelets_per_week +# 5. How many eggs does Claire eat in 4 weeks? (depends on 4, support: ["How many dozens of +eggs will she eat in 4 weeks?"]) +eggs_4_weeks = eggs_per_week * 4 +# 6. How many dozens of eggs does Claire eat in 4 weeks? (depends on 5 and 1, support: []) +dozens_4_weeks = eggs_4_weeks / eggs_in_dozen +# 7. Final Answer: How many dozens of eggs will she eat in 4 weeks? (depends on 6, support: []) +answer = dozens_4_weeks +Input +Model Output +Figure 4. The prompt for GSM8K (a Math Word Problems dataset) from standard, CoT (Wei et al., 2022), and Faithful CoT prompting +(ours). The ground-truth answer is 7, and only our method correctly computes the answer. +the mentioned entities, for example, relation(Vincent, +Dorothy)=son denotes that Vincent is Dorothy’s son. In the +Problem Solving stage, our solver is a simple logical infer- +ence engine that relies on a set of transitivity rules provided +by (Anonymous, 2022) among possible family relationships, +e.g., son@daughter=grandson (the son of one’s daughter +is one’s grandson). Our solver recursively applies these +rules on CSL to derive A, and determine that Vincent is +Gabrielle’s grandson. +4. Experimental setup +4.1. Datasets +Here, we summarize the evaluation datasets used for each +domain. We select the same number (6 to 10, depending on +the task) of exemplars as in Wei et al. (2022) to form our +few-shot prompt, which can be found in the Supplementary +Materials. Unless otherwise stated, we use the official splits: +training set for exemplar selection, validation set for prompt +tuning, and test set for evaluation.6 +Math Word Problems (MWP). +We follow Wei et al. +(2022) and consider the same five MWP benchmarks: +6See Appendix E for dataset statistics, examples, data cleaning +method, splits, prompt construction strategy, etc. +GSM8K (Cobbe et al., 2021), SVAMP (Patel et al., 2021), +MultiArith (Roy & Roth, 2015), ASDiv (Miao et al., 2020), +and AQuA (Ling et al., 2017). For all datasets, the input +question is phrased in NL. The answer is a string-valued +mathematical expression for AQuA, and one or more inte- +ger(s) for all other datasets. We use the same 8-shot prompt +for all datasets except AQuA. +Multi-hop QA. +We consider the three datasets: Strate- +gyQA (Geva et al., 2021), a dataset of open-domain ques- +tions that require an implicit multi-step strategy to answer, +e.g., “Did Aristotle use a laptop?” involves answering “1. +When did Aristotle live?”, “2. When was the laptop in- +vented?”, and “3. Is #2 before #1?”; Date Understanding +from BIG-bench (BIG-Bench collaboration, 2021), which +asks the model to infer a date from a context, by performing +computation on relative periods of time; and finally, Sports +Understanding from BIG-bench, which asks the model to +decide whether an artificially constructed statement related +to sports is plausible or implausible. Since the latter two +datasets do not have a training set, we follow Wei et al. +(2022) and select 10 examples from the test set to form the +prompt and use the rest for evaluation. +Planning. +We use the SayCan dataset (Ahn et al., 2022), +which assumes a scenario of a robot operating in a kitchen, +helping the user with household tasks, e.g., “bring a coke + +Faithful Chain-of-Thought Reasoning +Table 1. Accuracy of different prompting methods on 10 reasoning datasets from 4 domains. We compare our method, Faithful CoT, with +standard promoting (Brown et al., 2020), CoT prompting (Wei et al., 2022), as well as previously published few-shot SOTA results. The +best results within each decoding strategy are in boldface, and the new SOTA results across all strategies are underlined. +Math Word Problems +Planning +Multi-hop QA +Logical Inference +Method +GSM8K +SVAMP +MultiArith +ASDiv +AQuA +SayCan +StrategyQA +Date +Sport +CLUTRR +Few-shot SOTA +78.0 +86.8 +100.0 +87.8 +52.0 +88.3 +81.6 +65.3 +98.5 +- +Greedy Decoding +Standard +19.7 +69.9 +44.0 +74.0 +29.5 +85.8 +67.1 +49.0 +71.7 +41.1 +CoT +63.1 +76.4 +96.2 +78.6 +45.3 +87.4 +73.2 +59.9 +97.9 +40.8 +Faithful CoT (ours) +72.1 +83.5 +98.8 +79.9 +47.2 +89.3 +63.0 +80.8 +99.1 +58.9 +Self-Consistency Decoding +CoT +78.0 +86.8 +100.0 +84.2 +52.0 +89.3 +79.8 +63.8 +98.0 +45.7 +Faithful CoT (ours) +79.8 +88.9 +99.2 +84.4 +61.4 +93.2 +65.2 +85.5 +99.0 +71.9 +to the table”. There are a number of locations and objects +that the robot can interact with. The robot can only perform +a fixed set of actions, including find, pick, and put. The +task is to map a user query in NL to a plan of predefined +actions. Following Wei et al. (2022), we manually write 7 +exemplars, since no training set is provided. +Logical inference. +We use the CLUTRR (Sinha et al., +2019) benchmark described in Section 3.4. The dataset has +multiple splits based on the number of intermediate steps +K required to reach the answer. We construct the prompt +using 8 exemplars with K ∈ {2, 3}, and test the models on +the remaining examples with K up to 10. +4.2. Evaluation Metrics +We evaluate the model performance with the accuracy of +the final answer. Following previous work (Wei et al., 2022; +Wang et al., 2022; Chen et al., 2022), for all MWP datasets +(except AQuA) where the answer contains integer(s), a cor- +rect answer is defined as the exact match between the pre- +diction and the ground truth both rounded up to the near- +est integer (with the math.ceil() function in Python); for +StrategyQA and Sports Understanding where the answer +is a Boolean value, it is defined as the exact match be- +tween the prediction and the ground truth both evaluated +as a Boolean variable; for SayCan, the generated plan is +considered correct if it is among the ground truth plans; for +all other datasets, we rely on the exact match between the +prediction string and the ground truth string. +4.3. Language Model +In Translation, we always use OpenAI Codex (Chen et al., +2021) (code-davinci-002, with 175B parameters) as the +underlying LM, since it is so far the only code-generation +model with a public API.7 +7See Appendix A for implementation details. +4.4. Baselines +We compare our method to two other baselines, shown in +Figure 4: standard few-shot prompting, popularized by +Brown et al. (2020), with demonstrations of only the ques- +tion and the answer (green); and CoT prompting (Wei et al., +2022), which additionally provides a reasoning chain in NL +(blue). We also show the published SOTA few-shot results.8 +All prompting methods are compared under two decoding +strategies: greedy decoding, where the LM samples the +most probable next token from the vocabulary (i.e., temper- +ature = 0.0); and self-consistency decoding (Wang et al., +2022), where the LM generates multiple reasoning chains +and chooses the final chain based on majority voting on the +evaluated answer (we use a temperature of 0.4 and 40 gen- +erations for all datasets).9 We reproduce the baseline results +ourselves in cases when they are not reported or when we +clean the test set. +5. Results +Our results on all datasets are shown in Table 1. We see +that Faithful CoT outperforms CoT across most datasets and +domains for both greedy and self-consistency decoding. +With greedy decoding, our method outperforms CoT on 9 +of the 10 benchmark datasets spanning the 4 domains. On +average, it improves over CoT in the MWP domain by 4.4, +in Planning by 1.9, in Multi-hop QA by 4.0, and in Logical +Inference by a surprising 18.1. +Our method also outperforms CoT under self-consistency +decoding on 8 out of the 10 datasets. Compared to greedy +decoding, the average accuracy gain becomes larger for +Planning (1.9 → 3.9) and Logical Inference (18.1 → 26.2), +but smaller for MWP (4.4 → 2.5) and Multi-hop QA (4.0 +8See Appendix B for sources of the SOTA results. +9Note that we do not report the performance of standard prompt- +ing with self-consistency decoding, since when the number of sam- +pled outputs is large enough, this converges to standard prompting +with greedy decoding (Wang et al., 2022). + +Faithful Chain-of-Thought Reasoning +GSM8K +Date +SayCan CLUTRR +0 +20 +40 +60 +80 +100 +Accuracy +Full +No rationale +No NL but nudge +No NL +No solver +Figure 5. Ablation study results: accuracy when we remove dif- +ferent parts of the prompt. See Section 6.1 for details. +→ 2.7). Also, it is worth noting that our method achieves +the new few-shot SOTA results on 7 datasets. +On StrategyQA (from the multi-hop QA domain), however, +the performance of our method is still far from CoT and +even standard prompting. To understand why, we specif- +ically compare the examples where CoT makes a correct +prediction but our method fails. As shown in Figure 9 in +the Appendix, we find that the likely primary cause is the +sparsity of Datalog in the pretraining data for Codex, as an +overwhelming 29% of errors are syntax-related. Moreover, +including Datalog in the prompt also interferes with NL +generation, making it harder for Codex to produce relevant +subquestions (17%), retrieve knowledge correctly (10%), +and come up with valid reasoning from the knowledge to the +answer (10%). Another potential cause is the nature of the +task, as the difficulty for many StrategyQA questions does +not lie in symbolic operations (value/string comparisons) +but rather in retrieving correct facts and chaining them up, +which makes the advantages of our deterministic solver +less obvious. Still, with further pretraining on Datalog, we +believe that there is room for improvement with our method. +6. Analysis +In this section, we analyze the role of different components +in our pipeline, to better understand where its capabilities +come from and where it still struggles.10 Unless otherwise +stated, we choose one dataset from each domain for analysis +– GSM8K, Date Understanding, SayCan, and CLUTRR – +using greedy decoding. +6.1. Ablation Study +Given the strong performance of Faithful CoT, we now ad- +dress a natural question: how much does each part of the +prompt contribute to the accuracy? We perform an abla- +tion study where we remove different parts of the prompt +and see how the performance changes. In addition to the +original prompt (“Full”), we test four variations, illustrated +with the example from Figure 4: +10See Appendix C for full results where figures do not show +accuracy scores, and additional analysis of constraints and LMs. +Table 2. Robustness to the choice of exemplars across 6 runs. +Exemplars +GSM8K +Date +SayCan +CLUTRR +Set 0 (Table 1) +71.6 +80.8 +89.3 +58.9 +Set 1 +72.3 +81.3 +90.3 +59.0 +Set 2 +70.8 +85.0 +85.4 +57.2 +Set 3 +71.6 +82.5 +88.3 +58.0 +Set 4 +70.6 +77.4 +88.3 +55.5 +Set 5 +68.5 +85.0 +89.3 +56.0 +Mean +70.9 +82.0 +88.5 +57.4 +Std. +1.3 +2.9 +1.7 +1.5 +No rationale. We remove the rationales, i.e., everything +in the brackets from the NL comments, e.g., “independent, +support: [‘There are 15 trees’]”. +No NL but nudge. We remove all NL comments except the +“nudge” line: e.g., “# To answer this question, we write a +Python program to answer the following subquestions”. +No NL. We remove all NL comments. +No solver. Instead of calling the external solver, we add +“Answer: {answer}” to the end of every exemplar and let the +LM predict the answer itself. +Figure 5 shows the results of all prompt variations. On +GSM8K, Date Understanding, and SayCan, NL comments +contribute little to the performance, and sometimes even +slightly hurt it. On CLUTRR, however, their role is crucial, +since the exclusion of each component (rationale, nudge, +subquestions) results in a clear accuracy drop. In particular, +comparing No NL but nudge and No NL, the nudge line +itself brings a striking improvement by 31.3 points. +The external solver relieves the burden of problem solving +from the LM. Without it, the accuracy suffers a huge decline +on GSM8K, Date Understanding, and CLUTRR (-50.8, - +22.9, and -19.4 respectively), while on SayCan it improves +by 2.9 nonetheless. One potential influencing factor is that +SayCan might be too homogeneous, as it contains a set of +only 3 predefined actions. This can make the task relatively +easy, which allows all model variants to achieve around +90% accuracy and renders the solver unnecessary. Another +potential reason is the level of correspondence between the +final answer and the reasoning chain for different datasets: +as shown in Figure 3, the answer in SayCan is a sequence of +actions (e.g., find(redbull)), each directly corresponding +to one step in the reasoning chain (e.g., at redbull trash). +However, the answer in the other three datasets is only a +single number or string, which can only be derived after +executing all the steps in the reasoning chain. Therefore, +the latter type of tasks further necessitates the presence of +an external solver. +6.2. Robustness to Exemplars +We now answer the next question: how much does the +choice of exemplars matter? To do this, we annotate 20 +examples in total, randomly sample k (7-10, depending on + +Faithful Chain-of-Thought Reasoning +the dataset) to construct the prompt, and repeat the process +five times. Table 2 shows the performance of all six runs, +including the original (from Table 1). The mean accuracy is +close to the original (-1.5 to +1.2), still above the baselines +by a large margin (7 to 17) on all datasets except the arguably +easiest SayCan, considering the standard deviation (1.3 to +2.9). This strongly suggests that the benefits of Faithful CoT +are minimally influenced by the choice of exemplars. +6.3. Error Analysis +To further investigate where our method still fails, we in- +spect 100 errors11 from model predictions on each of the +four datasets and manually annotate the error categories. We +only present the results on GSM8K here, shown in Figure 6; +see Appendix F for those on the other datasets. +We categorize the errors on GSM8K into 6 types, inversely +sorted with frequency: +Wrong Subquestion (49%): The LM produces a wrong +NL subquestion, which eventually leads to the incorrect +answer. While this is the majority error type in our sam- +ple, it is worth noting that in a typical human-in-the-loop +collaboration, these errors are easily fixable. Even if the +user is unfamiliar with programming, they can inspect the +NL subquestions and potentially correct the model error by +simply deleting or editing a wrong subquestion. +Wrong Code (24%): The NL subquestion is correct, but +the code fails to answer the subquestion correctly. For ex- +ample, the code uses a variable that has not been previously +defined. +Semantic Understanding Error (12%): The LM incor- +rectly interprets certain semantic subtleties in the query. +This is the most complex and most interesting error cate- +gory. For example, consider the following problem: +If Martin eats Cheerios every day for breakfast, +he’ll lose 1.25 pounds/week. If he eats donuts ev- +ery day for breakfast, he’ll gain 1.75 pounds/week. +What will be the difference in his weight at the end +of 5 weeks between the two breakfast options? +The generated code, included in Appendix F.1, does not +assign opposite polarities (signs) for “pounds lost” vs. +“pounds gained”. For other examples in this category, we +notice errors like missing that a pair of something has 2 +items in it, missing to subtract 2 for “two years ago” when +it occurs as a subjunctive, and so on. Fixing these errors, in +general, will require more than providing additional exam- +ples in the prompt. +Generation Cutoff (7%): The generation stops midway, +11To encourage sample diversity, we embed all the errors using +text-davinci-002 and cluster the embeddings using spectral +clustering. This produces around 70 clusters of different sizes, +from which we gather 100 samples using importance sampling. +Figure 6. Error analysis for GSM8K. For a detailed description of +the error categories, see Section 6.3. +mainly due to the LM producing the same steps over and +over again. These errors could be easily detected in postpro- +cessing and possibly fixed by re-prompting the LM. +Wrong Gold Label (5%): We find 5 (out of our 100) exam- +ples that are genuine annotation errors in the gold labels. +Missing Subquestion (3%): The LM misses a relevant sub- +question needed for the rest of the reasoning chain to work. +These errors are also potentially fixable via human-in-the- +loop interaction, where the user can insert a subquestion +into the reasoning chain. +7. Conclusion +We propose Faithful CoT, a framework that decomposes +complex reasoning into Translation and Problem Solving. +During Translation, an LM produces a reasoning chain in +the form of interleaved natural and symbolic language. The +Problem-Solving stage calls an external solver that executes +the reasoning chain and derives the final answer. This pro- +cess guarantees that the reasoning chain is a faithful explana- +tion of how the model arrives at the answer. We demonstrate +the efficacy of our approach on 4 types of complex reasoning +problems: Math Word Problems, Multi-hop QA, Planning, +and Logical Inference. Our method sets new SOTA perfor- +mance on 7 of the 10 datasets, while additionally providing +a faithful explanation for the final answer. These results give +empirical evidence that improving model interpretability, +by guaranteeing the faithfulness of an explanation, does not +come at the expense of overall performance; in fact, we +see a strong synergy in between. Through a comprehensive +analysis on the strengths and weaknesses of our method, we +show its robustness to the choice of exemplars, the pivotal +role of the solver, as well as frequent error patterns where it +still struggles. +One limitation of our work is that the Translation stage is +still opaque, leaving an open question about whether it is +possible to improve its faithfulness as well. Moreover, it will +be helpful to perform a human evaluation on the correctness +of the generated reasoning chains. Finally, the NL com- +ments in the reasoning chain can serve as an interface for +users without a programming background to interactively +debug the model, which should be explored in future work. + +Missing Subquestion 3.0 % +Wrong Gold Label 5.0 % +Generation Cutoff 7.0 % +Semantic 12.0 % +Understanding Error +49.0 % +Wrong Subquestion +Wrong Code 24.0 %Faithful Chain-of-Thought Reasoning +Acknowledgements +This research is based upon work supported in part by +the DARPA KAIROS Program (contract FA8750-19-2- +1004), the DARPA LwLL Program (contract FA8750-19- +2-0201), the IARPA BETTER Program (contract 2019- +19051600004), the IARPA HIATUS Program (contract +2022-22072200005), and the NSF (Award 1928631). Ap- +proved for Public Release, Distribution Unlimited. The +views and conclusions contained herein are those of the +authors and should not be interpreted as necessarily repre- +senting the official policies, either expressed or implied, of +ODNI, DARPA, IARPA, NSF, or the U.S. Government. +We appreciate the support from OpenAI on increasing the +rate limit for the Codex API. We also thank Jiani Huang, +Ziyang Li, Litao Yan, Andrew Head, and Mayur Naik for +their valuable feedback. +References +Ahn, M., Brohan, A., Brown, N., Chebotar, Y., Cortes, O., +David, B., Finn, C., Fu, C., Gopalakrishnan, K., Hausman, +K., Herzog, A., Ho, D., Hsu, J., Ibarz, J., Ichter, B., +Irpan, A., Jang, E., Ruano, R. J., Jeffrey, K., Jesmonth, +S., Joshi, N. J., Julian, R., Kalashnikov, D., Kuang, Y., +Lee, K.-H., Levine, S., Lu, Y., Luu, L., Parada, C., Pastor, +P., Quiambao, J., Rao, K., Rettinghouse, J., Reyes, D., +Sermanet, P., Sievers, N., Tan, C., Toshev, A., Vanhoucke, +V., Xia, F., Xiao, T., Xu, P., Xu, S., Yan, M., and Zeng, +A. 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URL http://arxiv.org/abs/2203.11171. +arXiv:2203.11171 [cs]. +Wei, J., Wang, X., Schuurmans, D., Bosma, M., Ichter, B., +Xia, F., Chi, E. H., Le, Q. V., and Zhou, D. Chain of +Thought Prompting Elicits Reasoning in Large Language +Models. oct 2022. URL https://openreview.net/ +forum?id=_VjQlMeSB_J. +Zhou, D., Schärli, N., Hou, L., Wei, J., Scales, N., Wang, +X., Schuurmans, D., Bousquet, O., Le, Q., and Chi, E. +Least-to-Most Prompting Enables Complex Reasoning +in Large Language Models, may 2022. URL http:// +arxiv.org/abs/2205.10625. arXiv:2205.10625 [cs]. + +Faithful Chain-of-Thought Reasoning +A. Implementation Details +In all our experiments, +we use OpenAI GPT-3 (text-davinci-001 and text-davinci-002) and Codex +(code-davinci-001 and code-davinci-002) models through the Python API available at beta.openai.com, from Sept, +2022 to Jan, 2023. The inference cost per example is $0 for all Codex models since they are in limited beta period, and +$0.01 - $0.03 for GPT-3 models depending on the dataset. It takes 2-15 seconds to run inference on one example with Codex +models under a rate limit of 150,000 tokens/minute, and 1-8 seconds with GPT-3 models under 250,000 tokens/minute, also +depending on the dataset. For example, on the GSM8K test set of 1,319 examples, it takes 3.5h to finish the inference with +Codex and 2.3h with GPT-3. +We use the following hyper-parameters throughout all experiments: +temperature: 0.0 for greedy decoding, 0.4 for self-consistent decoding; +max_tokens: 1000; +n: 1 for greedy decoding, 40 for self-consistent decoding; +frequency_penalty: 0; +presence_penalty: 0. +Any unspecified hyper-parameters are set to the default value on https://beta.openai.com/docs/api-reference/ +completions/create. +B. Extended Results and Analysis +In this section, we present more results that do not fit into the main text. +B.1. Examples of Unfaithful CoT Output +Figure 7 shows examples of unfaithful output from the CoT method (Wei et al., 2022) on three datasets: GSM8K (MWP), +StrategyQA (Multi-hop QA), and SayCan (Planning). Note that here we only show model outputs that are apparently +unfaithful. In other cases, there is no known inconsistency between the reasoning chain and the answer, but this still does +not imply that the reasoning chain is a faithful explanation, since the ground truth (how the model predicts the answer) is +unknown. +Q: John plans to sell all his +toys and use the money to +buy video games. He has 13 +lego sets and he sells them +for $15 each. He ends up +buying 8 videogames for $20 +each and has $5 left. How +many lego sets does he still +have? +A: He started with 13 lego sets. He sold +them for $15 each. So he must have +gotten 13 * $15 = $195 from selling +them. He used this money to buy 8 +video games for $20 each. This means +he spent 8 * $20 = $160 on the video +games. He had $5 left, so in total he +must have had $195 + $5 = $200. The +answer is 0. +Math Word Problem (GSM8K) +Input +CoT output +Visit the table and the +counter. +The user has asked me to visit the table +and the counter, I will do so in that order. +1. find(table), 2. pick(table), 3. +find(counter), 4. put(counter), 5. done() +Planning (SayCan) +CoT output +Q: Could a bee hummingbird +balance a scale with a single +pea on it? +Multi-hop QA (StrategyQA) +A: The bee hummingbird is the smallest +bird in the world. The average weight of +a bee hummingbird is about 2 grams. A +single pea weighs about 0.1 grams. +Thus, a bee hummingbird could balance +a scale with a single pea on it. So the +answer is yes. +CoT output +Input +Input +Figure 7. Examples of unfaithful output from CoT prompting (Wei et al., 2022) on three datasets. The answer (green) does not follow +from the reasoning chain (blue). +The GSM8K example is explained in Section 1. In the StrategyQA example, though the reasoning chain correctly identifies +that a hummingbird weighs much more than a pea, the answer is still “yes”; in the SayCan example, the reasoning chain +only mentions “visit the table and the counter”, but the plan contains unnecessary “pick” and “put” operations. + +Faithful Chain-of-Thought Reasoning +Wei et al. (2022) claim that CoT “provides an interpretable window into the behavior of the model, suggesting how it might +have arrived at a particular answer and providing opportunities to debug where the reasoning path went wrong". As we +have pointed out in Section 1, since CoT does not guarantee faithfulness, how the model arrives at the answer could differ +drastically from what is shown in the reasoning chain. Furthermore, it is still hard for the user to debug the model: even if +they manually correct the reasoning chain and let the model regenerate the answer, it might still be wrong, since there is no +causality between the reasoning chain and the answer. +B.2. Few-shot SOTA Sources +The published few-shot SOTA results we compare to in Section 5 are from the following studies: +GSM8K, SVAMP, MultiArith, ASDiv, AQuA, StrategyQA: Wang et al. (2022); +SayCan, Date Understanding, Sports Understanding: (Wei et al., 2022); +CLUTRR: No existing work reports few-shot performance on CLUTRR with K up to 10. +C. Extended Analysis +C.1. Ablation Study +Table 3 shows the full results of the ablation study from Section 6.1. +Table 3. Ablation study results that accompany Figure 5. We report accuracy when we remove different parts of the prompt. +Exemplars +GSM8K +Date +SayCan +CLUTRR +Full +71.6 +80.8 +87.4 +58.9 +No rationale +74.8 +83.0 +87.4 +50.9 +No NL but nudge +72.3 +80.2 +87.4 +39.6 +No NL +72.0 +79.7 +89.3 +8.3 +No solver +21.4 +57.9 +90.3 +39.5 +C.2. Effect of LM +In this analysis, we want to answer the question: how much does the choice of LM matter? All experiments above are +done using code-davinci-002. Here we examine the effect of using different LMs as the translator, as shown in Figure 8 +and Table 4. Clearly, code-davinci-002 is far superior to all other models. While the exact differences between these +closed-source models are not yet clear, we speculate the following causes. Given that our prompt is a mixture of code and +NL comments: +code-davinci-002 is pretrained on NL and then code; +code-davinci-001 is pretrained on code only, which might explain its inability to work with NL comments; +text-davinci-001 is pretrained on NL only, which might explain its inability to work with code; +text-davinci-002 is pretrained on both NL and code and receives further instruction tuning in NL, which might have +drifted it from code again. +Table 4. Accuracy of Faithful COT with different LMs as the Translator, accompanying Figure 8. Due to the prompt length limit, +text-davinci-001 only allows us to run experiments on CLUTRR. +Exemplars +GSM8K +Date +SayCan +CLUTRR +code-davinci-002 +71.6 +80.8 +89.3 +58.9 +text-davinci-002 +62.1 +76.6 +79.6 +43.9 +code-davinci-001 +26.5 +43.2 +66.0 +23.4 +text-davinci-001 +N/A +N/A +N/A +17.3 + +Faithful Chain-of-Thought Reasoning +GSM8K +Date +SayCan +CLUTRR +0 +20 +40 +60 +80 +100 +Accuracy +code-davinci-002 +text-davinci-002 +code-davinci-001 +text-davinci-001 +Figure 8. Accuracy of Faithful COT with different LMs as the Translator. Due to the prompt length limit, text-davinci-001 only allows +us to experiment on CLUTRR. +Table 5. Accuracy change after enforcing different constraints on the generation. The “None” row shows the original performance +without any constraint (from Table 1). Each row below adds a different set of constraints: G stands for “graph validity”, O for “no +over-dependency”, and U for “no under-dependency”. Results are on all MWP datasets under self-consistent decoding. +Constraint +GSM8K +SVAMP +MultiArith +ASDiv +AQuA +None +79.5 +88.9 +98.8 +81.9 +61.4 ++ G +0.0 +0.0 +0.0 +-0.1 +-0.8 ++ O +-0.9 +-0.1 +-0.1 ++0.4 +-3.9 ++ U +-1.0 +-3.6 +0.0 +-1.2 ++1.2 ++ GO +-1.7 +-0.4 +-0.1 ++0.2 +-3.9 ++ GU +-1.0 +-3.7 +0.0 +-1.2 ++0.8 ++ OU +-4.0 +-5.4 +-0.1 +-2.6 +-4.3 ++ GOU +-5.0 +-5.9 +-0.1 +-3.2 +-5.5 +C.3. Enforcing Constraints +Since our generated reasoning chain contains structured components (e.g., dependency graphs), another natural question +to ask is: will it be helpful to enforce certain constraints on the generation? Using MWP datasets as a case study, we +examine the effect of three such constraints: +Graph validity. The dependency graph must be a Directed Cyclic Graph (DAG), e.g., it is not allowed for a subquestion to +depend on itself. +No over-dependency. The code cannot depend on any variable that its corresponding subquestion has not mentioned, e.g. +in Figure 4, since Q5 says “depend on 4”, then the corresponding code should not use the variable eggs_in_dozen, since it +is not the output of Q4. +No under-dependency. The code must depend on all variables that its corresponding subquestion has mentioned, e.g. in +the same example, since Q5 says “depend on 4”, then the corresponding code must use the variable eggs_in_dozen. +We investigate the effect of adding constraints on the generations under self-consistent decoding. Starting with our original +results (without any constraint), we add a different set of constraints at each time and report the accuracy change in Table 5. +Individually, the graph validity constraint results in little to no change in the performance, but the other two constraints +lead to a more unstable change–mostly a decrease–across datasets. Adding two or more constraints further lowers the +performance in almost all cases except on MultiArith (the easiest dataset), revealing the tradeoff between accuracy and +satisfying the constraints. It also indicates that a proportion of generations (1.0% to 8.9%) in our existing results do not +satisfy all constraints. However, it may still be worth enforcing some of these constraints (e.g., graph validity) at the cost of +performance, in order for users to better control and interact with the model. +D. Broader Impacts +With the recent success of generative large LMs, they are now being used to solve complex reasoning problems. When +using the output of an LM for reasoning, there is a danger that if the reasoning appears realistic, then the final answer or + +Faithful Chain-of-Thought Reasoning +conclusion will also be considered reliable. As we highlighted in Figure 7, this is often not true, since an LM may produce a +reasoning chain that looks plausible, but the final answer is still wrong. This work is a step in the direction of making the +use of LMs more trustworthy by using the LM for just expressing its reasoning in a symbolic program and executing the +program independently. In this work, we have ensured the faithfulness of the reasoning chain w.r.t how the final answer +is produced in a variety of domains, but admittedly the Translation phase is still opaque. Therefore, our pipeline is still +not entirely interpretable and can sometimes produce erroneous answers, which may pose a risk for users that rely on +our method for decision making. Another potential impact of our work is that since the reasoning chain interleaves NL +comments and symbolic programs, it may allow users without background knowledge in programming to easily understand +the model output and debug the model when it makes errors. +E. Dataset Details +E.1. Statistics +We show the dataset details in Table 6, including the statistics, the number of few-shot exemplars used in the prompt, and +example inputs and outputs. +In particular, we notice that in one of our baselines Wei et al. (2022), the reported number of exemplars used in the prompt is +inconsistent between the main text (10) and the appendix (6). To ensure fair comparison, we rerun the baseline with 10 +exemplars for our results in Table 1, which is what we use for our method. +Table 6. Datasets used for evaluation. “# Shot” stands for the number of few-shot examples in the prompt (following Wei et al. (2022)) +and “# Test” stands for the number of test examples. +Domain +Dataset +# Shot +# Test +Example +Math +Word +Problems +GSM8K +8 +1,319 +Q: Natalia sold clips to 48 of her friends in April, and then she sold half as many clips in +May. How many clips did Natalia sell altogether in April and May? +A: 72 +SVAMP +8 +1,000 +Q: Each pack of dvds costs 76 dollars. If there is a discount of 25 dollars on each pack. +How much do you have to pay to buy each pack? +A: 51 +MultiArith +8 +600 +Q: For Halloween Debby and her sister combined the candy they received. Debby had 32 +pieces of candy while her sister had 42. If they ate 35 pieces the first night, how many +pieces do they have left? +A: 39 +ASDiv +8 +2,096 +Q: Seven red apples and two green apples are in the basket. How many apples are in the +basket? +A: 9 +AQuA +8 +254 +Q: A car finishes a journey in 20 hours at the speed of 60 km/hr. If the same distance is to +be covered in 10 hours, how much speed does the car gain? +A: “120 kmph” +Multi-hop +QA +StrategyQA +6 +2,290 +Q: Did Aristotle use a laptop? +A: False +Date +Understanding +10 +359 +Q: Yesterday was April 30, 2021. What is the date tomorrow in MM/DD/YYYY? +A: “05/02/2021” +Sports +Understanding +10 +977 +Q: Is the following sentence plausible: “Lebron James hit the turnaround jumper”? +A: True +Planning +SayCan +7 +103 +Q: Could you get me a drink with caffeine? +A: +“1.find(redbull) +2.pick(redbull) +3.find(user) +4.put(redbull) +5.done().” +Logical +Inference +CLUTRR +8 +1,042 +Q: [Carlos] is [Clarence]’s brother. [Carlos] and his sister, [Annie], went shopping. [Annie] +asked her mom [Valerie] if she wanted anything, but [Valerie] said no. How is [Valerie] +related to [Clarence]? +A: “mother” +E.2. URLs and Licenses +We use the same distribution of datasets following Wei et al. (2022): +Math Word Problems + +Faithful Chain-of-Thought Reasoning +• GSM8K (Cobbe et al., 2021): https://github.com/openai/grade-school-math, MIT license: https://github. +com/openai/grade-school-math/blob/master/LICENSE. +• SVAMP (Patel et al., 2021): https://github.com/arkilpatel/SVAMP, MIT license: https://github.com/ +arkilpatel/SVAMP/blob/main/LICENSE. +• MultiArith (Roy & Roth, 2015), license: CC BY 4.0. +• ASDiv (Miao et al., 2020): https://github.com/chaochun/nlu-asdiv-dataset. +• AQuA (Ling et al., 2017): https://github.com/deepmind/AQuA, license: https://github.com/deepmind/AQuA/ +blob/master/LICENSE. +Multi-hop QA +• StrategyQA (Geva et al., 2021): we use the open-domain setting (question-only set) from (BIG-Bench collaboration, +2021): https://github.com/google/BIG-bench/tree/main/bigbench/benchmark_tasks/strategyqa. +• Date Understanding and Sports Understanding from BIG-Bench (BIG-Bench collaboration, 2021): Apache License +v.2: https://github.com/google/BIG-bench/blob/main/LICENSE. +Planning +• SayCan (Ahn et al., 2022): SayCan dataset can be accessed at https://say-can.github.io/ under CC BY 4.0 +license. +Logical Reasoning +• CLUTRR (Sinha et al., 2019): https://github.com/facebookresearch/clutrr, license: https://github.com/ +facebookresearch/clutrr/blob/main/LICENSE. +E.3. Data Cleaning +We perform manual cleaning on ASDiv, Date Understanding, Sports Understanding, and SayCan as we discover a number +of annotation issues. In our experiment, we rerun all baselines on our cleaned version of the test sets. They are provided in +the Supplementary Materials to assist future research. +Specifically, we clean each of the datasets as follows: +ASDiv: We start with the test set used by Wei et al. (2022), which removes all questions with float-valued and string-valued +answers. However, in their released version, we notice an error in the answer extraction step for questions with more than +one value in the answer (e.g., “what is the width and length of X?”, where the answer consists of two values). In their +implementation, only the first value is extracted as the ground truth answer, which is then compared against model outputs. +This might artificially inflate the final accuracy. To fix this, we extract all values in the answer as a set and compare model +outputs against it. +Date Understanding: We find a number of wrong answers in the test set. For example, for the question “Jane and John +married on Jan 2, 1958. It is their 5-year anniversary today. What is the date today in MM/DD/YYYY?”, the provided +answer is “01/02/1961”, whereas the correct answer should be “01/02/1963”. We manually correct these answers, and the +resulting test set has the same number of examples as the original one. +Sports Understanding: We notice a few ambiguities with the Sports Understanding dataset. For instance, running out of +bounds is illegal in many sports. The phrase "Domantas Sabonis ran out of bounds" is labeled as implausible, however, +Domantas Sabonis is a basketball player, and basketball players can indeed run out of bounds on the court. We remove 8 +questions with such action-based ambiguities. Additionally, since the release of this dataset, a few new athletes have risen to +fame with identical names to those mentioned in the dataset. For example, the question "Chris Paul struck out the side" is +implausible, as the referenced “Chris Paul” is a famous basketball player. However, “Chris Paul” is also the name of a new +MLB baseball player, in which case this statement is plausible. We remove 5 questions with such name-based ambiguities. + +Faithful Chain-of-Thought Reasoning +SayCan: We discover a few issues in the test set: (1) the environment setup (e.g., the list of objects, the list of locations, +and the initial location of each object) is not the same for all examples; (2) the annotation of the ground truth answer is +often incomplete (i.e., for a given task like “visit all locations”, there exist many possible plans in terms of the order of +locations visited, but not all of them are included in the annotation); (3) there are ambiguous descriptions in certain queries, +for example, in “Could you get me something refreshing?”, it is unclear what drinks are considered “refreshing”. For these +questions, we complete the annotation whenever possible, and filter out the rest. The resulting test set contains 103 examples +out of the original 120. +E.4. Dataset Splits +As stated in Section 4.1, we use the official splits whenever possible: training set for exemplar selection, validation set for +prompt tuning, and test set for evaluation. In cases where they are available, we adopt the following strategies for each +dataset: +GSM8K: it only has training and test sets. We form the validation set by randomly sampling 1,000 examples from the +training set. +Other MWP datasets: for AQuA, we use the official training/validation/test split. For the other datasets, only the test sets +are used, since we have the same prompt for GSM8K and them. +Date Understanding and Sports Understanding: they only have test sets. We follow Wei et al. (2022) to select the same +number of examples from the test set to form the few-shot prompt and use the remaining examples as a new test set. +SayCan: Following Wei et al. (2022), we manually write 7 few-shot exemplars, since no training set is provided. We +evaluate the models on our cleaned version of the test set, described in the previous subsection. +CLUTRR: this dataset is split into multiple folds. There is a training fold with K ∈ {2, 3} (where K is the number of +intermediate steps required to reach the answer), and one test fold for each K from 2 to 10. We construct the few-shot +prompt using exemplars from the training fold, and test our method on the concatenation of all test folds. +F. Error Analysis Details +Section 6.3 detailed error analysis and conclusions for GSM8K. In this section, we continue our error analyses for the rest of +the datasets we evaluate in the paper. +F.1. GSM8K +Table 7 lists generated code for the semantic understanding error example cited in Section 6.3. +Table 7. Generated code for question in Appendix F. +# 1. How many pounds will Martin lose per week if he eats Cheerios every day for breakfast? (independent, support: ["he’ll lose 1.25 +pounds/week"]) +pounds_lost_cheerios = 1.25 +# 2. How many pounds will Martin gain per week if he eats donuts every day for breakfast? (independent, support: ["he’ll gain 1.75 +pounds/week"]) pounds_gained_donuts = 1.75 +# 3. How many weeks are there in 5 weeks? (independent, support: ["External knowledge: there are 7 days in a week"]) +weeks_in_5_weeks = 5 +# 4. How many pounds will Martin lose in 5 weeks if he eats Cheerios every day for breakfast? (depends on 1 and 3, support: []) +pounds_lost_cheerios_5_weeks = pounds_lost_cheerios * weeks_in_5_weeks +# 5. How many pounds will Martin gain in 5 weeks if he eats donuts every day for breakfast? (depends on 2 and 3, support: []) +pounds_gained_donuts_5_weeks = pounds_gained_donuts * weeks_in_5_weeks +# 6. What will be the difference in his weight at the end of 5 weeks between the two breakfast options? (depends on 4 and 5, support: []) +difference_5_weeks = pounds_gained_donuts_5_weeks - pounds_lost_cheerios_5_weeks +# 7. Final Answer: What will be the difference in his weight at the end of 5 weeks between the two breakfast options? (depends on 6, +support: []) +answer = difference_5_weeks + +Faithful Chain-of-Thought Reasoning +F.2. StrategyQA +As mentioned in Section 5, Figure 9 shows the error type distribution on a sample of 70 instances from StrategyQA, where +we specifically compare the cases where the prediction of CoT is correct whereas ours is wrong. +Figure 9. Error analysis for StrategyQA. For a detailed description of the error categories, see Section F.2. +There are 8 error types, from most to least frequent: +Syntax (29%): these are syntax-related errors that result in an invalid Datalog program. For example, the LM redefines a +relation that is previously declared, uses an invalid string comparison operation, generates an ungrounded variable in the +answer definition, and so on. With further pretraining on Datalog, we believe Codex is able to overcome these errors to a +reasonable extent. +Infinite Loop (24%): the LM generates an infinite loop of subquestions, which are oftentimes rather philosophical. For +example, for the question “Are potatoes native to the European continent?”, the LM generates a list of 40 subquestions +(potentially more, if not truncated by the token limit): +1. Where are potatoes native to? +Potatoes are native to the Andes. +2. Which continent is the Andes in? +The Andes is in South America. +3. Which continent is South America in? +South America is in the Americas. +4. Which continent is the Americas in? +The Americas is in the Western Hemisphere. +5. Which hemisphere is the Western Hemisphere in? +The Western Hemisphere is in the Earth. +6. Which continent is the Earth in? +The Earth is in the Solar System. +...... +11. Which continent is the Universe in? +The Universe is in the Multiverse. +12. Which continent is the Multiverse in? +The Multiverse is in the Omniverse. +13. Which continent is the Omniverse in? +The Omniverse is in the Metaverse. +14. Which continent is the Metaverse in? +The Metaverse is in the Hyperverse. +...... +39. Which continent is the Quattuordecaverse in? + +Unknown +1.4 % +Knowledge Representation 2.9 % +Answer Definition 5.7 % +Syntax Errors +28.6 % +Wrong Knowledge10.0 % +Wrong 10.0 % +Reasoning +Wrong 17.1 % +Subquestions +24.3 % +Infinite LoopFaithful Chain-of-Thought Reasoning +The Quattuordecaverse is in the Quindecaverse. +40. Which continent is the Quindecaverse in? +We notice that this type of errors also happen if the prompt does not include Datalog code but just the NL subquestions, but +much less often. +Wrong Subquestions (17%): The LM generates an irrelevant subquestion to the original question. For instance, given the +question, “Would toast for a vegan have margarine instead of butter?”, one subquestion that is generated is “What is the +difference between a vegan and a vegetarian?”, which does not help with reaching the final answer. +Wrong Reasoning (10%): The subquestions are correct, but the reasoning from the retrieved facts to the final answer is +problematic. For example, to answer the question, “Do solo pianists require a conductor?”, the model correctly generates +the relevant subquestions and their respective answers: +1. What is the definition of a solo pianist? +A solo pianist is a pianist who performs alone. +2. What is the definition of a conductor? +A conductor is a person who leads an orchestra. +3. What is the definition of an orchestra? +An orchestra is a large instrumental ensemble that contains string, brass, woodwind, and percussion instruments. +But in deriving the final answer, the reasoning is wrong: “Now, we derive the final answer: Do solo pianists require a +conductor? The answer is Yes only if a solo pianist is not a conductor.” +Wrong Knowledge (10%): the LM fails to retrieve the correct knowledge to answer the subquestions. For example, +given the original question “Is the largest city in New Mexico also known as Yootó?”, the model correctly generates the +subquestions “What is the largest city in New Mexico?” (answer: Albuquerque) and “Is Albuquerque also known as Yootó?”. +However, when answering the second subquestion, it retrieves a wrong piece of knowledge (“Albuquerque is also known as +Yootó.”, whereas in reality, it should be “Santa Fe” that is known as Yootó). +Answer Definition (6%): In our prompt, we always derive the answer in the format of “The answer is Yes only if ...”, which +is followed by a Datalog rule containing conditions that should be satisfied for the answer to be true. However, the LM +sometimes generates this as “The answer is No only if ...”, which outputs the reversed answer. +Knowledge Representation (3%): The retrieved knowledge in NL is correct, but the representation of it in Datalog is +wrong. For example, for the piece of knowledge “The Lucy Show is not the same TV series as JAG (TV series)”, the model +represents it as follows: +.decl Same_TV_series(TV_series1:symbol, TV_series2:symbol) +Same_TV_series("The Lucy Show", "JAG (TV series)").” +which actually means the reverse (they are the same). +Unknown (1%): There is a very small proportion of errors (1 out of 70) where we are unsure of the cause. Specifically, we +expect the solver to output True, but it outputs False instead. +F.3. Date Understanding +Unlike GSM8K, we only have 69 errors out of the 359 test examples, so we annotate them all, as shown in Figure 10. +The error categories for date understanding are similar to GSM8K, except that we do not see any generation errors in the +samples, but we see questions with ambiguous phrasing allowing both the gold and predicted answers to be correct based on +interpretation. +F.4. SayCan +Since SayCan only has 120 test examples and Faithful CoT produces 7 errors, we annotate all 7 of them, as shown in +Figure 11. These 7 examples can be categorized into two types: + +Faithful Chain-of-Thought Reasoning +Figure 10. Error analysis for Date Understanding. For a detailed description of the error categories, see Section F.3. +Figure 11. Error analysis for SayCan. For a detailed description of the error categories, see Section F.4. +Additional Subgoals (64%): Cases where the model generated unnecessary subgoals in the decomposition of the original +task, leading the planner astray. This is illustrated by the request “Clear the jalapeno chips off the counter”: +(:goal +(and +(not (at jalapeno-chips counter)) +(not (at jalapeno-chips table)) +(not (at jalapeno-chips trash)) +(not (at jalapeno-chips bowl)) +(not (at jalapeno-chips user)) +) +) +Wrong Object (36%): Here the model generates the wrong object/object types in the goal. For example, a request such as +“I opened a pepsi earlier. How would you bring me an open can?” fails because the model generates actions with water +instead of Pepsi. +F.5. CLUTRR +For CLUTRR, we group all error cases by K, the number of steps in their gold reasoning chain, as a proxy for problem +complexity, and perform importance sampling on these groups to select 100 examples. Our annotation of these examples +reveals 5 error categories, as shown in Figure 12: +Inversed Relation (41%): This stands out as the majority of the errors. These errors are caused by the reversal of directional +relationships for the actors in the problem, i.e., predicting “mother” or “nephew” when the answer is “daughter” or “uncle” +respectively. +Wrong Relation (30%): Here the model extracts the relation incorrectly (not even the inverse). For example, for the +subquestion “How is [Donald] related to [Jason]?” with the correctly identified support “[Jason] is father of their father”, the + +Invalid Graph 4.3 % +Missed Subquestion +Wrong Subquestions 1.4 % +18.8 % +Wrong 15.9 % +Gold Label +4.3 % +Ambiguous Problem +Statement +Wrong Code 55.1 %Wrong Object +36.4 % +Additional Subgoals +63.6 %Faithful Chain-of-Thought Reasoning +Figure 12. Error analysis for CLUTRR. For a detailed description of the error categories, see Section F.5. +model produces relation(Donald, Jason) = son when the correct relation should be “grandson”. +Nonexistent Relation (4%): The model hallucinates a non-existent relation (e.g. “adopted” for daughter). +Wrong Path (12%): Here, the model does not generate a correct reasoning path from target entity A to target entity B in the +question. +Wrong Gold Label (13%): These are annotation errors in the CLUTRR dataset. In one example, for the sentence, +“[Gloria] asked her mother [Laura] if she could go outside and play with her friends.”, the annotation says Laura is Gloria’s +grandmother. +G. Prompts +Due to the space limit, we show one exemplar in the prompt for each dataset here. Our full prompts can be found in the +Supplementary Materials. +Among all the MWP datasets, our prompt for AQuA is different from the rest, because the answers are in a multiple-choice +format instead of integers. To produce a multiple-choice answer, we take a two-step approach by first producing a numerical +answer in the same way as for the other math datasets. Then, we perform an additional step of converting the numerical +answer into an answer choice by again prompting the language model to generate which answer choice is closest to the +previously produced numerical answer. An exemplar of this 2-step prompt is shown in Table 8. + +Nonexistent Relation 4.0 % +Wrong Path 12.0 % +Wrong Relation +30.0 % +Wrong 13.0 % +Gold Label +Inversed Relation 41.0 %Faithful Chain-of-Thought Reasoning +Table 8. An exemplar from our prompt for AQuA. +EXEMPLAR FOR AQUA +Step 1: Answer Prediction +# Question: In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 +km/hr and the time of flight increased by 30 minutes. The duration of the flight is: +# Answer option: [’A)1 hour’, ’B)2 hours’, ’C)3 hours’, ’D)4 hours’, ’E)5 hours’] +# Write Python code to solve the following questions. Store your result as a variable named ’answer’. +# 1. What was the duration of the flight? (independent, support: ["The duration of the flight is"]) +duration = Symbol(’duration’, positive=True) +# 2. What is the delay of the flight? (independent, support: ["the time of flight increased by 30 minutes"]) +delay = 30 / 60 +# 3. What was the total flight distance? (independent, support: ["In a flight of 600 km"]) +total_distance = 600 +# 4. What was the original speed? (depends on 1 and 3, support: ["External knowledge: speed is distance over time"]) +original_speed = total_distance / duration +# 5. What was the reduced speed? (depends on 1, 2, and 3, support: []) +reduced_speed = total_distance / (duration + delay) +# 6. What was the duration of the flight if the original speed was 200 km/hr faster than the reduced speed? (depends on 4, 5, and 1, +support: []) +solution = solve_it(original_speed - reduced_speed - 200, duration) +answer = solution[duration] +Step 2: Multiple Choice Conversion +# Question: In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 +km/hr and the time of flight increased by 30 minutes. The duration of the flight is: +# Answer option: [’A)1 hour’, ’B)2 hours’, ’C)3 hours’, ’D)4 hours’, ’E)5 hours’] +# Prediction: 1.00000000000000 +# Closest Option: A +Table 9. An exemplar from our prompt for GSM8K, SVAMP, MultiArith, and ASDiv. +EXEMPLAR FOR GSM8K, SVAMP, MULTIARITH, AND ASDIV +# Q: There are 15 trees in the grove. Grove workers will plant trees in the grove today. After they are done, there will be 21 trees. How +many trees did the grove workers plant today? +# To answer this question, write a Python program to answer the following subquestions: +# 1. How many trees are there in the beginning? (independent, support: ["There are 15 trees"]) +trees_begin = 15 +# 2. How many trees are there in the end? (independent, support: ["there will be 21 trees"]) +trees_end = 21 +# 3. How many trees did the grove workers plant today? (depends on 1 and 2, support: []) +trees_today = trees_end - trees_begin +# 4. Final Answer: How many trees did the grove workers plant today? (depends on 3, support: []) +answer = trees_today + +Faithful Chain-of-Thought Reasoning +Table 10. An exemplar from our prompt for StrategyQA. +EXEMPLAR FOR STRATEGYQA +// Q: Would a pear sink in water? +// To answer this question, we answer the following subquestions: +// 1. What is the density of a pear? +// The density of a pear is about 0.6g/cm3. +// 2. What is the density of water? +// Water has a density of 1g/cm3. +// Then, we represent these answers in Datalog: +// 1. The density of a pear is about 0.6g/cm3. +.decl Has_density(Object:symbol, Density:float) +Has_density("pear", 0.6). +// 2. Water has a density of 1g/cm3. +Has_density("water", 1). +// Now, we derive the final answer: Would a pear sink in water? +// The answer is Yes only if the density of a pear is more than the density of water. +.decl Answer() +Answer() :- Has_density("pear", density1), Has_density("water", density2), density1 > density2. +.output Answer +Table 11. An exemplar from our prompt for Date Understanding. +EXEMPLAR FOR DATE UNDERSTANDING +# Q: Yesterday was April 30, 2021. What is the date tomorrow in MM/DD/YYYY? +# To answer this question, we write a program to answer the following subquestions: +# import relevant packages +from datetime import date, time, datetime +from dateutil.relativedelta import relativedelta +# 1. What is the date yesterday? (independent, support: ["Yesterday was April 30, 2021"]) +date_yesterday = date(2021,4,30) +# 2. What is the date today? (depends on 1, support: ["Yesterday was April 30, 2021"]) +date_today = date_yesterday + relativedelta(days=1) +# 3. What is the date tomorrow? (depends on 2, support: []) +date_tomorrow = date_today + relativedelta(days=1) +# 4. Final Answer: What is the date tomorrow in MM/DD/YYYY? (depends on 3, support: []) +answer = date_tomorrow.strftime("%m/%d/%Y") +Table 12. An exemplar from our prompt for Sports Understanding. +EXEMPLAR FOR SPORTS UNDERSTANDING +# Q: Is the following statement plausible? Sam Darnold passed the puck +# To answer this question, write a Python program to answer the following subquestions: +# 1. Sam Darnold is a player in which sport? (independent, support: ["Sam Darnold is an NFL Quarterback", "NFL is the National +Football League"]) +player_sport = "football" +# 2. The phrase "passed the puck" implies playing which sport? (independent, support: ["Players pass the puck in hockey"]) +playing_sport = "hockey" +# 3. Is the following statement plausible? Sam Darnold passed the puck (depends on 1 and 2, support: ["Sam Darnold is an NFL +Quarterback", "NFL is the National Football League", "Players pass the puck in hockey"]) +plausibility = (player_sport == playing_sport) +# 4. Is the following statement plausible? Sam Darnold passed the puck (depends on 3, support: []) +answer = int(plausibility) + +Faithful Chain-of-Thought Reasoning +Table 13. An exemplar from our prompt for SayCan. +EXEMPLAR FOR SAYCAN +User query: Bring me something not sweet to eat. +Goal in PDDL: +(:goal +; I need to find a snack +(exists (?s - snack) +; it has to satisfy the following conditions +(and +; the snack must not be sweet +(not (is-sweet ?s)) +; bring it to the user +(at ?s user) +) +) +) +Table 14. An exemplar from our prompt for CLUTRR. +EXEMPLAR FOR CLUTRR +# Context: [Jason] always had some great adventure planned for his granddaughter [Guillermina] when she came to visit. So, naturally, +when [Myrna] told her daughter [Guillermina] that they would be going to visit [Jason] she could hardly contain herself. +# Question: How is [Jason] related to [Myrna]? +# To answer this question, we write a program to answer the following subquestions: +# 1. How is [Jason] related to [Guillermina]? (independent, support: "[Jason] always had some great adventure planned for his +granddaughter [Guillermina] when she came to visit.") +relation(Jason, Guillermina) = grandfather +# 2. How is [Guillermina] related to [Myrna]? (independent, support: "So, naturally, when [Myrna] told her daughter [Guillermina] that +they would be going to visit [Jason] she could hardly contain herself.") +relation(Guillermina, Myrna) = daughter +# 3. Final answer: How is [Jason] related to [Myrna]? (depends on 1, 2) +relation(Jason, Myrna) = relation(Jason, Guillermina) @ relation(Guillermina, Myrna) + diff --git a/ntFQT4oBgHgl3EQfqDbZ/content/tmp_files/load_file.txt b/ntFQT4oBgHgl3EQfqDbZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..85027cbd008c0ccc1f04d9d9a6acb84d56077ffb --- /dev/null +++ b/ntFQT4oBgHgl3EQfqDbZ/content/tmp_files/load_file.txt @@ -0,0 +1,1845 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf,len=1844 +page_content='Faithful Chain-of-Thought Reasoning Qing Lyu * 1 Shreya Havaldar * 1 Adam Stein * 1 Li Zhang 1 Delip Rao 1 Eric Wong 1 Marianna Apidianaki 1 Chris Callison-Burch 1 Abstract While Chain-of-Thought (CoT) prompting boosts Language Models’ (LM) performance on a gamut of complex reasoning tasks, the generated rea- soning chain does not necessarily reflect how the model arrives at the answer (aka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' faithful- ness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We propose Faithful CoT, a faithful-by- construction framework that decomposes a rea- soning task into two stages: Translation (Natural Language query → symbolic reasoning chain) and Problem Solving (reasoning chain → an- swer), using an LM and a deterministic solver respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We demonstrate the efficacy of our approach on 10 reasoning datasets from 4 diverse domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' It outperforms traditional CoT prompt- ing on 9 out of the 10 datasets, with an average accuracy gain of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 on Math Word Problems, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 on Planning, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 on Multi-hop Question Answer- ing (QA), and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 on Logical Inference, under greedy decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Together with self-consistency decoding, we achieve new state-of-the-art few- shot performance on 7 out of the 10 datasets, showing a strong synergy between faithfulness and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Introduction Complex reasoning tasks, such as commonsense reasoning and math reasoning, have long been the Achilles heel of LMs (Bengio, 2019), until a recent line of work on Chain- of-Thought (CoT) reasoning (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=') brought striking performance gains, by prompt- ing an LM to generate a reasoning chain along with the answer, given only a few in-context exemplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In addition to performance improvement, CoT is also claimed to “provide an interpretable window into the be- havior of the model” (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' However, it lacks one fundamental property of explanation, faith- fulness, meaning “an explanation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', the reasoning chain) Equal contribution 1Department of Computer and Information Science, University of Pennsylvania, Philadelphia, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 1Code and data to be released at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/ veronica320/Faithful-COT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Q: John plans to sell all his toys and use the money to buy video games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He has 13 lego sets and he sells them for $15 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He ends up buying 8 videogames for $20 each and has $5 left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many lego sets does he still have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: He started with 13 lego sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He sold them for $15 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' So he must have gotten 13 * $15 = $195 from selling them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He used this money to buy 8 video games for $20 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This means he spent 8 * $20 = $160 on the video games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He had $5 left, so in total he must have had $195 + $5 = $200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The answer is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Input CoT output Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An example of unfaithful output from CoT prompting (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022) on GSM8K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The answer (green) does not follow from the reasoning chain (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' should accurately represent the reasoning process behind the model’s prediction” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', how the model arrives at the final answer) (Jacovi & Goldberg, 2020, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In existing CoT methods, the final answer does not necessarily follow from the previously generated reasoning chain, so there is no guarantee on faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Figure 1 exemplifies such an unfaithful generation from our inspection of the model output from Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022) on GSM8K: the answer “0” is not even mentioned in the reasoning chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This, along with more examples from Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1, illustrates that existing CoT methods do not provide true interpretability of how the model predicts the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The lack of faithfulness in CoT can be dangerous in high- stake applications because it may mislead people into believ- ing that the model is inherently interpretable, while there is indeed no causal relationship between the reasoning chain and the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Even worse, when an unfaithful explana- tion looks plausible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', convincing to humans) (Jacovi & Goldberg, 2020), this makes it easier for people (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', le- gal practitioners) to over-trust the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', a recidivism predictor) even if it has implicit biases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', against racial minorities) (Pruthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Slack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' To address this concern, we propose Faithful CoT, a faithful-by-construction framework where the answer is the result of deterministically executing the reasoning chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Specifically, we break down a complex reasoning task into two sequential stages: Translation and Problem Solving (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' During Translation, an LM translates a Natural Language (NL) query into a reasoning chain, which inter- leaves NL and a task-dependent Symbolic Language (SL), such as Python, Datalog, or Planning Domain Definition Language (PDDL), as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Then, in the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='13379v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='CL] 31 Jan 2023 Faithful Chain-of-Thought Reasoning Query (NL) Translation Language Model Deterministic Solver Reasoning Chain (NL + SL) Problem Solving Answer Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An overview of our 2-stage pipeline, consisting of Trans- lation, where an LM translates a query (in NL/Natural Language) into a reasoning chain (which interleaves NL and SL/Symbolic Language), and Problem Solving, where an external solver ex- ecutes the reasoning chain to derive the answer, thus ensuring faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Problem Solving stage, the reasoning chain is executed by a deterministic solver, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', a Python/Datalog interpreter, or a PDDL planner, to derive the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We evaluate our approach on 10 reasoning datasets from 4 diverse domains: Math Word Problems (MWP), Planning, Multi-hop QA, and Logical Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We compare it with standard prompting (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2020) and CoT prompting, using the same underlying LM (Codex (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2021)) and the same decoding strategies (greedy and self-consistent decoding (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Results show that on a majority of the datasets, our approach outperforms both baselines, with an average accuracy gain over CoT of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 on MWP, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 on Planning, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 on Multi-hop QA, and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 on Logical Inference, using greedy decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Together with self-consistent decoding, we achieve new state-of-the-art few-shot performance on 7 out of the 10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Our key contributions are as follows: (a) We propose Faithful CoT, a faithful-by-construction prompting framework, which decomposes reasoning into Translation and Problem Solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The reasoning chain in- terleaves user-understandable natural language comments and executable symbolic language programs, thus providing interpretability of how the model arrives at the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (b) Our approach is generalizable to multiple domains be- yond arithmetic reasoning and simple symbolic reasoning, thanks to its flexible integration with any choice of SL and external solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We set the new SOTA performance on 7 out of the 10 reasoning datasets, showing a strong synergy between faithfulness and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (c) We provide an extensive analysis of the strengths and weaknesses of our method, showing its robustness to the choice of exemplars as well as the critical role of the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Related Work Faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In interpretability, faithfulness (also called fidelity or reliability) means that an explanation should “ac- curately represent the reasoning process behind the model’s prediction”, which is a fundamental requirement of an expla- nation (Harrington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Gilpin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Jacovi & Goldberg, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 It should be con- trasted with plausibility (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' persuasiveness or under- standability), which refers to “how convincing an expla- nation is to humans” (Herman, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Jacovi & Goldberg, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In the context of CoT prompting, a faithful reasoning chain needs to accurately reflect how the model arrives at the final answer, whereas a plausible reasoning chain is one that looks reasonable and coherent to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The vanilla CoT (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022) generates the reasoning chain in pure NL, which may often look plausible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' nevertheless, the final answer does not need to causally follow from the reasoning chain, thus not guaranteeing faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Chain-of-Thought-style prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In CoT-style prompting, given a complex question Q, an LM is prompted to generate a reasoning chain C along with the final answer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Specifically, the prompt consists of a few examples of (Q, C, A) triples, called in-context exemplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This allows pre-trained LMs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2020)) to solve unseen questions with much higher accuracy than standard prompting, where the exemplars do not contain the reason- ing chain C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We create a taxonomy of existing CoT-style prompting meth- ods into three types: all-at-once, ensemble-based, and modu- larized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' All-at-once prompting means that the LM produces C and A as one continuous string, without any dependencies or constraints in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Scratchpad (Nye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2021), the vanilla CoT (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), and “Let’s think step by step” (Kojima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), are all examples of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Ensemble-based prompting is designed to overcome the local optimality issue of the one-shot generation in previous methods by sampling multiple (C, A) pairs and choosing the best answer via strategies like majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Ex- amples include Self-Consistent CoT (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), Minerva (Lewkowycz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), and DIVERSE (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), which differ mainly in the voting granularity and the underlying LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Modularized methods break down Q into subproblems and then conquer them individually (Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In particular, Least- 2Note that this differs from the notion of faithfulness in the Natural Language Generation (NLG) literature, primarily in what constitutes the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In interpretability, we talk about the faithfulness of an explanation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' the model’s underlying reasoning mechanism – the ground truth is usually opaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In NLG, we talk about the faithfulness of the generated text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', a translated sentence, or a summary) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' some explicit source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', the source sentence, or the full document) – the ground truth is transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Faithful Chain-of-Thought Reasoning Would a pear sink in water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in the parking lot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' I spilled my coke on the table, could you throw it away and bring something to clean with?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' [Gabrielle] drove her daughter [Dorothy] to the hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=" [Dorothy]'s son [Vincent] showed up shortly after." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How is [Vincent] related to [Gabrielle]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many cars are there in the beginning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["there are 3 cars in the parking lot"]) n_cars_begin = 3 # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many cars arrive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["2 more cars arrive"]) n_cars_arrive = 2 # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Final answer: How many cars are in the parking lot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 1, 2) n_cars_total = n_cars_begin + n_cars_arrive // 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the density of a pear?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // The density of a pear is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 g/cm^3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the density of water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // Water has a density of 1 g/cm^3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // Then, we represent these answers in Datalog: // 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The density of a pear is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 g/cm^3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='decl Has_density(Object:symbol, Density:float) Has_density("pear", 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Water has a density of 1 g/cm^3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Has_density("water", 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // Now, we derive the final answer: Would a pear sink in water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // The answer is Yes only if the density of a pear is more than the density of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='decl Answer() Answer() :- Has_density("pear", density1), Has_density("water", density2), density1 > density2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='output Answer No 5 grandson 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' find(coke) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' pick(coke) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' find(trash) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' put(coke) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' find(sponge) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' pick(sponge) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' find(user) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' put(sponge) Query Reasoning Chain Answer Goal in PDDL: (:goal ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' I need to do both things (and ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' first throw away the coke (at coke trash) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' then bring the user a sponge as a cleaning tool (at sponge user) ) ) # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How is [Vincent] related to [Dorothy]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: " [Dorothy]\'s son [Vincent] showed up shortly after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='") relation(Vincent, Dorothy) = son # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How is [Dorothy] related to [Gabrielle]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: "[Gabrielle] drove her daughter [Dorothy] to the hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='") relation(Dorothy, Gabrielle) = daughter # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Final answer: How is [Vincent] related to [Gabrielle]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 1, 2) relation(Vincent, Gabrielle) = relation(Vincent, Dorothy) @ relation(Dorothy, Gabrielle) Deterministic Solver Translator (LM) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Examples from each task (Math Word Problems, Multi-hop QA, Planning, Logical Inference) showing our 2-stage Translation and Problem Solving pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' to-Most prompting (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022) uses an LM to first reduce the question to subquestions and then sequentially answer them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' However, the reasoning chain is entirely in NL, so there is still no faithfulness guarantee that the answer follows from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We note that our work is concurrent with Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022) and Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022), both generating the reasoning chain in Python code and calling a Python interpreter to derive the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' While we do not compare with them empirically since they are not yet published, we do want to highlight the following differences: (a) We demonstrate the general- izability of our approach to multiple symbolic languages beyond Python and multiple domains beyond arithmetic reasoning and simple symbolic reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (b) In particular, we innovatively recast a diverse set of realistic tasks (Plan- ning, Multi-hop QA, and Logical Inference) into a symbolic representation, which allows us to tackle them with a single framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (c) Our reasoning chain interleaves NL and SL in a structured fashion, which allows the user to better understand and potentially interact with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Method Our method, Faithful CoT, is a 2-stage pipeline, as seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Like previous CoT-style work, our prompt consists of (Q, C, A) triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Notable differences lie in our unique interleaving of NL (natural language) and SL (symbolic language) in C, as well as the way we derive the final answer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In the Translation stage, given a complex query Q in NL, we prompt an LM to translate it into a reasoning chain C, which interleaves NL and a task-specific SL (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Python, Datalog, or PDDL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 In the Problem Solving stage, we call a deterministic external solver, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', a Python interpreter, a Datalog executor, or PDDL planner, depending on the task, to obtain the answer A from the reasoning chain C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' As shown in Figure 3, we define CNL to be the NL com- ponent (black) and CSL to be the SL component (blue) in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Though we separate the two components notationally, they are interleaved in the generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Using this approach, C is guaranteed to be a faithful model explanation, since our final A is the result of deterministically executing CSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Moreover, CNL allows the user to better understand the 3Our prompts can be found in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Faithful Chain-of-Thought Reasoning reasoning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 We apply this method to 4 types of complex reasoning tasks: MWP, Multi-hop QA, Planning, and Logical Infer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Next, we will illustrate how our method works for each of them, with examples from Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Math Word Problems (MWP) Given a grade-school math question Q written in NL (“If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in the parking lot?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', shown in green in Figure 3), we want to obtain A as a real-valued number (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In the Translation stage, we prompt the LM to take in Q and generate a reasoning chain C, which interleaves CNL and CSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Specifically, the CNL component consists of three types of information: (a) Subquestions: Q is broken down into multiple smaller- scale subquestions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' how many cars are there in the beginning?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' how many cars arrive?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', and “3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' how many cars are in the parking lot?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (b) Dependency Graph: Each subquestion can either be answered directly via context (subquestions 1 and 2 are “independent”) or rely on answers to previous subquestions (subquestion 3 “depends on 1 and 2”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (c) Rationales: Each subquestion is accompanied with ra- tionale(s) to support the answer (the “support” field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The rationales can be either a subset of the original context (“2 more cars arrive”) or any external knowledge (“there are 7 days in a week”) relevant to the subquestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Each subquestion and its corresponding dependencies and rationales inform the subsequent generation of CSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In our example in Figure 3, CSL consists of Python code generated to answer each subquestion in CNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' During the Problem Solving stage, we execute CSL using our solver, a Python interpreter, to derive A (5 cars in the end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Multi-hop QA Given a complex question Q that involves multiple steps of reasoning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “Would a pear sink in water?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', shown in red in Figure 3), we want to obtain the answer A as a Boolean value or string value variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Similar to our MWP task formulation, C interleaves CNL (NL comments), and CSL (symbolic program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Depending on the nature of the task, the format of the reasoning chain C is slightly different: for some datasets, the LM first generates all subquestions and their answers in NL, and then represents these answers as SL to derive A (see Figure 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' for others, the LM interleaves the NL subquestions and the SL program, similar to the case of MWP (see Table 11 and Table 12 for examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 4While no constraints are enforced between CNL and CSL in our main experiments, we analyze this in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In terms of SL, we use both Python and Datalog, also de- pending on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' As Multi-hop QA problems involve multi-step reasoning to solve, CSL often utilizes Boolean algebra and string comparisons (in Python) along with rela- tion definitions and logic programming (in Datalog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We use their corresponding interpreter as our deterministic solver to execute CSL and obtain A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In the example from Figure 3, the LM first generates the sub- questions, “1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the density of a pear?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' and “2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the density of water?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', which are individually answered in NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The answers (“Water has a density of 1g/cm3”) are converted to Datalog statements (Has_density(“water”, 1)), which are then combined to formalize the truth condi- tion of the final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Finally, we execute the Datalog program to determine that a pear would not sink in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Planning In a user-robot interaction scenario, given a household task query Q from a user, we want to come up with a plan of actions A that the robot should take in order to accomplish the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, in Figure 3, given user query “I spilled my coke on the table, could you throw it away and bring something to clean with?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', a possible plan can be “find(coke), pick(coke), find(trash), put(coke) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='..”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In the Translation stage, an LM translates Q into C, consisting of CNL (which breaks down Q into subtasks) and CSL (which represents the subtasks as a symbolic goal in PDDL5 — a language to define and solve classical planning problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Figure 3 shows this translation, with CSL in blue and CNL in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Finally, we call a PDDL Planner as the determin- istic solver to obtain A, a plan to accomplish the goal CSL under the predefined scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Logical Inference Given a logical inference problem Q written in NL, we want to obtain A as a string-valued variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For exam- ple, the CLUTRR (Sinha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2019) dataset involves in- ferring the family relationship (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “grandson”) between two people from a short story (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “[Gabrielle] drove her daughter [Dorothy] to the hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' [Dorothy]’s son [Vin- cent] showed up shortly after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How is [Vincent] related to [Gabrielle]?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', shown in yellow in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' During the Translation stage, we prompt the LM to generate C, consist- ing of CNL and CSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Similar to previous tasks, CNL breaks down Q into subquestions (“How is [Vincent] related to [Dorothy]” and “How is [Dorothy] related to [Gabrielle]”), as well as provide input extracts as rationales to support the answer (“[Dorothy]’s son [Vincent] showed up shortly after”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Each subquestion in CNL is answered in CSL via a logical expression representing the relation between 5https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='org/wiki/Planning_Domain_ Definition_Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A goal is a special construct in PDDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Faithful Chain-of-Thought Reasoning Q: There are 15 trees in the grove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Grove workers will plant trees in the grove today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' After they are done, there will be 21 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: We start with 15 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Later we have 21 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The difference must be the number of trees they planted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' So, they must have planted 21 - 15 = 6 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The answer is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (7 more examples) Q: Claire makes a 3 egg omelet every morning for breakfast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many dozens of eggs will she eat in 4 weeks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: Claire makes a 3 egg omelet every morning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In one week she will eat 3 * 7 = 21 eggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In 4 weeks she will eat 4 * 21 = 84 eggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The answer is 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Input Model Output Standard prompting Chain of Thought (COT) prompting (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022) Faithful COT prompting (our method) Q: There are 15 trees in the grove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Grove workers will plant trees in the grove today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' After they are done, there will be 21 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: The answer is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (7 more examples) Q: Claire makes a 3 egg omelet every morning for breakfast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many dozens of eggs will she eat in 4 weeks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: The answer is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Input Model Output # Q: There are 15 trees in the grove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Grove workers will plant trees in the grove today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' After they are done, there will be 21 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' # To answer this question, we write a Python program to answer the following subquestions: # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees are there in the beginning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["There are 15 trees"]) trees_begin = 15 # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees are there in the end?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["there will be 21 trees"]) trees_end = 21 # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 1 and 2, support: []) trees_today = trees_end - trees_begin # 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Final Answer: How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 3, support: []) answer = trees_today ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (7 more examples) # Q: Claire makes a 3 egg omelet every morning for breakfast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many dozens of eggs will she eat in 4 weeks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' # To answer this question, write a Python program to answer the following subquestions: # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many eggs are in one dozen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["External knowledge: there are 12 eggs in a dozen"]) eggs_in_dozen = 12 # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many eggs are in one omelet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["Claire makes a 3 egg omelet every morning"]) eggs_in_omelet = 3 # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many omelets does Claire make in one week?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["External knowledge: there are 7 days in a week"]) omelets_per_week = 7 # 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many eggs does Claire eat in one week?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 2 and 3, support: []) eggs_per_week = eggs_in_omelet * omelets_per_week # 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many eggs does Claire eat in 4 weeks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 4, support: ["How many dozens of eggs will she eat in 4 weeks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='"]) eggs_4_weeks = eggs_per_week * 4 # 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many dozens of eggs does Claire eat in 4 weeks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 5 and 1, support: []) dozens_4_weeks = eggs_4_weeks / eggs_in_dozen # 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Final Answer: How many dozens of eggs will she eat in 4 weeks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 6, support: []) answer = dozens_4_weeks Input Model Output Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The prompt for GSM8K (a Math Word Problems dataset) from standard, CoT (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), and Faithful CoT prompting (ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The ground-truth answer is 7, and only our method correctly computes the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' the mentioned entities, for example, relation(Vincent, Dorothy)=son denotes that Vincent is Dorothy’s son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In the Problem Solving stage, our solver is a simple logical infer- ence engine that relies on a set of transitivity rules provided by (Anonymous, 2022) among possible family relationships, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', son@daughter=grandson (the son of one’s daughter is one’s grandson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Our solver recursively applies these rules on CSL to derive A, and determine that Vincent is Gabrielle’s grandson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Experimental setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Datasets Here, we summarize the evaluation datasets used for each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We select the same number (6 to 10, depending on the task) of exemplars as in Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022) to form our few-shot prompt, which can be found in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Unless otherwise stated, we use the official splits: training set for exemplar selection, validation set for prompt tuning, and test set for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 Math Word Problems (MWP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We follow Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022) and consider the same five MWP benchmarks: 6See Appendix E for dataset statistics, examples, data cleaning method, splits, prompt construction strategy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' GSM8K (Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2021), SVAMP (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2021), MultiArith (Roy & Roth, 2015), ASDiv (Miao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2020), and AQuA (Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For all datasets, the input question is phrased in NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The answer is a string-valued mathematical expression for AQuA, and one or more inte- ger(s) for all other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We use the same 8-shot prompt for all datasets except AQuA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Multi-hop QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We consider the three datasets: Strate- gyQA (Geva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2021), a dataset of open-domain ques- tions that require an implicit multi-step strategy to answer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “Did Aristotle use a laptop?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' involves answering “1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' When did Aristotle live?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' When was the laptop in- vented?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', and “3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Is #2 before #1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Date Understanding from BIG-bench (BIG-Bench collaboration, 2021), which asks the model to infer a date from a context, by performing computation on relative periods of time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' and finally, Sports Understanding from BIG-bench, which asks the model to decide whether an artificially constructed statement related to sports is plausible or implausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Since the latter two datasets do not have a training set, we follow Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022) and select 10 examples from the test set to form the prompt and use the rest for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We use the SayCan dataset (Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), which assumes a scenario of a robot operating in a kitchen, helping the user with household tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “bring a coke Faithful Chain-of-Thought Reasoning Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Accuracy of different prompting methods on 10 reasoning datasets from 4 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We compare our method, Faithful CoT, with standard promoting (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2020), CoT prompting (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), as well as previously published few-shot SOTA results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The best results within each decoding strategy are in boldface, and the new SOTA results across all strategies are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Math Word Problems Planning Multi-hop QA Logical Inference Method GSM8K SVAMP MultiArith ASDiv AQuA SayCan StrategyQA Date Sport CLUTRR Few-shot SOTA 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 Greedy Decoding Standard 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 CoT 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 Faithful CoT (ours) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 Self-Consistency Decoding CoT 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7 Faithful CoT (ours) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 to the table”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' There are a number of locations and objects that the robot can interact with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The robot can only perform a fixed set of actions, including find, pick, and put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The task is to map a user query in NL to a plan of predefined actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Following Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022), we manually write 7 exemplars, since no training set is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Logical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We use the CLUTRR (Sinha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2019) benchmark described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The dataset has multiple splits based on the number of intermediate steps K required to reach the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We construct the prompt using 8 exemplars with K ∈ {2, 3}, and test the models on the remaining examples with K up to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Evaluation Metrics We evaluate the model performance with the accuracy of the final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Following previous work (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), for all MWP datasets (except AQuA) where the answer contains integer(s), a cor- rect answer is defined as the exact match between the pre- diction and the ground truth both rounded up to the near- est integer (with the math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='ceil() function in Python);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' for StrategyQA and Sports Understanding where the answer is a Boolean value, it is defined as the exact match be- tween the prediction and the ground truth both evaluated as a Boolean variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' for SayCan, the generated plan is considered correct if it is among the ground truth plans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' for all other datasets, we rely on the exact match between the prediction string and the ground truth string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Language Model In Translation, we always use OpenAI Codex (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2021) (code-davinci-002, with 175B parameters) as the underlying LM, since it is so far the only code-generation model with a public API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7 7See Appendix A for implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Baselines We compare our method to two other baselines, shown in Figure 4: standard few-shot prompting, popularized by Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2020), with demonstrations of only the ques- tion and the answer (green);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' and CoT prompting (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), which additionally provides a reasoning chain in NL (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We also show the published SOTA few-shot results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 All prompting methods are compared under two decoding strategies: greedy decoding, where the LM samples the most probable next token from the vocabulary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', temper- ature = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' and self-consistency decoding (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022), where the LM generates multiple reasoning chains and chooses the final chain based on majority voting on the evaluated answer (we use a temperature of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 and 40 gen- erations for all datasets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 We reproduce the baseline results ourselves in cases when they are not reported or when we clean the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Results Our results on all datasets are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We see that Faithful CoT outperforms CoT across most datasets and domains for both greedy and self-consistency decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' With greedy decoding, our method outperforms CoT on 9 of the 10 benchmark datasets spanning the 4 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' On average, it improves over CoT in the MWP domain by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4, in Planning by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9, in Multi-hop QA by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0, and in Logical Inference by a surprising 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Our method also outperforms CoT under self-consistency decoding on 8 out of the 10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Compared to greedy decoding, the average accuracy gain becomes larger for Planning (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 → 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9) and Logical Inference (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 → 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2), but smaller for MWP (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5) and Multi-hop QA (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 8See Appendix B for sources of the SOTA results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 9Note that we do not report the performance of standard prompt- ing with self-consistency decoding, since when the number of sam- pled outputs is large enough, this converges to standard prompting with greedy decoding (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Faithful Chain-of-Thought Reasoning GSM8K Date SayCan CLUTRR 0 20 40 60 80 100 Accuracy Full No rationale No NL but nudge No NL No solver Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Ablation study results: accuracy when we remove dif- ferent parts of the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' See Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Also, it is worth noting that our method achieves the new few-shot SOTA results on 7 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' On StrategyQA (from the multi-hop QA domain), however, the performance of our method is still far from CoT and even standard prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' To understand why, we specif- ically compare the examples where CoT makes a correct prediction but our method fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' As shown in Figure 9 in the Appendix, we find that the likely primary cause is the sparsity of Datalog in the pretraining data for Codex, as an overwhelming 29% of errors are syntax-related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Moreover, including Datalog in the prompt also interferes with NL generation, making it harder for Codex to produce relevant subquestions (17%), retrieve knowledge correctly (10%), and come up with valid reasoning from the knowledge to the answer (10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Another potential cause is the nature of the task, as the difficulty for many StrategyQA questions does not lie in symbolic operations (value/string comparisons) but rather in retrieving correct facts and chaining them up, which makes the advantages of our deterministic solver less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Still, with further pretraining on Datalog, we believe that there is room for improvement with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Analysis In this section, we analyze the role of different components in our pipeline, to better understand where its capabilities come from and where it still struggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='10 Unless otherwise stated, we choose one dataset from each domain for analysis – GSM8K, Date Understanding, SayCan, and CLUTRR – using greedy decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Ablation Study Given the strong performance of Faithful CoT, we now ad- dress a natural question: how much does each part of the prompt contribute to the accuracy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We perform an abla- tion study where we remove different parts of the prompt and see how the performance changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In addition to the original prompt (“Full”), we test four variations, illustrated with the example from Figure 4: 10See Appendix C for full results where figures do not show accuracy scores, and additional analysis of constraints and LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Robustness to the choice of exemplars across 6 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Exemplars GSM8K Date SayCan CLUTRR Set 0 (Table 1) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 Set 1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 Set 2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 Set 3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 Set 4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 Set 5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 Mean 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 No rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We remove the rationales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', everything in the brackets from the NL comments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “independent, support: [‘There are 15 trees’]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' No NL but nudge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We remove all NL comments except the “nudge” line: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “# To answer this question, we write a Python program to answer the following subquestions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' No NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We remove all NL comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' No solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Instead of calling the external solver, we add “Answer: {answer}” to the end of every exemplar and let the LM predict the answer itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Figure 5 shows the results of all prompt variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' On GSM8K, Date Understanding, and SayCan, NL comments contribute little to the performance, and sometimes even slightly hurt it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' On CLUTRR, however, their role is crucial, since the exclusion of each component (rationale, nudge, subquestions) results in a clear accuracy drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In particular, comparing No NL but nudge and No NL, the nudge line itself brings a striking improvement by 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The external solver relieves the burden of problem solving from the LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Without it, the accuracy suffers a huge decline on GSM8K, Date Understanding, and CLUTRR (-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8, - 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9, and -19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 respectively), while on SayCan it improves by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' One potential influencing factor is that SayCan might be too homogeneous, as it contains a set of only 3 predefined actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This can make the task relatively easy, which allows all model variants to achieve around 90% accuracy and renders the solver unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Another potential reason is the level of correspondence between the final answer and the reasoning chain for different datasets: as shown in Figure 3, the answer in SayCan is a sequence of actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', find(redbull)), each directly corresponding to one step in the reasoning chain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', at redbull trash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' However, the answer in the other three datasets is only a single number or string, which can only be derived after executing all the steps in the reasoning chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Therefore, the latter type of tasks further necessitates the presence of an external solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Robustness to Exemplars We now answer the next question: how much does the choice of exemplars matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' To do this, we annotate 20 examples in total, randomly sample k (7-10, depending on Faithful Chain-of-Thought Reasoning the dataset) to construct the prompt, and repeat the process five times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Table 2 shows the performance of all six runs, including the original (from Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The mean accuracy is close to the original (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 to +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2), still above the baselines by a large margin (7 to 17) on all datasets except the arguably easiest SayCan, considering the standard deviation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This strongly suggests that the benefits of Faithful CoT are minimally influenced by the choice of exemplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Error Analysis To further investigate where our method still fails, we in- spect 100 errors11 from model predictions on each of the four datasets and manually annotate the error categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We only present the results on GSM8K here, shown in Figure 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' see Appendix F for those on the other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We categorize the errors on GSM8K into 6 types, inversely sorted with frequency: Wrong Subquestion (49%): The LM produces a wrong NL subquestion, which eventually leads to the incorrect answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' While this is the majority error type in our sam- ple, it is worth noting that in a typical human-in-the-loop collaboration, these errors are easily fixable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Even if the user is unfamiliar with programming, they can inspect the NL subquestions and potentially correct the model error by simply deleting or editing a wrong subquestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wrong Code (24%): The NL subquestion is correct, but the code fails to answer the subquestion correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For ex- ample, the code uses a variable that has not been previously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Semantic Understanding Error (12%): The LM incor- rectly interprets certain semantic subtleties in the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This is the most complex and most interesting error cate- gory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, consider the following problem: If Martin eats Cheerios every day for breakfast, he’ll lose 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='25 pounds/week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' If he eats donuts ev- ery day for breakfast, he’ll gain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='75 pounds/week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What will be the difference in his weight at the end of 5 weeks between the two breakfast options?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The generated code, included in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1, does not assign opposite polarities (signs) for “pounds lost” vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' “pounds gained”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For other examples in this category, we notice errors like missing that a pair of something has 2 items in it, missing to subtract 2 for “two years ago” when it occurs as a subjunctive, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Fixing these errors, in general, will require more than providing additional exam- ples in the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Generation Cutoff (7%): The generation stops midway, 11To encourage sample diversity, we embed all the errors using text-davinci-002 and cluster the embeddings using spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This produces around 70 clusters of different sizes, from which we gather 100 samples using importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Error analysis for GSM8K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For a detailed description of the error categories, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' mainly due to the LM producing the same steps over and over again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' These errors could be easily detected in postpro- cessing and possibly fixed by re-prompting the LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wrong Gold Label (5%): We find 5 (out of our 100) exam- ples that are genuine annotation errors in the gold labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Missing Subquestion (3%): The LM misses a relevant sub- question needed for the rest of the reasoning chain to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' These errors are also potentially fixable via human-in-the- loop interaction, where the user can insert a subquestion into the reasoning chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Conclusion We propose Faithful CoT, a framework that decomposes complex reasoning into Translation and Problem Solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' During Translation, an LM produces a reasoning chain in the form of interleaved natural and symbolic language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The Problem-Solving stage calls an external solver that executes the reasoning chain and derives the final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This pro- cess guarantees that the reasoning chain is a faithful explana- tion of how the model arrives at the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We demonstrate the efficacy of our approach on 4 types of complex reasoning problems: Math Word Problems, Multi-hop QA, Planning, and Logical Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Our method sets new SOTA perfor- mance on 7 of the 10 datasets, while additionally providing a faithful explanation for the final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' These results give empirical evidence that improving model interpretability, by guaranteeing the faithfulness of an explanation, does not come at the expense of overall performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' in fact, we see a strong synergy in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Through a comprehensive analysis on the strengths and weaknesses of our method, we show its robustness to the choice of exemplars, the pivotal role of the solver, as well as frequent error patterns where it still struggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' One limitation of our work is that the Translation stage is still opaque, leaving an open question about whether it is possible to improve its faithfulness as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Moreover, it will be helpful to perform a human evaluation on the correctness of the generated reasoning chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Finally, the NL com- ments in the reasoning chain can serve as an interface for users without a programming background to interactively debug the model, which should be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Missing Subquestion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Wrong Gold Label 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Generation Cutoff 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Semantic 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Understanding Error 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Wrong Subquestion Wrong Code 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 %Faithful Chain-of-Thought Reasoning Acknowledgements This research is based upon work supported in part by the DARPA KAIROS Program (contract FA8750-19-2- 1004), the DARPA LwLL Program (contract FA8750-19- 2-0201), the IARPA BETTER Program (contract 2019- 19051600004), the IARPA HIATUS Program (contract 2022-22072200005), and the NSF (Award 1928631).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Ap- proved for Public Release, Distribution Unlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily repre- senting the official policies, either expressed or implied, of ODNI, DARPA, IARPA, NSF, or the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We appreciate the support from OpenAI on increasing the rate limit for the Codex API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We also thank Jiani Huang, Ziyang Li, Litao Yan, Andrew Head, and Mayur Naik for their valuable feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' References Ahn, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='11171 [cs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Schuurmans, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Bosma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Ichter, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Xia, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Chi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', and Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Chain of Thought Prompting Elicits Reasoning in Large Language Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' oct 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' URL https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='net/ forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='id=_VjQlMeSB_J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Schärli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Hou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Wei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Scales, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Schuurmans, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Bousquet, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', and Chi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Least-to-Most Prompting Enables Complex Reasoning in Large Language Models, may 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' URL http:// arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='org/abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='10625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='10625 [cs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Faithful Chain-of-Thought Reasoning A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Implementation Details In all our experiments, we use OpenAI GPT-3 (text-davinci-001 and text-davinci-002) and Codex (code-davinci-001 and code-davinci-002) models through the Python API available at beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com, from Sept, 2022 to Jan, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The inference cost per example is $0 for all Codex models since they are in limited beta period, and $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='01 - $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='03 for GPT-3 models depending on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' It takes 2-15 seconds to run inference on one example with Codex models under a rate limit of 150,000 tokens/minute, and 1-8 seconds with GPT-3 models under 250,000 tokens/minute, also depending on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, on the GSM8K test set of 1,319 examples, it takes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5h to finish the inference with Codex and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3h with GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We use the following hyper-parameters throughout all experiments: temperature: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 for greedy decoding, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 for self-consistent decoding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' max_tokens: 1000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' n: 1 for greedy decoding, 40 for self-consistent decoding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' frequency_penalty: 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' presence_penalty: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Any unspecified hyper-parameters are set to the default value on https://beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/docs/api-reference/ completions/create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Extended Results and Analysis In this section, we present more results that do not fit into the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Examples of Unfaithful CoT Output Figure 7 shows examples of unfaithful output from the CoT method (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022) on three datasets: GSM8K (MWP), StrategyQA (Multi-hop QA), and SayCan (Planning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Note that here we only show model outputs that are apparently unfaithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In other cases, there is no known inconsistency between the reasoning chain and the answer, but this still does not imply that the reasoning chain is a faithful explanation, since the ground truth (how the model predicts the answer) is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Q: John plans to sell all his toys and use the money to buy video games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He has 13 lego sets and he sells them for $15 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He ends up buying 8 videogames for $20 each and has $5 left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many lego sets does he still have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: He started with 13 lego sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He sold them for $15 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' So he must have gotten 13 * $15 = $195 from selling them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He used this money to buy 8 video games for $20 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This means he spent 8 * $20 = $160 on the video games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' He had $5 left, so in total he must have had $195 + $5 = $200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The answer is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Math Word Problem (GSM8K) Input CoT output Visit the table and the counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The user has asked me to visit the table and the counter, I will do so in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' find(table), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' pick(table), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' find(counter), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' put(counter), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' done() Planning (SayCan) CoT output Q: Could a bee hummingbird balance a scale with a single pea on it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Multi-hop QA (StrategyQA) A: The bee hummingbird is the smallest bird in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The average weight of a bee hummingbird is about 2 grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A single pea weighs about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Thus, a bee hummingbird could balance a scale with a single pea on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' So the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' CoT output Input Input Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Examples of unfaithful output from CoT prompting (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022) on three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The answer (green) does not follow from the reasoning chain (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The GSM8K example is explained in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In the StrategyQA example, though the reasoning chain correctly identifies that a hummingbird weighs much more than a pea, the answer is still “yes”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' in the SayCan example, the reasoning chain only mentions “visit the table and the counter”, but the plan contains unnecessary “pick” and “put” operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Faithful Chain-of-Thought Reasoning Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022) claim that CoT “provides an interpretable window into the behavior of the model, suggesting how it might have arrived at a particular answer and providing opportunities to debug where the reasoning path went wrong".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' As we have pointed out in Section 1, since CoT does not guarantee faithfulness, how the model arrives at the answer could differ drastically from what is shown in the reasoning chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Furthermore, it is still hard for the user to debug the model: even if they manually correct the reasoning chain and let the model regenerate the answer, it might still be wrong, since there is no causality between the reasoning chain and the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Few-shot SOTA Sources The published few-shot SOTA results we compare to in Section 5 are from the following studies: GSM8K, SVAMP, MultiArith, ASDiv, AQuA, StrategyQA: Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' SayCan, Date Understanding, Sports Understanding: (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' CLUTRR: No existing work reports few-shot performance on CLUTRR with K up to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Extended Analysis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Ablation Study Table 3 shows the full results of the ablation study from Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Ablation study results that accompany Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We report accuracy when we remove different parts of the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Exemplars GSM8K Date SayCan CLUTRR Full 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 No rationale 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 No NL but nudge 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 No NL 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 No solver 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Effect of LM In this analysis, we want to answer the question: how much does the choice of LM matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' All experiments above are done using code-davinci-002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Here we examine the effect of using different LMs as the translator, as shown in Figure 8 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Clearly, code-davinci-002 is far superior to all other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' While the exact differences between these closed-source models are not yet clear, we speculate the following causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Given that our prompt is a mixture of code and NL comments: code-davinci-002 is pretrained on NL and then code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' code-davinci-001 is pretrained on code only, which might explain its inability to work with NL comments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' text-davinci-001 is pretrained on NL only, which might explain its inability to work with code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' text-davinci-002 is pretrained on both NL and code and receives further instruction tuning in NL, which might have drifted it from code again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Accuracy of Faithful COT with different LMs as the Translator, accompanying Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Due to the prompt length limit, text-davinci-001 only allows us to run experiments on CLUTRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Exemplars GSM8K Date SayCan CLUTRR code-davinci-002 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 text-davinci-002 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 code-davinci-001 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 text-davinci-001 N/A N/A N/A 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 Faithful Chain-of-Thought Reasoning GSM8K Date SayCan CLUTRR 0 20 40 60 80 100 Accuracy code-davinci-002 text-davinci-002 code-davinci-001 text-davinci-001 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Accuracy of Faithful COT with different LMs as the Translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Due to the prompt length limit, text-davinci-001 only allows us to experiment on CLUTRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Accuracy change after enforcing different constraints on the generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The “None” row shows the original performance without any constraint (from Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Each row below adds a different set of constraints: G stands for “graph validity”, O for “no over-dependency”, and U for “no under-dependency”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Results are on all MWP datasets under self-consistent decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Constraint GSM8K SVAMP MultiArith ASDiv AQuA None 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 + G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 + O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 + U 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 + GO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 + GU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 + OU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 + GOU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Enforcing Constraints Since our generated reasoning chain contains structured components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', dependency graphs), another natural question to ask is: will it be helpful to enforce certain constraints on the generation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Using MWP datasets as a case study, we examine the effect of three such constraints: Graph validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The dependency graph must be a Directed Cyclic Graph (DAG), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', it is not allowed for a subquestion to depend on itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' No over-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The code cannot depend on any variable that its corresponding subquestion has not mentioned, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' in Figure 4, since Q5 says “depend on 4”, then the corresponding code should not use the variable eggs_in_dozen, since it is not the output of Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' No under-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The code must depend on all variables that its corresponding subquestion has mentioned, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' in the same example, since Q5 says “depend on 4”, then the corresponding code must use the variable eggs_in_dozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We investigate the effect of adding constraints on the generations under self-consistent decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Starting with our original results (without any constraint), we add a different set of constraints at each time and report the accuracy change in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Individually, the graph validity constraint results in little to no change in the performance, but the other two constraints lead to a more unstable change–mostly a decrease–across datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Adding two or more constraints further lowers the performance in almost all cases except on MultiArith (the easiest dataset), revealing the tradeoff between accuracy and satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' It also indicates that a proportion of generations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0% to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9%) in our existing results do not satisfy all constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' However, it may still be worth enforcing some of these constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', graph validity) at the cost of performance, in order for users to better control and interact with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Broader Impacts With the recent success of generative large LMs, they are now being used to solve complex reasoning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' When using the output of an LM for reasoning, there is a danger that if the reasoning appears realistic, then the final answer or Faithful Chain-of-Thought Reasoning conclusion will also be considered reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' As we highlighted in Figure 7, this is often not true, since an LM may produce a reasoning chain that looks plausible, but the final answer is still wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This work is a step in the direction of making the use of LMs more trustworthy by using the LM for just expressing its reasoning in a symbolic program and executing the program independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In this work, we have ensured the faithfulness of the reasoning chain w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='t how the final answer is produced in a variety of domains, but admittedly the Translation phase is still opaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Therefore, our pipeline is still not entirely interpretable and can sometimes produce erroneous answers, which may pose a risk for users that rely on our method for decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Another potential impact of our work is that since the reasoning chain interleaves NL comments and symbolic programs, it may allow users without background knowledge in programming to easily understand the model output and debug the model when it makes errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Dataset Details E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Statistics We show the dataset details in Table 6, including the statistics, the number of few-shot exemplars used in the prompt, and example inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In particular, we notice that in one of our baselines Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022), the reported number of exemplars used in the prompt is inconsistent between the main text (10) and the appendix (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' To ensure fair comparison, we rerun the baseline with 10 exemplars for our results in Table 1, which is what we use for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Datasets used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' “# Shot” stands for the number of few-shot examples in the prompt (following Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022)) and “# Test” stands for the number of test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Domain Dataset # Shot # Test Example Math Word Problems GSM8K 8 1,319 Q: Natalia sold clips to 48 of her friends in April, and then she sold half as many clips in May.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many clips did Natalia sell altogether in April and May?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: 72 SVAMP 8 1,000 Q: Each pack of dvds costs 76 dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' If there is a discount of 25 dollars on each pack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How much do you have to pay to buy each pack?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: 51 MultiArith 8 600 Q: For Halloween Debby and her sister combined the candy they received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Debby had 32 pieces of candy while her sister had 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' If they ate 35 pieces the first night, how many pieces do they have left?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: 39 ASDiv 8 2,096 Q: Seven red apples and two green apples are in the basket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many apples are in the basket?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: 9 AQuA 8 254 Q: A car finishes a journey in 20 hours at the speed of 60 km/hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' If the same distance is to be covered in 10 hours, how much speed does the car gain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: “120 kmph” Multi-hop QA StrategyQA 6 2,290 Q: Did Aristotle use a laptop?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: False Date Understanding 10 359 Q: Yesterday was April 30, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the date tomorrow in MM/DD/YYYY?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: “05/02/2021” Sports Understanding 10 977 Q: Is the following sentence plausible: “Lebron James hit the turnaround jumper”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: True Planning SayCan 7 103 Q: Could you get me a drink with caffeine?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: “1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='find(redbull) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='pick(redbull) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='find(user) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='put(redbull) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='done().” Logical Inference CLUTRR 8 1,042 Q: [Carlos] is [Clarence]’s brother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' [Carlos] and his sister, [Annie], went shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' [Annie] asked her mom [Valerie] if she wanted anything, but [Valerie] said no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How is [Valerie] related to [Clarence]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A: “mother” E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' URLs and Licenses We use the same distribution of datasets following Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022): Math Word Problems Faithful Chain-of-Thought Reasoning GSM8K (Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2021): https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/openai/grade-school-math, MIT license: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' com/openai/grade-school-math/blob/master/LICENSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' SVAMP (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2021): https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/arkilpatel/SVAMP, MIT license: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/ arkilpatel/SVAMP/blob/main/LICENSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' MultiArith (Roy & Roth, 2015), license: CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' ASDiv (Miao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2020): https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/chaochun/nlu-asdiv-dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' AQuA (Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2017): https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/deepmind/AQuA, license: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/deepmind/AQuA/ blob/master/LICENSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Multi-hop QA StrategyQA (Geva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2021): we use the open-domain setting (question-only set) from (BIG-Bench collaboration, 2021): https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/google/BIG-bench/tree/main/bigbench/benchmark_tasks/strategyqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Date Understanding and Sports Understanding from BIG-Bench (BIG-Bench collaboration, 2021): Apache License v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/google/BIG-bench/blob/main/LICENSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Planning SayCan (Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2022): SayCan dataset can be accessed at https://say-can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='io/ under CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Logical Reasoning CLUTRR (Sinha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', 2019): https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/facebookresearch/clutrr, license: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='com/ facebookresearch/clutrr/blob/main/LICENSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Data Cleaning We perform manual cleaning on ASDiv, Date Understanding, Sports Understanding, and SayCan as we discover a number of annotation issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In our experiment, we rerun all baselines on our cleaned version of the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' They are provided in the Supplementary Materials to assist future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Specifically, we clean each of the datasets as follows: ASDiv: We start with the test set used by Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022), which removes all questions with float-valued and string-valued answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' However, in their released version, we notice an error in the answer extraction step for questions with more than one value in the answer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', “what is the width and length of X?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', where the answer consists of two values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In their implementation, only the first value is extracted as the ground truth answer, which is then compared against model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This might artificially inflate the final accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' To fix this, we extract all values in the answer as a set and compare model outputs against it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Date Understanding: We find a number of wrong answers in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, for the question “Jane and John married on Jan 2, 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' It is their 5-year anniversary today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the date today in MM/DD/YYYY?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', the provided answer is “01/02/1961”, whereas the correct answer should be “01/02/1963”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We manually correct these answers, and the resulting test set has the same number of examples as the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Sports Understanding: We notice a few ambiguities with the Sports Understanding dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For instance, running out of bounds is illegal in many sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The phrase "Domantas Sabonis ran out of bounds" is labeled as implausible, however, Domantas Sabonis is a basketball player, and basketball players can indeed run out of bounds on the court.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We remove 8 questions with such action-based ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Additionally, since the release of this dataset, a few new athletes have risen to fame with identical names to those mentioned in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, the question "Chris Paul struck out the side" is implausible, as the referenced “Chris Paul” is a famous basketball player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' However, “Chris Paul” is also the name of a new MLB baseball player, in which case this statement is plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We remove 5 questions with such name-based ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Faithful Chain-of-Thought Reasoning SayCan: We discover a few issues in the test set: (1) the environment setup (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', the list of objects, the list of locations, and the initial location of each object) is not the same for all examples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2) the annotation of the ground truth answer is often incomplete (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', for a given task like “visit all locations”, there exist many possible plans in terms of the order of locations visited, but not all of them are included in the annotation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (3) there are ambiguous descriptions in certain queries, for example, in “Could you get me something refreshing?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', it is unclear what drinks are considered “refreshing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For these questions, we complete the annotation whenever possible, and filter out the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The resulting test set contains 103 examples out of the original 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Dataset Splits As stated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1, we use the official splits whenever possible: training set for exemplar selection, validation set for prompt tuning, and test set for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In cases where they are available, we adopt the following strategies for each dataset: GSM8K: it only has training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We form the validation set by randomly sampling 1,000 examples from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Other MWP datasets: for AQuA, we use the official training/validation/test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For the other datasets, only the test sets are used, since we have the same prompt for GSM8K and them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Date Understanding and Sports Understanding: they only have test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We follow Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022) to select the same number of examples from the test set to form the few-shot prompt and use the remaining examples as a new test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' SayCan: Following Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (2022), we manually write 7 few-shot exemplars, since no training set is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We evaluate the models on our cleaned version of the test set, described in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' CLUTRR: this dataset is split into multiple folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' There is a training fold with K ∈ {2, 3} (where K is the number of intermediate steps required to reach the answer), and one test fold for each K from 2 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We construct the few-shot prompt using exemplars from the training fold, and test our method on the concatenation of all test folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Error Analysis Details Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 detailed error analysis and conclusions for GSM8K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In this section, we continue our error analyses for the rest of the datasets we evaluate in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' GSM8K Table 7 lists generated code for the semantic understanding error example cited in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Generated code for question in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many pounds will Martin lose per week if he eats Cheerios every day for breakfast?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["he’ll lose 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='25 pounds/week"]) pounds_lost_cheerios = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='25 # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many pounds will Martin gain per week if he eats donuts every day for breakfast?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["he’ll gain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='75 pounds/week"]) pounds_gained_donuts = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='75 # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many weeks are there in 5 weeks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["External knowledge: there are 7 days in a week"]) weeks_in_5_weeks = 5 # 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many pounds will Martin lose in 5 weeks if he eats Cheerios every day for breakfast?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 1 and 3, support: []) pounds_lost_cheerios_5_weeks = pounds_lost_cheerios * weeks_in_5_weeks # 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many pounds will Martin gain in 5 weeks if he eats donuts every day for breakfast?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 2 and 3, support: []) pounds_gained_donuts_5_weeks = pounds_gained_donuts * weeks_in_5_weeks # 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What will be the difference in his weight at the end of 5 weeks between the two breakfast options?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 4 and 5, support: []) difference_5_weeks = pounds_gained_donuts_5_weeks - pounds_lost_cheerios_5_weeks # 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Final Answer: What will be the difference in his weight at the end of 5 weeks between the two breakfast options?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 6, support: []) answer = difference_5_weeks Faithful Chain-of-Thought Reasoning F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' StrategyQA As mentioned in Section 5, Figure 9 shows the error type distribution on a sample of 70 instances from StrategyQA, where we specifically compare the cases where the prediction of CoT is correct whereas ours is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Error analysis for StrategyQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For a detailed description of the error categories, see Section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' There are 8 error types, from most to least frequent: Syntax (29%): these are syntax-related errors that result in an invalid Datalog program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, the LM redefines a relation that is previously declared, uses an invalid string comparison operation, generates an ungrounded variable in the answer definition, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' With further pretraining on Datalog, we believe Codex is able to overcome these errors to a reasonable extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Infinite Loop (24%): the LM generates an infinite loop of subquestions, which are oftentimes rather philosophical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, for the question “Are potatoes native to the European continent?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', the LM generates a list of 40 subquestions (potentially more, if not truncated by the token limit): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Where are potatoes native to?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Potatoes are native to the Andes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is the Andes in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The Andes is in South America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is South America in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' South America is in the Americas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is the Americas in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The Americas is in the Western Hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which hemisphere is the Western Hemisphere in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The Western Hemisphere is in the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is the Earth in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The Earth is in the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='. 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is the Universe in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The Universe is in the Multiverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is the Multiverse in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The Multiverse is in the Omniverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is the Omniverse in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The Omniverse is in the Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is the Metaverse in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The Metaverse is in the Hyperverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='. 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is the Quattuordecaverse in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Unknown 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 % Knowledge Representation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 % Answer Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='7 % Syntax Errors 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 % Wrong Knowledge10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Wrong 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Reasoning Wrong 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 % Subquestions 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 % Infinite LoopFaithful Chain-of-Thought Reasoning The Quattuordecaverse is in the Quindecaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Which continent is the Quindecaverse in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' We notice that this type of errors also happen if the prompt does not include Datalog code but just the NL subquestions, but much less often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wrong Subquestions (17%): The LM generates an irrelevant subquestion to the original question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For instance, given the question, “Would toast for a vegan have margarine instead of butter?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', one subquestion that is generated is “What is the difference between a vegan and a vegetarian?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', which does not help with reaching the final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wrong Reasoning (10%): The subquestions are correct, but the reasoning from the retrieved facts to the final answer is problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, to answer the question, “Do solo pianists require a conductor?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', the model correctly generates the relevant subquestions and their respective answers: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the definition of a solo pianist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A solo pianist is a pianist who performs alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the definition of a conductor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' A conductor is a person who leads an orchestra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the definition of an orchestra?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An orchestra is a large instrumental ensemble that contains string, brass, woodwind, and percussion instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' But in deriving the final answer, the reasoning is wrong: “Now, we derive the final answer: Do solo pianists require a conductor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The answer is Yes only if a solo pianist is not a conductor.” Wrong Knowledge (10%): the LM fails to retrieve the correct knowledge to answer the subquestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, given the original question “Is the largest city in New Mexico also known as Yootó?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', the model correctly generates the subquestions “What is the largest city in New Mexico?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (answer: Albuquerque) and “Is Albuquerque also known as Yootó?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' However, when answering the second subquestion, it retrieves a wrong piece of knowledge (“Albuquerque is also known as Yootó.”, whereas in reality, it should be “Santa Fe” that is known as Yootó).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Answer Definition (6%): In our prompt, we always derive the answer in the format of “The answer is Yes only if .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='..”, which is followed by a Datalog rule containing conditions that should be satisfied for the answer to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' However, the LM sometimes generates this as “The answer is No only if .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='..”, which outputs the reversed answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Knowledge Representation (3%): The retrieved knowledge in NL is correct, but the representation of it in Datalog is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, for the piece of knowledge “The Lucy Show is not the same TV series as JAG (TV series)”, the model represents it as follows: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='decl Same_TV_series(TV_series1:symbol, TV_series2:symbol) Same_TV_series("The Lucy Show", "JAG (TV series)").” which actually means the reverse (they are the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Unknown (1%): There is a very small proportion of errors (1 out of 70) where we are unsure of the cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Specifically, we expect the solver to output True, but it outputs False instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Date Understanding Unlike GSM8K, we only have 69 errors out of the 359 test examples, so we annotate them all, as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The error categories for date understanding are similar to GSM8K, except that we do not see any generation errors in the samples, but we see questions with ambiguous phrasing allowing both the gold and predicted answers to be correct based on interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' SayCan Since SayCan only has 120 test examples and Faithful CoT produces 7 errors, we annotate all 7 of them, as shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' These 7 examples can be categorized into two types: Faithful Chain-of-Thought Reasoning Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Error analysis for Date Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For a detailed description of the error categories, see Section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Error analysis for SayCan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For a detailed description of the error categories, see Section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Additional Subgoals (64%): Cases where the model generated unnecessary subgoals in the decomposition of the original task, leading the planner astray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' This is illustrated by the request “Clear the jalapeno chips off the counter”: (:goal (and (not (at jalapeno-chips counter)) (not (at jalapeno-chips table)) (not (at jalapeno-chips trash)) (not (at jalapeno-chips bowl)) (not (at jalapeno-chips user)) ) ) Wrong Object (36%): Here the model generates the wrong object/object types in the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, a request such as “I opened a pepsi earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How would you bring me an open can?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' fails because the model generates actions with water instead of Pepsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' CLUTRR For CLUTRR, we group all error cases by K, the number of steps in their gold reasoning chain, as a proxy for problem complexity, and perform importance sampling on these groups to select 100 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Our annotation of these examples reveals 5 error categories, as shown in Figure 12: Inversed Relation (41%): This stands out as the majority of the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' These errors are caused by the reversal of directional relationships for the actors in the problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=', predicting “mother” or “nephew” when the answer is “daughter” or “uncle” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wrong Relation (30%): Here the model extracts the relation incorrectly (not even the inverse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For example, for the subquestion “How is [Donald] related to [Jason]?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' with the correctly identified support “[Jason] is father of their father”, the Invalid Graph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 % Missed Subquestion Wrong Subquestions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 % 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='8 % Wrong 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='9 % Gold Label 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='3 % Ambiguous Problem Statement Wrong Code 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='1 %Wrong Object 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='4 % Additional Subgoals 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6 %Faithful Chain-of-Thought Reasoning Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Error analysis for CLUTRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' For a detailed description of the error categories, see Section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' model produces relation(Donald, Jason) = son when the correct relation should be “grandson”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Nonexistent Relation (4%): The model hallucinates a non-existent relation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' “adopted” for daughter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wrong Path (12%): Here, the model does not generate a correct reasoning path from target entity A to target entity B in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Wrong Gold Label (13%): These are annotation errors in the CLUTRR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' In one example, for the sentence, “[Gloria] asked her mother [Laura] if she could go outside and play with her friends.”, the annotation says Laura is Gloria’s grandmother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Prompts Due to the space limit, we show one exemplar in the prompt for each dataset here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Our full prompts can be found in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Among all the MWP datasets, our prompt for AQuA is different from the rest, because the answers are in a multiple-choice format instead of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' To produce a multiple-choice answer, we take a two-step approach by first producing a numerical answer in the same way as for the other math datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Then, we perform an additional step of converting the numerical answer into an answer choice by again prompting the language model to generate which answer choice is closest to the previously produced numerical answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An exemplar of this 2-step prompt is shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Nonexistent Relation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Wrong Path 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Wrong Relation 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Wrong 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 % Gold Label Inversed Relation 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='0 %Faithful Chain-of-Thought Reasoning Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An exemplar from our prompt for AQuA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' EXEMPLAR FOR AQUA Step 1: Answer Prediction # Question: In a flight of 600 km, an aircraft was slowed down due to bad weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Its average speed for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The duration of the flight is: # Answer option: [’A)1 hour’, ’B)2 hours’, ’C)3 hours’, ’D)4 hours’, ’E)5 hours’] # Write Python code to solve the following questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Store your result as a variable named ’answer’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What was the duration of the flight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["The duration of the flight is"]) duration = Symbol(’duration’, positive=True) # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the delay of the flight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["the time of flight increased by 30 minutes"]) delay = 30 / 60 # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What was the total flight distance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["In a flight of 600 km"]) total_distance = 600 # 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What was the original speed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 1 and 3, support: ["External knowledge: speed is distance over time"]) original_speed = total_distance / duration # 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What was the reduced speed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 1, 2, and 3, support: []) reduced_speed = total_distance / (duration + delay) # 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What was the duration of the flight if the original speed was 200 km/hr faster than the reduced speed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 4, 5, and 1, support: []) solution = solve_it(original_speed - reduced_speed - 200, duration) answer = solution[duration] Step 2: Multiple Choice Conversion # Question: In a flight of 600 km, an aircraft was slowed down due to bad weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Its average speed for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The duration of the flight is: # Answer option: [’A)1 hour’, ’B)2 hours’, ’C)3 hours’, ’D)4 hours’, ’E)5 hours’] # Prediction: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='00000000000000 # Closest Option: A Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An exemplar from our prompt for GSM8K, SVAMP, MultiArith, and ASDiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' EXEMPLAR FOR GSM8K, SVAMP, MULTIARITH, AND ASDIV # Q: There are 15 trees in the grove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Grove workers will plant trees in the grove today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' After they are done, there will be 21 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' # To answer this question, write a Python program to answer the following subquestions: # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees are there in the beginning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["There are 15 trees"]) trees_begin = 15 # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees are there in the end?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["there will be 21 trees"]) trees_end = 21 # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 1 and 2, support: []) trees_today = trees_end - trees_begin # 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Final Answer: How many trees did the grove workers plant today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 3, support: []) answer = trees_today Faithful Chain-of-Thought Reasoning Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An exemplar from our prompt for StrategyQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' EXEMPLAR FOR STRATEGYQA // Q: Would a pear sink in water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // To answer this question, we answer the following subquestions: // 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the density of a pear?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // The density of a pear is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the density of water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // Water has a density of 1g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // Then, we represent these answers in Datalog: // 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The density of a pear is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='decl Has_density(Object:symbol, Density:float) Has_density("pear", 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Water has a density of 1g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Has_density("water", 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // Now, we derive the final answer: Would a pear sink in water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' // The answer is Yes only if the density of a pear is more than the density of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='decl Answer() Answer() :- Has_density("pear", density1), Has_density("water", density2), density1 > density2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='output Answer Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An exemplar from our prompt for Date Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' EXEMPLAR FOR DATE UNDERSTANDING # Q: Yesterday was April 30, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the date tomorrow in MM/DD/YYYY?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' # To answer this question, we write a program to answer the following subquestions: # import relevant packages from datetime import date, time, datetime from dateutil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='relativedelta import relativedelta # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the date yesterday?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["Yesterday was April 30, 2021"]) date_yesterday = date(2021,4,30) # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the date today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 1, support: ["Yesterday was April 30, 2021"]) date_today = date_yesterday + relativedelta(days=1) # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' What is the date tomorrow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 2, support: []) date_tomorrow = date_today + relativedelta(days=1) # 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Final Answer: What is the date tomorrow in MM/DD/YYYY?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 3, support: []) answer = date_tomorrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='strftime("%m/%d/%Y") Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An exemplar from our prompt for Sports Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' EXEMPLAR FOR SPORTS UNDERSTANDING # Q: Is the following statement plausible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Sam Darnold passed the puck # To answer this question, write a Python program to answer the following subquestions: # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Sam Darnold is a player in which sport?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["Sam Darnold is an NFL Quarterback", "NFL is the National Football League"]) player_sport = "football" # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' The phrase "passed the puck" implies playing which sport?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: ["Players pass the puck in hockey"]) playing_sport = "hockey" # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Is the following statement plausible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Sam Darnold passed the puck (depends on 1 and 2, support: ["Sam Darnold is an NFL Quarterback", "NFL is the National Football League", "Players pass the puck in hockey"]) plausibility = (player_sport == playing_sport) # 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Is the following statement plausible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Sam Darnold passed the puck (depends on 3, support: []) answer = int(plausibility) Faithful Chain-of-Thought Reasoning Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An exemplar from our prompt for SayCan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' EXEMPLAR FOR SAYCAN User query: Bring me something not sweet to eat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Goal in PDDL: (:goal ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' I need to find a snack (exists (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='s - snack) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' it has to satisfy the following conditions (and ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' the snack must not be sweet (not (is-sweet ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='s)) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' bring it to the user (at ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='s user) ) ) ) Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' An exemplar from our prompt for CLUTRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' EXEMPLAR FOR CLUTRR # Context: [Jason] always had some great adventure planned for his granddaughter [Guillermina] when she came to visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' So, naturally, when [Myrna] told her daughter [Guillermina] that they would be going to visit [Jason] she could hardly contain herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' # Question: How is [Jason] related to [Myrna]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' # To answer this question, we write a program to answer the following subquestions: # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How is [Jason] related to [Guillermina]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: "[Jason] always had some great adventure planned for his granddaughter [Guillermina] when she came to visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='") relation(Jason, Guillermina) = grandfather # 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' How is [Guillermina] related to [Myrna]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (independent, support: "So, naturally, when [Myrna] told her daughter [Guillermina] that they would be going to visit [Jason] she could hardly contain herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content='") relation(Guillermina, Myrna) = daughter # 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' Final answer: How is [Jason] related to [Myrna]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} +page_content=' (depends on 1, 2) relation(Jason, Myrna) = relation(Jason, Guillermina) @ relation(Guillermina, Myrna)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFQT4oBgHgl3EQfqDbZ/content/2301.13379v1.pdf'} diff --git a/odE2T4oBgHgl3EQfzwj2/content/tmp_files/2301.04135v1.pdf.txt b/odE2T4oBgHgl3EQfzwj2/content/tmp_files/2301.04135v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c00d2043608c8a47e6940448ee49cc3331e82e44 --- /dev/null +++ b/odE2T4oBgHgl3EQfzwj2/content/tmp_files/2301.04135v1.pdf.txt @@ -0,0 +1,674 @@ +Average localization of resonances on the quantum repeller +J. Montes,1, 2, ∗ Gabriel G. Carlo,3, † and F. Borondo1, 2, ‡ +1Departamento de Qu´ımica, Universidad Aut´onoma de Madrid, Cantoblanco, 28049–Madrid, Spain +2Instituto de Ciencias Matem´aticas (ICMAT), Cantoblanco, 28049–Madrid, Spain +3Comisi´on Nacional de Energ´ıa At´omica, CONICET, Departamento de F´ısica, +Av. del Libertador 8250, 1429 Buenos Aires, Argentina +(Dated: January 12, 2023) +There has been a very recent surge in the interest on the localization properties of resonances +associated to partially open (scattering) systems, which are of the greatest relevance when studying +resonant cavities like those used in microlasers for example. Very recently it has been found that no +localization is present in a scaled form of these states. Moreover, a new kind of scarring on structures +different from periodic orbits is described for non normalized resonances. In this paper, we analyze +the localization of a scaled distribution function based on the quantum repeller representation for +the partially open quantum tribaker map, a paradigmatic system. We find localization, which in the +generic case can be associated to the shortest periodic orbits in non trivial ways. Also, normalized +states present enhancements that could not be conclusively associated to periodic orbits and that +become more evident when looking at the repeller. These findings leave the door open for new +perspectives on recent theoretical developments. +I. +INTRODUCTION +Quantum scattering has been an active research field +since long ago, being optical cavities one of the main +focus of attention. There is a long history of both the- +oretical and experimental studies [1, 2], but also recent +experiments [3], in which a lot of questions have been +settled, but others still remain open. This situation cor- +responds classically to placing an opening in the system, +letting the trajectories that arrive at a region of phase +space escape. +This escape can be complete and then +we speak about a complete opening, or partial, where +we define a convenient reflectivity function in order to +take into account a partial confinement of these trajec- +tories inside the cavity. This mechanism gives rise to a +classical (conditionally) invariant distribution that cor- +responds to a (partial) fractal set having (multi) fractal +dimension(s). At the quantum level and for the complete +opening case, the fractal Weyl law is a well established +result, which shows that the number of long-lived states +(or resonances) scales with the Planck constant as ℏ−d/2, +where d+1 is the fractal dimension of the classical repeller +[4–7]. If the opening is partial [8], we have multifractal +measures [9] that characterize a conditionally invariant +distribution that extends over all the phase space and +the usual fractal Weyl law has to be redefined. This has +been done for the very important case of maps (these +systems embody all the features of the chaotic cavities, +but are much more amenable to theoretical studies) [10]. +The morphology of the corresponding eigenfunctions is +still not completely understood, being their localization +properties [11] a source of active developments nowadays. +∗ E–mail address: jmontes.3@alumni.unav.es +† E–mail address: carlo@tandar.cnea.gov.ar +‡ E–mail address: f.borondo@uam.es +Very recently there has been a renovated interest in +elucidating some open questions about the structure of +resonances associated to (partially) open systems. +In +[12] the average phase space distribution of resonances +in chaotic systems with escape was conjectured to be de- +scribed by a classical measure. This measure is condition- +ally invariant and uniformly distributed on sets with the +same temporal distance to the quantum resolved chaotic +saddle. A family of conditionally invariant classical mea- +sures was found to describe the multifractal phase space +distribution, product structure along stable and unsta- +ble directions, and the dependence on the decay rate of +resonances [13] (better behaved for long and short lived +ones). Moreover, in [14] the intensity statistics was found +to universally follow an exponential distribution in the +partially open chaotic standard map, baker map, and +a random matrix model. In [15] a local randomization +on phase space for the baker map with escape was intro- +duced to obtain a semiclassical description of resonances, +though some quantitative differences were found. Finally, +in [16] the resonances were conjectured to be a product +of an essentially classical conditionally invariant measure +and universal fluctuations. Notably, scarring is present +in almost all of them, although along segments of rays +but not along periodic orbits (POs). +On the other hand, there has been a complementary +line of research in order to study the properties of reso- +nances based on the localization phenomena on POs, the +so called scarring [17]. This led to the development of the +semiclassical theory of short POs for open quantum maps +[18]. This is a constructive approach based on short POs +contained in the classical repeller, which motivates a ba- +sis of scar functions able to expand the quantum repeller +[19, 20], which is suitable to express the quantum non- +unitary operators [21, 22]. We have recently extended +the short POs theory to partially open quantum maps +[23, 24]. +Can these two visions on the same problem be put +arXiv:2301.04135v1 [quant-ph] 9 Jan 2023 + +2 +together? Is scarring in the long lived resonances of par- +tially open systems essentially different from that occur- +ring along POs in the closed scenario? How are the lim- +its corresponding to completely open and closed systems +reached? Which is the effect of the normalization/scaling +of these resonances by their average in terms of localiza- +tion? In this work we try to answer these questions, and +for that purpose we study the localization properties of +the longest lived resonances of the (partially) open trib- +aker map. +We use the Husimi functions associated to +the right eigenvectors scaled using their average, and the +repeller representation of the left and right eigenvectors +scaled using the corresponding quantum repeller. In this +way, we are able to clearly show the localization present +when using the latter mathematical objects which reveal +themselves as a very adequate tool to understand the sta- +tistical properties of the resonances. Moreover, thanks to +them scarring on short POs is easily identified. +The organization of this paper is as follows: in Sec. II +we describe the (partially) open tribaker map and the +localization measures used. In Sec. III we apply these +definitions to unveil the localization properties of reso- +nances and how short POs could be apparently hidden +in the set of long lived resonances. The conclusions are +outlined in Sec. IV. +II. +SYSTEM AND MEASURES +Maps are a powerful and convenient tool to study clas- +sical and quantum chaos [25–27]. +In particular, open +maps on the 2-torus have an associated fractal repeller, +that is an invariant measure which lies at the intersection +of the forward and backwards trapped sets (nonescaping +trajectories in the past or future). +When the opening +is partial, some amount of them is reflected following +a reflectivity function that we will take as a constant +R ∈ (0 : 1) for simplicity. In this case, the corresponding +measure extends over all the phase space, but still shows +multifractality. +The quantum version of maps on the torus requires +taking ⟨q + 1|ψ⟩ += +ei2πχq⟨q|ψ⟩, and ⟨p + 1|ψ⟩ += +ei2πχp⟨p|ψ⟩, with χq, χp ∈ [0, 1). The dimension of the +Hilbert space is N = (2πℏ)−1, the semiclassical limit is +reached when N → ∞, and the evolution operator is a +N × N matrix. Position and momentum eigenstates are +given by |qj⟩ = |(j + χq)/N⟩ and |pj⟩ = |(j + χp)/N⟩ +with j ∈ {0, . . . , N − 1}, transforming as ⟨pk|qj⟩ += +1 +√ +N e−2iπ(j+χq)(k+χp)/N +≡ (Gχq,χp +N +). A partially open +quantum map is non-unitary and has N right eigenvec- +tors |ΨR +j ⟩ and N left ones ⟨ΨL +j | for each resonance (eigen- +value) zj, with ⟨ΨL +j |ΨR +k ⟩ = δjk and ⟨ΨR +j |ΨR +j ⟩ = ⟨ΨL +j |ΨL +j ⟩ +being the norm. +The classical tribaker map is defined as +B(q, p) = +� +� +� +(3q, p/3) +if 0 ≤ q < 1/3 +(3q − 1, (p + 1)/3) if 1/3 ≤ q < 2/3 +(3q − 2, (p + 2)/3) if 2/3 ≤ q < 1 +(1) +If we place an opening in the region 1/3 < q < 2/3 all +trajectories arriving at it will suffer an intensity modifi- +cation given by the reflectivity function. The quantum +closed counterpart with antiperiodic boundary conditions +χq = χp = 1/2 (preserving time reversal and parity) in +position representation is [28, 29] +U B = G−1 +N +� +� +GN/3 +0 +0 +0 +GN/3 +0 +0 +0 +GN/3 +� +� . +(2) +The corresponding partially open map is given by +P = +� +� +11N/3 +0 +0 +0 +√ +R 11N/3 +0 +0 +0 +11N/3 +� +� , +(3) +applied to Eq. (2), in such a way that the original sym- +metries are preserved. +Motivated by the classical repeller, we can define the +corresponding quantum object by means of an operator +ˆhj constructed in terms of the right |ΨR +j ⟩ and left ⟨ΨL +j | +eigenstates [19, 20] +ˆhj = |ΨR +j ⟩⟨ΨL +j | +⟨ΨL +j |ΨR +j ⟩ . +(4) +When the expectation values of these projectors are +evaluated on coherent states |q, p⟩, |⟨q, p|ˆhj|q, p⟩| +∝ +� +HR +j (q, p)HL +j (q, p), being HR,L +j +the Husimi functions of +the R, L resonances, which for closed (unitary) systems +are simply the Husimi functions of the eigenstates. We +define the quantum repeller as the average of the first j of +them, ordered by decreasing modulus of their eigenvalues +(|zl| ⩾ |zm| with l ≤ m): +ˆQj ≡ 1 +j +j +� +j′=1 +ˆhj′, +(5) +and denote their phase space representation by means of +coherent states as +hj(q, p) = |⟨q, p|ˆhj|q, p⟩| +(6) +Qj(q, p) = |⟨q, p| ˆQj|q, p⟩|, +(7) +where hj(q, p) will be simply called the repeller represen- +tation of the resonances. We have explicitly taken the +modulus since these are complex functions in general. +Also, for the sake of an easy comparison with the scaled +Husimis that will be defined next, we give an equal weight +to each resonance in the repeller sum, although weighting +them with the corresponding eigenvalues should provide +a more faithful representation of the longest lived sector +of the quantum map. It is worth noticing that, in this +work, we focus on the long lived set of resonances and this +sum runs on just a subset of the whole set. It is useful +to define [14] the scaled Husimi ˜HR +i and repeller ˜hi func- +tions as HR +i /
j and hi/Qj respectively, where the +average < · · · > is taken over this j subset of eigenstates. + +3 +We explore two measures of localization for both scaled +functions. +The first one closely follows the conjecture +of universal intensity statistics formulated in [14], i.e. +that these functions obey an exponential distribution +P(w)dw = e−wdw, where w = ˜HR +i (q, p) or w = ˜hi(q, p). +We define the corresponding localization measure Σϵ as +the deviation from the exponential behavior taken as the +distance between both curves at every point in the his- +tograms, for each case. +The second measure of localization is intrinsically +phase space motivated. In fact, the norm ratio µ is taken +to be [20] +µ(˜hi) = +� +∥˜hi∥1/∥˜hi∥2 +∥ρc∥1/∥ρc∥2 +�2 +. +(8) +We use ˜hi(q, p) in its definition but is also valid for +˜HR +i (q, p). A coherent state at an arbitrary position (q, p) +in phase space is the normalization factor ρc = |q, p⟩⟨q, p|, +with the phase space norm given by +∥˜hi∥γ = +�� +T 2 +˜hi(q, p)γdqdp +�1/γ +. +(9) +The norm ratio is also independent of the h normaliza- +tion, with a minimum value of 1 for a maximally localized +distribution (a coherent state) and a maximum value of +N/2 for the uniform distribution. +III. +LOCALIZATION IN PHASE SPACE +In order to investigate the average localization on the +quantum repeller defined in Sec. II, we have computed +the two previously defined measures, Σϵ and µ, for the +closed (R = 1), partially open (with R = 0.1 and 0.05), +and the completely open (R = 0) tribaker map. In the +calculations, we have used the 32 longest lived resonances +to evaluate ˜HR +i and ˜hi, which in the closed case are ar- +bitrarily selected. +Results for Σϵ (calculated for 1000 +points distributions and 3 selected values of i) are shown +in Fig. 1. As can be seen, for the closed system there is +essentially no difference between the Husimis of the right +eigenstates and the distributions taken on the repeller, +as expected from the fact that there are no differences +between the right and left eigenstates. More remarkably, +we see meaningful differences with respect to the local- +ization for the closed system in the case of the repeller +of the partially open maps, and also for the complete +opening. The low to moderate R values that have been +considered for the partially open cases produce apprecia- +ble changes in the behavior of this measure with respect +to the closed case. +It needs to be underlined that for +the Husimi distributions of the right eigenstates there is +no notable change from the closed scenario at any of the +finite reflectivities shown. This agrees with the univer- +sality conjecture of [14] and shows that our calculations +are in that regime. +0 +0.5 +1 +1.5 +2 +104 +0 +20 +40 +60 +80 +100 +FIG. 1. (Color online) Σϵ as a function of N for the closed (a), +partially open with R = 0.1 (b), R = 0.05 (c), and completely +open (d) tribaker map. The symbols ∗, △ and ◦ stand for the +6th, 10th and 16th eigenstates (ordered as indicated in the +main text), respectively. The gray (red) points correspond to +the measure calculated for ˜ +HR +i , while the black points for ˜hi. +Next, we consider the µ measure whose results are +shown in Fig. 2. Again, as expected, there is essentially +no difference here between the values for ˜HR +i and ˜hi in the +closed scenario. However, the localization is now better +detected than with the Σϵ measure for finite reflectivities +and also for the completely open case. In fact, though the +latter seems to be more localized, the results for partial +openings are almost at the same level than this limiting +case. Moreover, a greater localization with respect to the +closed case is also detected in the Husimi distributions of +the right resonances not only for the repeller representa- +tion, which nevertheless is always much more localized. +Finally, this localization seems to survive in the semi- +classical limit given the slow pace at which it grows with +N. Notice that the Husimi averages and the quantum +repellers are more localized than individual normalized +eigenfunctions, indicating the expected delocalization in- +duced by normalization on the original resonances. +We now turn to analyze some specific examples in or- +der to identify the origin of this localization. In Fig. 3 +we show the phase space representation ˜HR +10(q, p) for the +largest N evaluated in this work and all the values of +R considered in our calculation (a detailed explanation +is given in the caption). No noticeable localization can +be identified with the naked eye due to the large size +of the Hilbert space. The same happens if we consider +the repeller representation ˜h10(q, p) displayed in Fig. 4 +for the same values of the parameter used in Fig. 3. We +notice that for the completely open case, some values of +the normalized distributions are excluded from statistics +in the opening. This is due to the extremely low values +of resonances in that area, and that explains the “holes” +visible in Figs. 3 (d) (smaller) and 4 (d) (bigger). This +also shows that the normalization by the average in order +to analyze the properties of resonances does not have a + +4 +1000 +3000 +5000 +7000 +0 +0.5 +1 +1.5 +2 +104 +0 +600 +1200 +1800 +2400 +3000 +3600 +FIG. 2. (Color online) µ as a function of N for the closed +(different y axis range than the other panels) (a), partially +open with R = 0.1 (b), R = 0.05 (c), and completely open +(d) tribaker map. The symbols ∗, △ and ◦ stand for the 6th, +10th and 16th eigenstates, respectively. The gray (red) points +correspond to the measure calculated for ˜ +HR +i , while the black +points for ˜hi. Full lines show the values of µ for the Husimi +averages
j and the quantum repellers Qj. +smooth transition to the limiting completely open case. +We now give more insight on the mechanism leading +to the detected localization by means of the repeller rep- +resentation. We select a Hilbert space of a smaller size +(N = 5685) in order to make the identification of the +features of phase space distributions easier. In Fig. 5 we +show the average Husimi function, the non-normalized +Husimi representation for the 16th eigenfunction (in de- +creasing eigenvalue modulus order as usual), the normal- +ized version of it, and the superposition of these two lat- +ter distributions (see details of the different panels in the +FIG. 3. +(Color online) ˜ +HR +10(q, p) with N = 16185 for the +closed (a), partially open with R = 0.1 (b), R = 0.05 (c), and +completely open (d) tribaker map. Lower to higher values go +from blue (black) to red (gray). +FIG. 4. (Color online) ˜h10(q, p) with N = 16185 for the closed +(a), partially open with R = 0.1 (b), R = 0.05 (c), and com- +pletely open (d) tribaker map. Lower to higher values go from +blue (black) to red (gray). +FIG. 5. Color online)
j (a), HR +16(q, p) (b), ˜ +HR +16(q, p) +(c), and HR +16(q, p) + ˜ +HR +16(q, p) (d). Lower to higher values go +from blue (black) to red (gray). Fixed points of short POs +are marked by means of white circles. In all cases we have +taken N = 5685 and R = 0.05. +caption). In it, we can identify scarring by a short PO +of period 8 (whose fixed points are marked by means of +white circles) and its symmetry related partner. The dif- +ferent intensities of each fixed point belonging to these +POs can be attributed to non trivial interference effects +among symmetric companions and possibly other POs +with less weight. The normalized version of this distribu- +tion masks the localization on this PO which is contained +in the repeller (the invariant generated when the open- +ing is complete). But nonetheless new localization can be +clearly appreciated. We could also identify fixed points of +short POs that in some cases are outside of the repeller, +but this conclusion is somewhat unclear. If we now check +the corresponding situation for the repeller representa- +tion which is shown in Fig. 6, we identify the same short + +0.8 +0.6 +0.4 +0.2 +0 +0 +0.2 +0.4 +0.6 +0.8 +q(d)(a)(b)0.8 +c) +0.6 +p +0.4 +0.2 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +q(d)(a)(b)0.8 +0.6 +0.4 +0.2 +0 +0 +0.2 +0.4 +0.6 +0.8 +q(d)(a)(b)5 +FIG. 6. (Color online) Qj (a), h16(q, p) (b), ˜h16(q, p) (c), and +h16(q, p) + ˜h16(q, p) (d). Lower to higher values go from blue +(black) to red (gray). Fixed points of short POs are marked +by means of white circles. In all cases we have taken N = 5685 +and R = 0.05. +PO as clearly scarring the non-normalized distribution. +Thanks to this representation it becomes clear that local- +ization outside the repeller is not significant at this level +of reflectivity. A different short PO than those found in +the Husimi representation corresponds to localization in +the normalized case, this PO has periodic points outside +of the repeller. In both cases we show the superposition +of the non normalized distributions and the normalized +ones for the sake of comparison, from which it becomes +clear that normalization dominates over the features of +individual non-normalized eigenfunctions. +IV. +CONCLUSIONS +In this work we have studied the localization on scaled +phase space distributions of resonances corresponding to +the partially open tribaker map. We have found that for +the longest lived ones, which define the quantum repeller, +localization is present and persistent as N grows. This is +hinted by deviations with respect to a conjectured univer- +sal intensity statistics [14] for the scaled representations +on this quantum analogue of the classical invariant set. +In fact, the repeller representation is the complete anal- +ogy of the usual Husimi functions in closed systems, to +which it converges as the partial opening becomes closer +to R = 1. It is fundamental to consider both the un- +stable and stable manifolds counterparts when studying +quantum localization, being this a feature of the repeller +representation by definition. The Qj property of almost +faithfully expanding the long-lived sector of the quantum +open evolution operator is the other side of the same con- +cept, this property being exact when using eigenvalues to +weight each term in the sum (for the longest lived reso- +nances both choices are similar). In our view, it is in this +sense that localization should be regarded in open sys- +tems, no matter if they are of partial or complete nature. +As a matter of fact there are no appreciable differences +with respect to the universal exponential distribution +conjectured in [14] when looking at the scaled Husimis +of the right resonances, but there are when inspecting +the representation based on the repeller and the norm ra- +tio seems to indicate more localization than in the closed +system for both of them. But localization of scaled distri- +butions seems to happen outside of the completely open +repeller while for non normalized ones it does on short +POs belonging to it. This brings up the question about +the meaning of normalization for localization properties +of eigenfunctions. Undoubtedly, the Shnirelman’s theo- +rem [30] should have a generalization for partially open +systems, but also scarring. +Some recent studies suggest that the scarring persists +in the semiclassical limit [31] for closed systems. This +seems to be the case also for non normalized distribu- +tions when an opening is present, though this short POs +localization is not of the same kind we have found in +the scaled distributions. Remarkably, from our results it +seems that its presence and persistence will be stronger +in this latter scenario. To settle the question if this does +survive or not in the semiclassical limit, more investiga- +tions need to be carried out and we leave it for future +work. +V. +ACKNOWLEDGMENTS +Support from CONICET is gratefully acknowledged. +This research has also been partially funded by the Span- +ish Ministry of Science, Innovation and Universities, Go- +bierno de Espa˜na under Contract No. PGC2018-093854- +B-I00, and by ICMAT Severo Ochoa Programme for Cen- +tres of Excellence in R & D (CEX2019-000904-S). +[1] H. Cao and J. Wiersig, Rev. Mod. Phys. 87, 61 (2015). +[2] M. Novaes, J. Phys. A: Math. Theor. 46, 143001 (2013). +[3] S. Bittner, K. Kim, Y. Zeng, Q.J. Wang, and H. Cao, +New J. Phys. 22, 083002 (2020). +[4] W.T. Lu, S. Sridhar and M. Zworski, Phys. Rev. Lett. +91, 154101 (2003). +[5] S. Nonnenmacher, Nonlinearity 24, R123 (2011). +[6] J.A. Ramilowski, S.D. Prado, F. Borondo and D. Farrelly, +Phys. Rev. E 80, 055201(R) (2009); A. Ebersp¨acher, J. +Main and G. Wunner, Phys. Rev. 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Berry, Physica D 1 267 (1980). +[27] M. Degli Espositi, B. Winn, J.Phys.A: Math.Gen.38, +5895-5912 (2005). +[28] M. Saraceno, Ann. Phys. 199, 37 (1990); M. Saraceno +and R.O. Vallejos, Chaos 6, 193 (1996); A. �Lozi´nski, +P. Pako´nski and K. +˙Zyczkowski, Phys. Rev. E 66, +065201(R) (2002). +[29] M. Saraceno and A. Voros, Physica D 79, 206 (1994). +[30] A.I. Shnirelman (in Russian), Usp. Math. Nauk 29, 181 +(1974). +[31] E.G. Vergini, Phys. Rev. Lett. 108, 264101 (2012); E.G. +Vergini, EPL 110, 10010 (2015). + diff --git a/odE2T4oBgHgl3EQfzwj2/content/tmp_files/load_file.txt b/odE2T4oBgHgl3EQfzwj2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1cf2eb656455ff0699ced4e7cd0ea00b5c6d0ce4 --- /dev/null +++ b/odE2T4oBgHgl3EQfzwj2/content/tmp_files/load_file.txt @@ -0,0 +1,484 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf,len=483 +page_content='Average localization of resonances on the quantum repeller J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Montes,1, 2, ∗ Gabriel G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Carlo,3, † and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Borondo1, 2, ‡ 1Departamento de Qu´ımica, Universidad Aut´onoma de Madrid, Cantoblanco, 28049–Madrid, Spain 2Instituto de Ciencias Matem´aticas (ICMAT), Cantoblanco, 28049–Madrid, Spain 3Comisi´on Nacional de Energ´ıa At´omica, CONICET, Departamento de F´ısica, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' del Libertador 8250, 1429 Buenos Aires, Argentina (Dated: January 12, 2023) There has been a very recent surge in the interest on the localization properties of resonances associated to partially open (scattering) systems, which are of the greatest relevance when studying resonant cavities like those used in microlasers for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Very recently it has been found that no localization is present in a scaled form of these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Moreover, a new kind of scarring on structures different from periodic orbits is described for non normalized resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In this paper, we analyze the localization of a scaled distribution function based on the quantum repeller representation for the partially open quantum tribaker map, a paradigmatic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We find localization, which in the generic case can be associated to the shortest periodic orbits in non trivial ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Also, normalized states present enhancements that could not be conclusively associated to periodic orbits and that become more evident when looking at the repeller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' These findings leave the door open for new perspectives on recent theoretical developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' INTRODUCTION Quantum scattering has been an active research field since long ago, being optical cavities one of the main focus of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' There is a long history of both the- oretical and experimental studies [1, 2], but also recent experiments [3], in which a lot of questions have been settled, but others still remain open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This situation cor- responds classically to placing an opening in the system, letting the trajectories that arrive at a region of phase space escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This escape can be complete and then we speak about a complete opening, or partial, where we define a convenient reflectivity function in order to take into account a partial confinement of these trajec- tories inside the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This mechanism gives rise to a classical (conditionally) invariant distribution that cor- responds to a (partial) fractal set having (multi) fractal dimension(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' At the quantum level and for the complete opening case, the fractal Weyl law is a well established result, which shows that the number of long-lived states (or resonances) scales with the Planck constant as ℏ−d/2, where d+1 is the fractal dimension of the classical repeller [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' If the opening is partial [8], we have multifractal measures [9] that characterize a conditionally invariant distribution that extends over all the phase space and the usual fractal Weyl law has to be redefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This has been done for the very important case of maps (these systems embody all the features of the chaotic cavities, but are much more amenable to theoretical studies) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The morphology of the corresponding eigenfunctions is still not completely understood, being their localization properties [11] a source of active developments nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' ∗ E–mail address: jmontes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='3@alumni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='unav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='es † E–mail address: carlo@tandar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='cnea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='ar ‡ E–mail address: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='borondo@uam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='es Very recently there has been a renovated interest in elucidating some open questions about the structure of resonances associated to (partially) open systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In [12] the average phase space distribution of resonances in chaotic systems with escape was conjectured to be de- scribed by a classical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This measure is condition- ally invariant and uniformly distributed on sets with the same temporal distance to the quantum resolved chaotic saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' A family of conditionally invariant classical mea- sures was found to describe the multifractal phase space distribution, product structure along stable and unsta- ble directions, and the dependence on the decay rate of resonances [13] (better behaved for long and short lived ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Moreover, in [14] the intensity statistics was found to universally follow an exponential distribution in the partially open chaotic standard map, baker map, and a random matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In [15] a local randomization on phase space for the baker map with escape was intro- duced to obtain a semiclassical description of resonances, though some quantitative differences were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Finally, in [16] the resonances were conjectured to be a product of an essentially classical conditionally invariant measure and universal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Notably, scarring is present in almost all of them, although along segments of rays but not along periodic orbits (POs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' On the other hand, there has been a complementary line of research in order to study the properties of reso- nances based on the localization phenomena on POs, the so called scarring [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This led to the development of the semiclassical theory of short POs for open quantum maps [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This is a constructive approach based on short POs contained in the classical repeller, which motivates a ba- sis of scar functions able to expand the quantum repeller [19, 20], which is suitable to express the quantum non- unitary operators [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We have recently extended the short POs theory to partially open quantum maps [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Can these two visions on the same problem be put arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='04135v1 [quant-ph] 9 Jan 2023 2 together?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Is scarring in the long lived resonances of par- tially open systems essentially different from that occur- ring along POs in the closed scenario?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' How are the lim- its corresponding to completely open and closed systems reached?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Which is the effect of the normalization/scaling of these resonances by their average in terms of localiza- tion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In this work we try to answer these questions, and for that purpose we study the localization properties of the longest lived resonances of the (partially) open trib- aker map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We use the Husimi functions associated to the right eigenvectors scaled using their average, and the repeller representation of the left and right eigenvectors scaled using the corresponding quantum repeller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In this way, we are able to clearly show the localization present when using the latter mathematical objects which reveal themselves as a very adequate tool to understand the sta- tistical properties of the resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Moreover, thanks to them scarring on short POs is easily identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The organization of this paper is as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' II we describe the (partially) open tribaker map and the localization measures used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' III we apply these definitions to unveil the localization properties of reso- nances and how short POs could be apparently hidden in the set of long lived resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The conclusions are outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' SYSTEM AND MEASURES Maps are a powerful and convenient tool to study clas- sical and quantum chaos [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In particular, open maps on the 2-torus have an associated fractal repeller, that is an invariant measure which lies at the intersection of the forward and backwards trapped sets (nonescaping trajectories in the past or future).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' When the opening is partial, some amount of them is reflected following a reflectivity function that we will take as a constant R ∈ (0 : 1) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In this case, the corresponding measure extends over all the phase space, but still shows multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The quantum version of maps on the torus requires taking ⟨q + 1|ψ⟩ = ei2πχq⟨q|ψ⟩, and ⟨p + 1|ψ⟩ = ei2πχp⟨p|ψ⟩, with χq, χp ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The dimension of the Hilbert space is N = (2πℏ)−1, the semiclassical limit is reached when N → ∞, and the evolution operator is a N × N matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Position and momentum eigenstates are given by |qj⟩ = |(j + χq)/N⟩ and |pj⟩ = |(j + χp)/N⟩ with j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' , N − 1}, transforming as ⟨pk|qj⟩ = 1 √ N e−2iπ(j+χq)(k+χp)/N ≡ (Gχq,χp N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' A partially open quantum map is non-unitary and has N right eigenvec- tors |ΨR j ⟩ and N left ones ⟨ΨL j | for each resonance (eigen- value) zj, with ⟨ΨL j |ΨR k ⟩ = δjk and ⟨ΨR j |ΨR j ⟩ = ⟨ΨL j |ΨL j ⟩ being the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The classical tribaker map is defined as B(q, p) = � � � (3q, p/3) if 0 ≤ q < 1/3 (3q − 1, (p + 1)/3) if 1/3 ≤ q < 2/3 (3q − 2, (p + 2)/3) if 2/3 ≤ q < 1 (1) If we place an opening in the region 1/3 < q < 2/3 all trajectories arriving at it will suffer an intensity modifi- cation given by the reflectivity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The quantum closed counterpart with antiperiodic boundary conditions χq = χp = 1/2 (preserving time reversal and parity) in position representation is [28, 29] U B = G−1 N � � GN/3 0 0 0 GN/3 0 0 0 GN/3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (2) The corresponding partially open map is given by P = � � 11N/3 0 0 0 √ R 11N/3 0 0 0 11N/3 � � , (3) applied to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (2), in such a way that the original sym- metries are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Motivated by the classical repeller, we can define the corresponding quantum object by means of an operator ˆhj constructed in terms of the right |ΨR j ⟩ and left ⟨ΨL j | eigenstates [19, 20] ˆhj = |ΨR j ⟩⟨ΨL j | ⟨ΨL j |ΨR j ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (4) When the expectation values of these projectors are evaluated on coherent states |q, p⟩, |⟨q, p|ˆhj|q, p⟩| ∝ � HR j (q, p)HL j (q, p), being HR,L j the Husimi functions of the R, L resonances, which for closed (unitary) systems are simply the Husimi functions of the eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We define the quantum repeller as the average of the first j of them, ordered by decreasing modulus of their eigenvalues (|zl| ⩾ |zm| with l ≤ m): ˆQj ≡ 1 j j � j′=1 ˆhj′, (5) and denote their phase space representation by means of coherent states as hj(q, p) = |⟨q, p|ˆhj|q, p⟩| (6) Qj(q, p) = |⟨q, p| ˆQj|q, p⟩|, (7) where hj(q, p) will be simply called the repeller represen- tation of the resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We have explicitly taken the modulus since these are complex functions in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Also, for the sake of an easy comparison with the scaled Husimis that will be defined next, we give an equal weight to each resonance in the repeller sum, although weighting them with the corresponding eigenvalues should provide a more faithful representation of the longest lived sector of the quantum map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' It is worth noticing that, in this work, we focus on the long lived set of resonances and this sum runs on just a subset of the whole set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' It is useful to define [14] the scaled Husimi ˜HR i and repeller ˜hi func- tions as HR i /
j and hi/Qj respectively, where the average < · · · > is taken over this j subset of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 3 We explore two measures of localization for both scaled functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The first one closely follows the conjecture of universal intensity statistics formulated in [14], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' that these functions obey an exponential distribution P(w)dw = e−wdw, where w = ˜HR i (q, p) or w = ˜hi(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We define the corresponding localization measure Σϵ as the deviation from the exponential behavior taken as the distance between both curves at every point in the his- tograms, for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The second measure of localization is intrinsically phase space motivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In fact, the norm ratio µ is taken to be [20] µ(˜hi) = � ∥˜hi∥1/∥˜hi∥2 ∥ρc∥1/∥ρc∥2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (8) We use ˜hi(q, p) in its definition but is also valid for ˜HR i (q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' A coherent state at an arbitrary position (q, p) in phase space is the normalization factor ρc = |q, p⟩⟨q, p|, with the phase space norm given by ∥˜hi∥γ = �� T 2 ˜hi(q, p)γdqdp �1/γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (9) The norm ratio is also independent of the h normaliza- tion, with a minimum value of 1 for a maximally localized distribution (a coherent state) and a maximum value of N/2 for the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' LOCALIZATION IN PHASE SPACE In order to investigate the average localization on the quantum repeller defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' II, we have computed the two previously defined measures, Σϵ and µ, for the closed (R = 1), partially open (with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='05), and the completely open (R = 0) tribaker map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In the calculations, we have used the 32 longest lived resonances to evaluate ˜HR i and ˜hi, which in the closed case are ar- bitrarily selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Results for Σϵ (calculated for 1000 points distributions and 3 selected values of i) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' As can be seen, for the closed system there is essentially no difference between the Husimis of the right eigenstates and the distributions taken on the repeller, as expected from the fact that there are no differences between the right and left eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' More remarkably, we see meaningful differences with respect to the local- ization for the closed system in the case of the repeller of the partially open maps, and also for the complete opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The low to moderate R values that have been considered for the partially open cases produce apprecia- ble changes in the behavior of this measure with respect to the closed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' It needs to be underlined that for the Husimi distributions of the right eigenstates there is no notable change from the closed scenario at any of the finite reflectivities shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This agrees with the univer- sality conjecture of [14] and shows that our calculations are in that regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='5 2 104 0 20 40 60 80 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (Color online) Σϵ as a function of N for the closed (a), partially open with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='1 (b), R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='05 (c), and completely open (d) tribaker map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The symbols ∗, △ and ◦ stand for the 6th, 10th and 16th eigenstates (ordered as indicated in the main text), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The gray (red) points correspond to the measure calculated for ˜ HR i , while the black points for ˜hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Next, we consider the µ measure whose results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Again, as expected, there is essentially no difference here between the values for ˜HR i and ˜hi in the closed scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' However, the localization is now better detected than with the Σϵ measure for finite reflectivities and also for the completely open case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In fact, though the latter seems to be more localized, the results for partial openings are almost at the same level than this limiting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Moreover, a greater localization with respect to the closed case is also detected in the Husimi distributions of the right resonances not only for the repeller representa- tion, which nevertheless is always much more localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Finally, this localization seems to survive in the semi- classical limit given the slow pace at which it grows with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Notice that the Husimi averages and the quantum repellers are more localized than individual normalized eigenfunctions, indicating the expected delocalization in- duced by normalization on the original resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We now turn to analyze some specific examples in or- der to identify the origin of this localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 3 we show the phase space representation ˜HR 10(q, p) for the largest N evaluated in this work and all the values of R considered in our calculation (a detailed explanation is given in the caption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' No noticeable localization can be identified with the naked eye due to the large size of the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The same happens if we consider the repeller representation ˜h10(q, p) displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 4 for the same values of the parameter used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We notice that for the completely open case, some values of the normalized distributions are excluded from statistics in the opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This is due to the extremely low values of resonances in that area, and that explains the “holes” visible in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 3 (d) (smaller) and 4 (d) (bigger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This also shows that the normalization by the average in order to analyze the properties of resonances does not have a 4 1000 3000 5000 7000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='5 2 104 0 600 1200 1800 2400 3000 3600 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (Color online) µ as a function of N for the closed (different y axis range than the other panels) (a), partially open with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='1 (b), R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='05 (c), and completely open (d) tribaker map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The symbols ∗, △ and ◦ stand for the 6th, 10th and 16th eigenstates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The gray (red) points correspond to the measure calculated for ˜ HR i , while the black points for ˜hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Full lines show the values of µ for the Husimi averages
j and the quantum repellers Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' smooth transition to the limiting completely open case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We now give more insight on the mechanism leading to the detected localization by means of the repeller rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We select a Hilbert space of a smaller size (N = 5685) in order to make the identification of the features of phase space distributions easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 5 we show the average Husimi function, the non-normalized Husimi representation for the 16th eigenfunction (in de- creasing eigenvalue modulus order as usual), the normal- ized version of it, and the superposition of these two lat- ter distributions (see details of the different panels in the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (Color online) ˜ HR 10(q, p) with N = 16185 for the closed (a), partially open with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='1 (b), R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='05 (c), and completely open (d) tribaker map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Lower to higher values go from blue (black) to red (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (Color online) ˜h10(q, p) with N = 16185 for the closed (a), partially open with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='1 (b), R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='05 (c), and com- pletely open (d) tribaker map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Lower to higher values go from blue (black) to red (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Color online)
j (a), HR 16(q, p) (b), ˜ HR 16(q, p) (c), and HR 16(q, p) + ˜ HR 16(q, p) (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Lower to higher values go from blue (black) to red (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Fixed points of short POs are marked by means of white circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In all cases we have taken N = 5685 and R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' caption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In it, we can identify scarring by a short PO of period 8 (whose fixed points are marked by means of white circles) and its symmetry related partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The dif- ferent intensities of each fixed point belonging to these POs can be attributed to non trivial interference effects among symmetric companions and possibly other POs with less weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The normalized version of this distribu- tion masks the localization on this PO which is contained in the repeller (the invariant generated when the open- ing is complete).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' But nonetheless new localization can be clearly appreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We could also identify fixed points of short POs that in some cases are outside of the repeller, but this conclusion is somewhat unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' If we now check the corresponding situation for the repeller representa- tion which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 6, we identify the same short 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='8 q(d)(a)(b)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='8 c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='6 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='8 1 q(d)(a)(b)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='8 q(d)(a)(b)5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' (Color online) Qj (a), h16(q, p) (b), ˜h16(q, p) (c), and h16(q, p) + ˜h16(q, p) (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Lower to higher values go from blue (black) to red (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Fixed points of short POs are marked by means of white circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In all cases we have taken N = 5685 and R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' PO as clearly scarring the non-normalized distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Thanks to this representation it becomes clear that local- ization outside the repeller is not significant at this level of reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' A different short PO than those found in the Husimi representation corresponds to localization in the normalized case, this PO has periodic points outside of the repeller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In both cases we show the superposition of the non normalized distributions and the normalized ones for the sake of comparison, from which it becomes clear that normalization dominates over the features of individual non-normalized eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' CONCLUSIONS In this work we have studied the localization on scaled phase space distributions of resonances corresponding to the partially open tribaker map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' We have found that for the longest lived ones, which define the quantum repeller, localization is present and persistent as N grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This is hinted by deviations with respect to a conjectured univer- sal intensity statistics [14] for the scaled representations on this quantum analogue of the classical invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In fact, the repeller representation is the complete anal- ogy of the usual Husimi functions in closed systems, to which it converges as the partial opening becomes closer to R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' It is fundamental to consider both the un- stable and stable manifolds counterparts when studying quantum localization, being this a feature of the repeller representation by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' The Qj property of almost faithfully expanding the long-lived sector of the quantum open evolution operator is the other side of the same con- cept, this property being exact when using eigenvalues to weight each term in the sum (for the longest lived reso- nances both choices are similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' In our view, it is in this sense that localization should be regarded in open sys- tems, no matter if they are of partial or complete nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' As a matter of fact there are no appreciable differences with respect to the universal exponential distribution conjectured in [14] when looking at the scaled Husimis of the right resonances, but there are when inspecting the representation based on the repeller and the norm ra- tio seems to indicate more localization than in the closed system for both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' But localization of scaled distri- butions seems to happen outside of the completely open repeller while for non normalized ones it does on short POs belonging to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This brings up the question about the meaning of normalization for localization properties of eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Undoubtedly, the Shnirelman’s theo- rem [30] should have a generalization for partially open systems, but also scarring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Some recent studies suggest that the scarring persists in the semiclassical limit [31] for closed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This seems to be the case also for non normalized distribu- tions when an opening is present, though this short POs localization is not of the same kind we have found in the scaled distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Remarkably, from our results it seems that its presence and persistence will be stronger in this latter scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' To settle the question if this does survive or not in the semiclassical limit, more investiga- tions need to be carried out and we leave it for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' ACKNOWLEDGMENTS Support from CONICET is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' This research has also been partially funded by the Span- ish Ministry of Science, Innovation and Universities, Go- bierno de Espa˜na under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' PGC2018-093854- B-I00, and by ICMAT Severo Ochoa Programme for Cen- tres of Excellence in R & D (CEX2019-000904-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Cao and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Wiersig, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Phys.' 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Kunzmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' B¨acker, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Ketzmerick, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' E 103, 042204 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Clauß and R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Clauß, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Fritzsch, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' B¨acker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 129, 193901 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Saraceno, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' 103, 054102 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' [20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} +page_content=' Ermann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfzwj2/content/2301.04135v1.pdf'} 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in the van der Waals antiferromagnet CrPS4 +Dennis K. de Wal,1, ∗ Arnaud Iwens,1 Tian Liu,1 Ping Tang,2 Gerrit E. W. Bauer,1, 2, 3 and Bart J. van Wees1 +1Zernike Institude for Advanced Materials, University of Groningen, Groningen, the Netherlands +2Advanced Institute for Materials Research (AIMR), Tohoku University, Sendai, Japan +3Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, Beijing, China +(Dated: January 10, 2023) +We demonstrate the potential of van der Waals magnets for spintronic applications by reporting +long-distance magnon spin transport in the electrically insulating antiferromagnet chromium thio- +phosphate (CrPS4) with perpendicular magnetic anisotropy. We inject and detect magnon spins +non-locally by Pt contacts and monitor the non-local resistance as a function of an in-plane mag- +netic field up to 7 Tesla. We observe a non-local resistance over distances up to at least a micron +below the Neel temperature (TN = 38 Kelvin) close to magnetic field strengths that saturate the +sublattice magnetizations. +Since the discovery of the long-range magnetic order in +mono- and bilayers of Cr2Ge2Te6 [1] and CrI3 [2] many +(anti)ferromagnetic van der Waals materials have been +identified in monolayer or few layer thicknesses. They are +attractive platforms for spintronics due to the rich spin +textures caused by the interplay of inter- and intralayer +exchange and magnetic anisotropies. +Many antiferromagnetic van der Waals materials are +electrically insulating at low temperatures, which implies +the absence of magnetization damping by free carriers. +They are therefore attractive for the study of collective +excitations of the magnetic order, i.e. spin waves and its +quanta, the magnons [3, 4]. Magnon transport has been +extensively studied in conventional magnets by, e.g., spin +pumping [5], the spin Seebeck effect (SSE) [6], and elec- +trical magnon spin injection/detection [7]. Long distance +magnon transport in the antiferromagnets hematite [8], +nickel oxide [9], and YFeO3 [10] has been demonstrated. +Ultrathin films of the low-damping ferrimagnetic yttrium +iron garnet (YIG), the material of choice for efficient +magnon transport, show the beneficial effects of two- +dimensional (2D) vs. three-dimensional (3D) transport +in the form of strongly enhanced magnon conductivities +[11]. Magnon spin transport driven by temperature gra- +dients (SSE) [12] has been reported in ferro- and antifer- +romagnetic van der Waals materials [13, 14]. However, +the local and non-local SSEs provide only convoluted in- +formation on the magnon transport properties. Thermal +magnon currents are generated by thermal gradients in +the entire sample, making it difficult to disentangle the +magnon relaxation length and magnon spin conductiv- +ity [7, 11]. Antiferromagnetic resonance of CrCl3 [15] re- +veals the existence of acoustic and optical magnon modes, +but does not resolve their roles in spin transport. In or- +der to assess the potential of van der Waals magnets for +spintronic applications, we therefore have to study the +propagation of magnons that are locally generated by +microwaves or, as we will show here, by electrical injec- +tion. +∗ d.k.de.wal@rug.nl +Heavy metal contacts such as Pt with a large (inverse) +spin Hall effect have become a standard instrument to +study magnetic materials. The spin Hall magnetoresis- +tance (SMR) in a Pt contact is a reliable method to mea- +sure the surface equilibrium magnetization [16], which +has already been used to study CrPS4 [17] and FePS3 +[18]. With two Pt contacts, the spins injected by an elec- +tric current in one terminal by the spin Hall effect prop- +agate in an electrically insulating magnet in the form of +magnons, which can be detected by another contact via +the inverse spin Hall effect [7]. Here we report, to the +best of our knowledge for the first time, such non-local +electrical measurements of magnon transport in a van der +Waals antiferromagnet, in our case CrPS4 +CrPS4 is an A-type antiferromagnet (see Figure 1(a)). +Individual layers are out-of-plane (oop) 2D ferromagnets, +but consecutive layers order antiferromagnetically at a +N´eel temperature TN ≃ 38 K. Its relative stability in +air facilitates the fabrication of devices. +An oop field +of Hspinflop ≈ 0.9 T (at 5 K) induces a spin-flop transi- +tion to a canted state, while the magnetization becomes +saturated into a “spin flip” state at 8.5 T. In-plane (ip) +fields result in magnetization saturation at nearly the +same field, indicating that the anisotropy field (HA ≈ +0.01 T) is much smaller than the exchange field (HE ≈ +4.25 T) [19, 20]. CrPS4 is therefore an excellent platform +to study magnons in controlled non-collinear spin tex- +tures because the moderate spin-flop and spin-flip critical +fields are accessible by standard lab equipment. +Figure 1(b) shows the calculated band-edge (k = 0) +frequencies of the acoustic and optical magnons of a +bilayer of CrPS4 with easy axis along z as a function +of ip magnetic fields using the parameters above. Fig. +1(c) sketches the magnetization precession amplitudes for +fields normal to the layers below the spin-flop transition +(H < Hspinflop) in which the N´eel vector remains along +z and the magnon modes carry opposite spins +ℏ/−ℏ, +in the z-direction. Fig. 1(d) sketches the excitations of +the canted spin texture at an ip magnetic field below the +spin-flip transition (H < HE⊥). The associated magnons +evolve from the zero-field spin up and down states with +a net magnetization along y as indicated by the purple +arXiv:2301.03268v1 [cond-mat.mes-hall] 9 Jan 2023 + +2 +FIG. 1. Spin texture and magnon modes in antiferromagnetic CrPS 4. (a) Atom and spins of a bilayer of CrPS4. Red and +blue arrows indicate the local magnetic moments of the Cr atoms (turquoise). The interlayer (intralayer) exchange coupling +is ferromagnetic (antiferromagnetic). (b) In-plane magnetic field dependence of the magnon band edges. (c) Optical (ω+) and +acoustic magnon modes at out-of-plane (oop) magnetic fields below the spin-flop transition (d) magnon modes at in-plane (ip) +magnetic fields below the spin-flip transition (H < HE⊥). The net magnetization of the ω⊥1 mode precesses (purple vector) +around the ip external field vector with equal modulus, while that of the ω⊥2 mode oscillates in the direction of the field. +FIG. 2. (a) Optical micrograph of a transport device with 7 +parallel Pt strips bonded by Ti/Au leads on top of CrPS4 film, +where a and b indicate the orientation.of the single crystal. +(b) Electrical measurement circuit, in which the red arrows +indicate electrically active spins in the Pt strips and ϕ is an +ip magnetic field angle. +arrows. +We fabricated three devices by depositing multiple par- +allel Pt strips on exfoliated CrPS4 flakes with a thickness +of ∼100 nm (see figure 2(a)). We study both the local +and non local resistances as a function of magnitude and +direction of an ip magnetic field. We measure the mag- +netoresistance of a single Pt strip (SMR) [16] as well as +magnon transport and the spin Seebeck effect non-locally +by two Pt strips (see Fig. 2(b)). +Via the SMR we monitor the surface magnetization +as a function of temperature, ip external field, and bias +current. The current I in a Pt strip generates a trans- +verse spin current that when partially reflected at the +Pt|CrPS4 interface induces an additional current by the +inverse spin Hall effect, effectively reducing the electrical +resistance. A polarization of the spin-Hall spin current +(red arrows in Fig. +2(b)) parallel (normal) to the lo- +cal moments of the magnet at the interface, minimizes +(maximizes) the dephasing by the exchange interaction +and therefore the electric resistance [16]. +An in-plane +magnetic field H = Hˆy (ip angle ϕ = 0) cants the oop +antiferromagnetic order by an angle +θ⊥ = arcsin +H +2HE + HA +(1) +with the z-axis. The electric resistance Rl of a Pt wire +along the x-axis therefore should be maximal for θ⊥ = 0 +and minimal for θ⊥ = π/2, i.e. at and beyond the spin- +flip transition. +On the other hand, magnon injection +is most efficient when magnetic moments and current- +induced spins are parallel, maximizing the non-local re- +sistance Rnl = Vdetector/Iinjector. Rnl > 0 by defining the +polarity of the voltage on the detector opposite to that +of the current in the injector (see figure 2(b)) . +The Joule heating by a charge current I generates a +temperature gradient over the interface and in the mag- +net, generating a spin current and associated inverse spin +Hall voltage (spin Seebeck effect) in the injector as well as +the detector with associated local and non-local voltage +signals. +We can separate the electrical and thermal signals by +recording the first and second harmonic responses to a +current bias that oscillates with frequency ω. The first + +(b) +0.05 +0.04 +1.5 +0.03 +0.02 +(ZHI) +0.01 +0.00. +1.0 +8.0 +8.2 +8.4 +8.6 +8.8 +W11 +(a) +W12 +3 +0.5 +0.0 +0 +2 +4 +6 +8 +10 +H (T) +N +N +M1 +M1 +M1 +M1 +Ho +Ho +(c) +(d) +y +y +y +M2 +y +M2 +X +X +X +M2 ++3 +m(a) +(b) +n +10 μm3 +harmonic response reflects the Ohmic signal V ∼ I, while +the thermal signals V ∼ I2 appear at double frequency. +Here we focus on the linear response R(1ω) +l/nl , with a brief +discussion of R(2ω) +l/nl in the Supplementary Material (SM). +The spin Hall effect and inverse spin Hall effect dictate +the following dependence on the ip field angle: +R(1ω) +l += R(1ω) +l,0 ++ ∆R(1ω) +l +sin2 ϕ +(2) +R(1ω) +nl += ∆R(1ω) +nl +cos2 ϕ +(3) +where R(1ω) +0,l/nl are constant offsets and ∆R(1ω) +l/nl are the +strenghts of the signals that depend on the ip field angle +ϕ (defined in Fig. 2). +We measured the local and non-local resistance in a +liquid-He cryostat at temperatures between 5 and 300 K +as a function of an ip magnetic field up to 7.9 T and as +a function of ip angle ϕ. Figure 2 shows the schematics +of device D1 and the local and non-local measurement +configurations. The measurements on D1 were carried +out on three different pairs of Pt contacts with edge-to- +edge distances of 330 nm, 420 nm and 780 nm. On device +D2, we measured the resistances for two different pairs +(∼300 nm and ∼450 nm) of Pt contacts. +Results for +device D3 with similar flake thickness and contacts with +an edge to edge spacing of 300 nm are shown in the SM. +The time-dependent voltage responses may be ex- +panded as V (t) = R1I(t)+R2I2(t)+· · · , where standard +low frequency (7 Hz - 17 Hz) lock-in techniques access the +constants R1 and R2 [7, 21] +Fig. +3 shows the observed R(1ω) +l +− R(1ω) +l,0 +(≈ 6 kΩ) +of the injector contact as function of the direction of +an ip magnetic field of 7 T at AC current bias of 60 +µA and at T += 24 K. The observed ϕ dependence +agrees well with the model for the SMR sketched above. +∆R(1ω) +l +/R(1ω) +l,0 +≃ 10−4 is of the same order of magnitude +as the SMR of CrPS4 in the oop configuration [17] and +that of other magnetic materials, which is a strong indi- +cation of an efficient interface exchange coupling and a +large spin-mixing conductance. +The modulation ∆R(1ω) +l +at T = 20 K in Fig. +4(a) +as a function of magnetic field strength agrees also with +expectations, while the lack of a bias-current dependence +confirms that we are in the linear response regime. We +observe saturation at fields > 6T (see Fig. +4), which +corresponds to the onset of the spin-flip state. +At fields of several tesla, the SMR decreases with tem- +perature but persists above TN and even up to room +temperature (not shown), which is consistent with re- +ports for CrPS4 [17] and the van der Waals material +Cr2Ge2Te6 [22]. +The robust SMR can possibly be as- +cribed to a TN that is enhanced by the interface spin +orbit coupling and/or a paramagnetic SMR by a field- +induced magnetization [23]. +We now focus on the non-local signal R(1ω) +nl +plotted +in the lower panel Fig. +3 for current bias of 60 µA, +H = 7 T, T = 24 K (measured together with R(1ω) +l +). +FIG. 3. Top panel: Local resistance modulation ∆R(1ω) +l +of +the Pt strip as function of in-plane angle ϕ (relative to the +wire normal) of an applied magnetic field of 7 T. The bias +current is 60 µA and sample temperature is 24 K. The red +curve is a fit by sin2 ϕ. A 5.97 kΩ offset resistance has been +subtracted. Bottom panel: Simulataneously measured non- +local resistance R(1ω) +nl +, fitted by cos2 ϕ (red curve). +∆R(1ω) +l +≃ 0.6 mΩ is about 30 times smaller than that +of Pt—YIG (thickness of 200 nm) [7]. +R(1ω) +nl +is maxi- +mum (minimum) at φ = 0◦ (φ = 90◦) which reflects the +angular dependence of the spin injection and detection +efficiencies by the spin Hall effects in Pt. +Fig. 4 reveals a remarkable dependence of the magnon +transport on magnetic field. At H ≤ 6 T, no R(1ω) +nl +is +observed within the experimental uncertainty. At fields +> 6 T, R(1ω) +nl +increases sharply and appears to saturate at +fields > 7 T. This rapid increase correlates with the satu- +ration of the bulk magnetization (see SM) and is therefore +associated with the spin-flip transition from a canted an- +tiferromagnetic (AFM) to a collinear ferromagnetic (FM) +state (see Fig. 1(c&d)). +The non-local resistance at current bias of 60 µA and +at 7 T in Fig. 4 is non-monotonous, with a maximum +around 25 K. We attribute this to two effects. On one +hand, the critical fields for the spin-flip transition de- +creases with temperature. +At low temperatures and 7 +T, the sample is still in the canted-AFM phase. +The +sharp increase in R(1ω) +nl +coincides with the formation of +the saturated FM phase at T ∼ 25 K. Moreover, the +equilibrium magnon density and resulting magnon con- +ductivity increase with temperature [24, 25]. At larger +temperatures, the magnetization and R(1ω) +nl +decrease and +vanish at TN. +Further, we assess the magnon transport in CrPS4 by +measuring as a function of distance d between the Pt + +0.6 +data +0.5 +sin2(Φ)-fit +△RIw (Q) +0.4 +0.3 +0.2 +0.1 +0.0 +-0.1 +-90 +0 +90 +180 +270 +1.0 +data +cos2()-fit +0.8 +(U)m +0.6 +0.4 +0.2 +0.0 +-0.2 +0.4 +-90 +0 +90 +180 +270 +Angle Φ (deg)4 +FIG. 4. +Field- and temperature-dependent results on device +D2, Top: Field dependence of R1ω +l +at different bias currents +at 25 Kelvin. middle: Same for R1ω +nl . Bottom: Temperature +dependence of R1ω +nl at 7 T. +FIG. 5. Non-local resistance as function of distance d between +the injector and detector contacts. For different bias currents +at sample temperature of 24 K, R(1ω) +nl +∼ 1/d is shown for three +different Pt strip pairs on device D1 with d being equal to 330 +nm, 420 nm and 780 nm, respectively. +contacts as shown in Fig. 5. The absence of a system- +atic dependence on the current bias again confirms that +we operate in the linear response regime. The model of +diffusive magnon transport in YIG leads to as decay of +R(1ω) +nl +with increasing d as a function of the magnon dif- +fusion length λ. For efficient spin injection at the Pt|YIG +interface, this described by [7]: +R(1ω) +nl += C +λ +exp (d/λ) +1 − exp (2d/λ), +(4) +where C is a constant. +Since we observe an alge- +braic R(1ω) +nl +∼ 1/d rather than exponential dependence, +magnon transport over the length scales d ≤ 1 µm is +Ohmic [7], i.e. purely diffusive while magnon decay sets +in at larger distances only. +The abrupt field dependence of the non-local resistance +differs sharply from the linear dependence of the SMR +(top panes of Fig. +4) that indicates a surface magne- +tization proportional to a static magnetic susceptibility. +A surprise of the present study is the absence of non- +local transport in the non-collinear phase. This behav- +ior is markedly different from previous studies of trans- +port that were carried out with magnetic fields parallel +to the N´eel vector, including the spin-flop transition [8– +10]. However, the associated theories do not address the +present configuration either. +The magnon band edges of CrPS4 in the canted phase +as plotted in Fig. +1(b) diagonalize the classical spin +Hamiltonian with eigenfrequencies [26] at low fields H ≤ +HE⊥ = 2HE + HA +ω+ = γ +� +(2HE sin2 θ⊥ + HA cos2 θ⊥)(2HE + HA) +ω− = γ +� +HA(2HE + HA) cos2 θ⊥ +(5) +and at high fields H > HE⊥ = 2HE + HA +ω+ = γ +� +(H − HA)H +ω− = γ +� +(H − 2HE)(H − 2HE − HA), +(6) +where θ⊥ is defined in equation 1 and includes the ex- +ternal ip field H. The small anisotropy causes the low +frequencies of the acoustic modes that at high magnetic +fields and low temperatures are dominantly populated. +The collinear ferromagnetic phase above HE⊥ can be +treated by a two-mode linearized Boltzmann equation +similar to YIG, while the large SMR implies that inter- +faces are transparent, so we may expect at high non-local +signal at H > HE⊥. As the Pt contacts do not inject +or detect spin polarizations in the z-direction. At zero +canting angle (θ⊥), the spin current injected by the Pt +contacts is fully absorbed by the antiferromagnet in the +form of a spin transfer torque to the magnetic sublat- +tices, while magnon injection and R(1ω) +nl +vanish. With in- +creasing canting angle, the magnon injection efficiency in- +creases proportional with the induced net magnetization. +However, the exchange interaction in a non-collinear con- +figuration also increasingly affects the non-local signal by +reducing the magnon decay length (P. Tang, in prepara- +tion). In the collinear phase both magnon injection and +magnon transport do not prevent the non-local signal. +The abruptness of the observed onset of non-local trans- +port at the spin-flip transition field is surprising, however. + +60uA25K +0.15 +80uA25K +R1(Q2) +0.10 +0.05 +0.00 +0.05 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +0.4 +60uA25K +80uA25K +0.3 +0.2 +0.1 +0.0 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +Field (T) +0.4 +40uA 7 T +60uA 7 T +80uA 7 T +0.3 +(uw) +0.2 +0.1 +0.0 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Temperature (K)0.8 +140uA +0.7 +80uA +60uA +0.6 +40uA +(m2) +0.5 +0.4 +0.3 +R +0.2 +0.1 +0.0 +300 +400 +500 +600 +700 +800 +Distance (nm)5 +The abrupt increase of non-local resistance near the spin +flip transition may be caused by the combined effects of +enhanced magnon injection into the low energy magnon +branch and the sudden suppression of magnon relaxation +when the system approaches the FM state. +Summarizing, we report non-local spin transport in +a van der Waals magnet, to the best of our knowl- +edge for the first time. +The spin conduit is the elec- +trically insulating antiferromagnet CrPS4 with perpen- +dicular anisotropy. We focus on a configuration that has +escaped attention even in conventional antiferromagnets, +with an in-plane magnetic field normal to the Pt spin +injector and detector that tilts the antiparallel spins into +the plane. Surprisingly, we do not observe spin trans- +port in the non-collinear phase. At the critical field that +forces the transition to a collinear ferromagnetic phase, +we observe an abrupt increase of the non-local spin signal +over distances that exceed a micron. These results herald +the potential of 2D van der Waals magnets for scalable +magnonic circuits. +ACKNOWLEDGMENTS +We acknowledge the technical support from J. G. Hol- +stein, H. Adema T. Schouten, H. H. de Vries and F. +H. van der Velde. +We acknowledge the financial sup- +port of the Zernike Institute for Advanced Materials and +the European Union’s Horizon 2020 research and inno- +vation program under Grant Agreement No. 785219 and +No. +881603 (Graphene Flagship Core 2 and Core 3). +This project is also financed by the NWO Spinoza prize +awarded to BJW by the NWO and has received funding +from the European Research Council (ERC) under the +European Union’s 2DMAGSPIN (Grant agreement No. +101053054). GB acknowledges funding by JSPS Kakenhi +grant no. 19H00645. +[1] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, +W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. +Louie, J. Xia, and X. Zhang, Nature 546, 265 (2017). +[2] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, +R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. +McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo- +Herrero, and X. Xu, Nature 546, 270 (2017). +[3] F. 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Melkov, Magnetization oscil- +lations and waves (CRC Press, Boca Raton, 1996). + diff --git a/uNE1T4oBgHgl3EQfkAQx/content/tmp_files/load_file.txt b/uNE1T4oBgHgl3EQfkAQx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7726d59a950c1c5d54dde0896e921cfa5237f97b --- /dev/null +++ b/uNE1T4oBgHgl3EQfkAQx/content/tmp_files/load_file.txt @@ -0,0 +1,555 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf,len=554 +page_content='Long distance magnon transport in the van der Waals antiferromagnet CrPS4 Dennis K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' de Wal,1, ∗ Arnaud Iwens,1 Tian Liu,1 Ping Tang,2 Gerrit E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Bauer,1, 2, 3 and Bart J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' van Wees1 1Zernike Institude for Advanced Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' University of Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' the Netherlands 2Advanced Institute for Materials Research (AIMR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Sendai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Japan 3Kavli Institute for Theoretical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' University of the Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' China (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 2023) We demonstrate the potential of van der Waals magnets for spintronic applications by reporting long-distance magnon spin transport in the electrically insulating antiferromagnet chromium thio- phosphate (CrPS4) with perpendicular magnetic anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We inject and detect magnon spins non-locally by Pt contacts and monitor the non-local resistance as a function of an in-plane mag- netic field up to 7 Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We observe a non-local resistance over distances up to at least a micron below the Neel temperature (TN = 38 Kelvin) close to magnetic field strengths that saturate the sublattice magnetizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Since the discovery of the long-range magnetic order in mono- and bilayers of Cr2Ge2Te6 [1] and CrI3 [2] many (anti)ferromagnetic van der Waals materials have been identified in monolayer or few layer thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' They are attractive platforms for spintronics due to the rich spin textures caused by the interplay of inter- and intralayer exchange and magnetic anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Many antiferromagnetic van der Waals materials are electrically insulating at low temperatures, which implies the absence of magnetization damping by free carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' They are therefore attractive for the study of collective excitations of the magnetic order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' spin waves and its quanta, the magnons [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Magnon transport has been extensively studied in conventional magnets by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=', spin pumping [5], the spin Seebeck effect (SSE) [6], and elec- trical magnon spin injection/detection [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Long distance magnon transport in the antiferromagnets hematite [8], nickel oxide [9], and YFeO3 [10] has been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Ultrathin films of the low-damping ferrimagnetic yttrium iron garnet (YIG), the material of choice for efficient magnon transport, show the beneficial effects of two- dimensional (2D) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' three-dimensional (3D) transport in the form of strongly enhanced magnon conductivities [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Magnon spin transport driven by temperature gra- dients (SSE) [12] has been reported in ferro- and antifer- romagnetic van der Waals materials [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' However, the local and non-local SSEs provide only convoluted in- formation on the magnon transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Thermal magnon currents are generated by thermal gradients in the entire sample, making it difficult to disentangle the magnon relaxation length and magnon spin conductiv- ity [7, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Antiferromagnetic resonance of CrCl3 [15] re- veals the existence of acoustic and optical magnon modes, but does not resolve their roles in spin transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' In or- der to assess the potential of van der Waals magnets for spintronic applications, we therefore have to study the propagation of magnons that are locally generated by microwaves or, as we will show here, by electrical injec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' ∗ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='wal@rug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='nl Heavy metal contacts such as Pt with a large (inverse) spin Hall effect have become a standard instrument to study magnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The spin Hall magnetoresis- tance (SMR) in a Pt contact is a reliable method to mea- sure the surface equilibrium magnetization [16], which has already been used to study CrPS4 [17] and FePS3 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' With two Pt contacts, the spins injected by an elec- tric current in one terminal by the spin Hall effect prop- agate in an electrically insulating magnet in the form of magnons, which can be detected by another contact via the inverse spin Hall effect [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Here we report, to the best of our knowledge for the first time, such non-local electrical measurements of magnon transport in a van der Waals antiferromagnet, in our case CrPS4 CrPS4 is an A-type antiferromagnet (see Figure 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Individual layers are out-of-plane (oop) 2D ferromagnets, but consecutive layers order antiferromagnetically at a N´eel temperature TN ≃ 38 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Its relative stability in air facilitates the fabrication of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' An oop field of Hspinflop ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='9 T (at 5 K) induces a spin-flop transi- tion to a canted state, while the magnetization becomes saturated into a “spin flip” state at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='5 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' In-plane (ip) fields result in magnetization saturation at nearly the same field, indicating that the anisotropy field (HA ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='01 T) is much smaller than the exchange field (HE ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='25 T) [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' CrPS4 is therefore an excellent platform to study magnons in controlled non-collinear spin tex- tures because the moderate spin-flop and spin-flip critical fields are accessible by standard lab equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Figure 1(b) shows the calculated band-edge (k = 0) frequencies of the acoustic and optical magnons of a bilayer of CrPS4 with easy axis along z as a function of ip magnetic fields using the parameters above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 1(c) sketches the magnetization precession amplitudes for fields normal to the layers below the spin-flop transition (H < Hspinflop) in which the N´eel vector remains along z and the magnon modes carry opposite spins +ℏ/−ℏ, in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 1(d) sketches the excitations of the canted spin texture at an ip magnetic field below the spin-flip transition (H < HE⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The associated magnons evolve from the zero-field spin up and down states with a net magnetization along y as indicated by the purple arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='03268v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='mes-hall] 9 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Spin texture and magnon modes in antiferromagnetic CrPS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' (a) Atom and spins of a bilayer of CrPS4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Red and blue arrows indicate the local magnetic moments of the Cr atoms (turquoise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The interlayer (intralayer) exchange coupling is ferromagnetic (antiferromagnetic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' (b) In-plane magnetic field dependence of the magnon band edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' (c) Optical (ω+) and acoustic magnon modes at out-of-plane (oop) magnetic fields below the spin-flop transition (d) magnon modes at in-plane (ip) magnetic fields below the spin-flip transition (H < HE⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The net magnetization of the ω⊥1 mode precesses (purple vector) around the ip external field vector with equal modulus, while that of the ω⊥2 mode oscillates in the direction of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' (a) Optical micrograph of a transport device with 7 parallel Pt strips bonded by Ti/Au leads on top of CrPS4 film, where a and b indicate the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='of the single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' (b) Electrical measurement circuit, in which the red arrows indicate electrically active spins in the Pt strips and ϕ is an ip magnetic field angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We fabricated three devices by depositing multiple par- allel Pt strips on exfoliated CrPS4 flakes with a thickness of ∼100 nm (see figure 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We study both the local and non local resistances as a function of magnitude and direction of an ip magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We measure the mag- netoresistance of a single Pt strip (SMR) [16] as well as magnon transport and the spin Seebeck effect non-locally by two Pt strips (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Via the SMR we monitor the surface magnetization as a function of temperature, ip external field, and bias current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The current I in a Pt strip generates a trans- verse spin current that when partially reflected at the Pt|CrPS4 interface induces an additional current by the inverse spin Hall effect, effectively reducing the electrical resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' A polarization of the spin-Hall spin current (red arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 2(b)) parallel (normal) to the lo- cal moments of the magnet at the interface, minimizes (maximizes) the dephasing by the exchange interaction and therefore the electric resistance [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' An in-plane magnetic field H = Hˆy (ip angle ϕ = 0) cants the oop antiferromagnetic order by an angle θ⊥ = arcsin H 2HE + HA (1) with the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The electric resistance Rl of a Pt wire along the x-axis therefore should be maximal for θ⊥ = 0 and minimal for θ⊥ = π/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' at and beyond the spin- flip transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' On the other hand, magnon injection is most efficient when magnetic moments and current- induced spins are parallel, maximizing the non-local re- sistance Rnl = Vdetector/Iinjector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Rnl > 0 by defining the polarity of the voltage on the detector opposite to that of the current in the injector (see figure 2(b)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The Joule heating by a charge current I generates a temperature gradient over the interface and in the mag- net, generating a spin current and associated inverse spin Hall voltage (spin Seebeck effect) in the injector as well as the detector with associated local and non-local voltage signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We can separate the electrical and thermal signals by recording the first and second harmonic responses to a current bias that oscillates with frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The first (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='02 (ZHI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='8 W11 (a) W12 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='0 0 2 4 6 8 10 H (T) N N M1 M1 M1 M1 Ho Ho (c) (d) y y y M2 y M2 X X X M2 +3 m(a) (b) n 10 μm3 harmonic response reflects the Ohmic signal V ∼ I, while the thermal signals V ∼ I2 appear at double frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Here we focus on the linear response R(1ω) l/nl , with a brief discussion of R(2ω) l/nl in the Supplementary Material (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The spin Hall effect and inverse spin Hall effect dictate the following dependence on the ip field angle: R(1ω) l = R(1ω) l,0 + ∆R(1ω) l sin2 ϕ (2) R(1ω) nl = ∆R(1ω) nl cos2 ϕ (3) where R(1ω) 0,l/nl are constant offsets and ∆R(1ω) l/nl are the strenghts of the signals that depend on the ip field angle ϕ (defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We measured the local and non-local resistance in a liquid-He cryostat at temperatures between 5 and 300 K as a function of an ip magnetic field up to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='9 T and as a function of ip angle ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Figure 2 shows the schematics of device D1 and the local and non-local measurement configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The measurements on D1 were carried out on three different pairs of Pt contacts with edge-to- edge distances of 330 nm, 420 nm and 780 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' On device D2, we measured the resistances for two different pairs (∼300 nm and ∼450 nm) of Pt contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Results for device D3 with similar flake thickness and contacts with an edge to edge spacing of 300 nm are shown in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The time-dependent voltage responses may be ex- panded as V (t) = R1I(t)+R2I2(t)+· · · , where standard low frequency (7 Hz - 17 Hz) lock-in techniques access the constants R1 and R2 [7, 21] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 3 shows the observed R(1ω) l − R(1ω) l,0 (≈ 6 kΩ) of the injector contact as function of the direction of an ip magnetic field of 7 T at AC current bias of 60 µA and at T = 24 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The observed ϕ dependence agrees well with the model for the SMR sketched above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' ∆R(1ω) l /R(1ω) l,0 ≃ 10−4 is of the same order of magnitude as the SMR of CrPS4 in the oop configuration [17] and that of other magnetic materials, which is a strong indi- cation of an efficient interface exchange coupling and a large spin-mixing conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The modulation ∆R(1ω) l at T = 20 K in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 4(a) as a function of magnetic field strength agrees also with expectations, while the lack of a bias-current dependence confirms that we are in the linear response regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We observe saturation at fields > 6T (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 4), which corresponds to the onset of the spin-flip state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' At fields of several tesla, the SMR decreases with tem- perature but persists above TN and even up to room temperature (not shown), which is consistent with re- ports for CrPS4 [17] and the van der Waals material Cr2Ge2Te6 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The robust SMR can possibly be as- cribed to a TN that is enhanced by the interface spin orbit coupling and/or a paramagnetic SMR by a field- induced magnetization [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We now focus on the non-local signal R(1ω) nl plotted in the lower panel Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 3 for current bias of 60 µA, H = 7 T, T = 24 K (measured together with R(1ω) l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Top panel: Local resistance modulation ∆R(1ω) l of the Pt strip as function of in-plane angle ϕ (relative to the wire normal) of an applied magnetic field of 7 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The bias current is 60 µA and sample temperature is 24 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The red curve is a fit by sin2 ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' A 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='97 kΩ offset resistance has been subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Bottom panel: Simulataneously measured non- local resistance R(1ω) nl , fitted by cos2 ϕ (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' ∆R(1ω) l ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='6 mΩ is about 30 times smaller than that of Pt—YIG (thickness of 200 nm) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' R(1ω) nl is maxi- mum (minimum) at φ = 0◦ (φ = 90◦) which reflects the angular dependence of the spin injection and detection efficiencies by the spin Hall effects in Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 4 reveals a remarkable dependence of the magnon transport on magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' At H ≤ 6 T, no R(1ω) nl is observed within the experimental uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' At fields > 6 T, R(1ω) nl increases sharply and appears to saturate at fields > 7 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' This rapid increase correlates with the satu- ration of the bulk magnetization (see SM) and is therefore associated with the spin-flip transition from a canted an- tiferromagnetic (AFM) to a collinear ferromagnetic (FM) state (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 1(c&d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The non-local resistance at current bias of 60 µA and at 7 T in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 4 is non-monotonous, with a maximum around 25 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We attribute this to two effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' On one hand, the critical fields for the spin-flip transition de- creases with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' At low temperatures and 7 T, the sample is still in the canted-AFM phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The sharp increase in R(1ω) nl coincides with the formation of the saturated FM phase at T ∼ 25 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Moreover, the equilibrium magnon density and resulting magnon con- ductivity increase with temperature [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' At larger temperatures, the magnetization and R(1ω) nl decrease and vanish at TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Further, we assess the magnon transport in CrPS4 by measuring as a function of distance d between the Pt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='6 data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='5 sin2(Φ)-fit △RIw (Q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='1 90 0 90 180 270 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='0 data cos2()-fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='8 (U)m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='4 90 0 90 180 270 Angle Φ (deg)4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Field- and temperature-dependent results on device D2, Top: Field dependence of R1ω l at different bias currents at 25 Kelvin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' middle: Same for R1ω nl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Bottom: Temperature dependence of R1ω nl at 7 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Non-local resistance as function of distance d between the injector and detector contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' For different bias currents at sample temperature of 24 K, R(1ω) nl ∼ 1/d is shown for three different Pt strip pairs on device D1 with d being equal to 330 nm, 420 nm and 780 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' contacts as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The absence of a system- atic dependence on the current bias again confirms that we operate in the linear response regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The model of diffusive magnon transport in YIG leads to as decay of R(1ω) nl with increasing d as a function of the magnon dif- fusion length λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' For efficient spin injection at the Pt|YIG interface, this described by [7]: R(1ω) nl = C λ exp (d/λ) 1 − exp (2d/λ), (4) where C is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Since we observe an alge- braic R(1ω) nl ∼ 1/d rather than exponential dependence, magnon transport over the length scales d ≤ 1 µm is Ohmic [7], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' purely diffusive while magnon decay sets in at larger distances only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The abrupt field dependence of the non-local resistance differs sharply from the linear dependence of the SMR (top panes of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 4) that indicates a surface magne- tization proportional to a static magnetic susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' A surprise of the present study is the absence of non- local transport in the non-collinear phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' This behav- ior is markedly different from previous studies of trans- port that were carried out with magnetic fields parallel to the N´eel vector, including the spin-flop transition [8– 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' However, the associated theories do not address the present configuration either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The magnon band edges of CrPS4 in the canted phase as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 1(b) diagonalize the classical spin Hamiltonian with eigenfrequencies [26] at low fields H ≤ HE⊥ = 2HE + HA ω+ = γ � (2HE sin2 θ⊥ + HA cos2 θ⊥)(2HE + HA) ω− = γ � HA(2HE + HA) cos2 θ⊥ (5) and at high fields H > HE⊥ = 2HE + HA ω+ = γ � (H − HA)H ω− = γ � (H − 2HE)(H − 2HE − HA), (6) where θ⊥ is defined in equation 1 and includes the ex- ternal ip field H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The small anisotropy causes the low frequencies of the acoustic modes that at high magnetic fields and low temperatures are dominantly populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The collinear ferromagnetic phase above HE⊥ can be treated by a two-mode linearized Boltzmann equation similar to YIG, while the large SMR implies that inter- faces are transparent, so we may expect at high non-local signal at H > HE⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' As the Pt contacts do not inject or detect spin polarizations in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' At zero canting angle (θ⊥), the spin current injected by the Pt contacts is fully absorbed by the antiferromagnet in the form of a spin transfer torque to the magnetic sublat- tices, while magnon injection and R(1ω) nl vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' With in- creasing canting angle, the magnon injection efficiency in- creases proportional with the induced net magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' However, the exchange interaction in a non-collinear con- figuration also increasingly affects the non-local signal by reducing the magnon decay length (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Tang, in prepara- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' In the collinear phase both magnon injection and magnon transport do not prevent the non-local signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The abruptness of the observed onset of non-local trans- port at the spin-flip transition field is surprising, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 60uA25K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='15 80uA25K R1(Q2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='05 8 6 4 2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='4 60uA25K 80uA25K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='0 8 6 4 2 0 2 4 6 8 Field (T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='4 40uA 7 T 60uA 7 T 80uA 7 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='3 (uw) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='0 10 15 20 25 30 35 40 45 50 Temperature (K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='8 140uA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='7 80uA 60uA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='6 40uA (m2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='3 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content='0 300 400 500 600 700 800 Distance (nm)5 The abrupt increase of non-local resistance near the spin flip transition may be caused by the combined effects of enhanced magnon injection into the low energy magnon branch and the sudden suppression of magnon relaxation when the system approaches the FM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Summarizing, we report non-local spin transport in a van der Waals magnet, to the best of our knowl- edge for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' The spin conduit is the elec- trically insulating antiferromagnet CrPS4 with perpen- dicular anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We focus on a configuration that has escaped attention even in conventional antiferromagnets, with an in-plane magnetic field normal to the Pt spin injector and detector that tilts the antiparallel spins into the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Surprisingly, we do not observe spin trans- port in the non-collinear phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' At the critical field that forces the transition to a collinear ferromagnetic phase, we observe an abrupt increase of the non-local spin signal over distances that exceed a micron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' These results herald the potential of 2D van der Waals magnets for scalable magnonic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge the technical support from J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Hol- stein, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Adema T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Schouten, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' de Vries and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' van der Velde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' We acknowledge the financial sup- port of the Zernike Institute for Advanced Materials and the European Union’s Horizon 2020 research and inno- vation program under Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 785219 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 881603 (Graphene Flagship Core 2 and Core 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' This project is also financed by the NWO Spinoza prize awarded to BJW by the NWO and has received funding from the European Research Council (ERC) under the European Union’s 2DMAGSPIN (Grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 101053054).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' GB acknowledges funding by JSPS Kakenhi grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' 19H00645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Gong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Ji, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} 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Gurevich and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} +page_content=' Melkov, Magnetization oscil- lations and waves (CRC Press, Boca Raton, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfkAQx/content/2301.03268v1.pdf'} diff --git a/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf b/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2ed4d66e916d4732dbb802342cdb693bcfb730b5 --- /dev/null +++ b/vdE3T4oBgHgl3EQfkwp-/content/2301.04600v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:61f525a224eb628cef511085d733fa75024c479e5f82261b90161e7a5a15edeb +size 209011 diff --git a/vdE3T4oBgHgl3EQfkwp-/vector_store/index.faiss b/vdE3T4oBgHgl3EQfkwp-/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f36d89b393b7d00c44256cddda77a28dffb08c79 --- /dev/null +++ b/vdE3T4oBgHgl3EQfkwp-/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7fc9c22f07a54f9f7aa67c3b7fbf3fd4616bae099e9bbf4c3a7e7fc8f6a300ae +size 1572909 diff --git a/vdE5T4oBgHgl3EQfMA4V/content/tmp_files/2301.05477v1.pdf.txt b/vdE5T4oBgHgl3EQfMA4V/content/tmp_files/2301.05477v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..147dc8e17afa6189575a278bfbaa5c7cecb8a404 --- /dev/null +++ b/vdE5T4oBgHgl3EQfMA4V/content/tmp_files/2301.05477v1.pdf.txt @@ -0,0 +1,387 @@ +arXiv:2301.05477v1 [math.AG] 13 Jan 2023 +A remark on a result of Huber and Kahn +Somayeh Habibi and Farhad Rahmati +January 16, 2023 +Abstract +A. Huber and B. Kahn construct a relative slice filtration on a prin- +cipal T-bundle X → Y for smooth Y . As a consequence of their result, +one can observe that the mixed Tateness of the motive M(Y ), associated +to the smooth scheme Y , implies that the motive M(X) is mixed Tate. +In this note we prove the inverse implication for a principal G-bundle, +for a split reductive group G. +Mathematics Subject Classification (2000): 14C25 +Keywords: Voevodsky motives; mixed Tate motives; slice filtration; G- +bundles +Introduction +In [3], A. Huber and B. Kahn establish their theory of slice filtration for motives +inside the Voevodsky’s motivic categories. As an application, they construct +a relative slice filtrarion on a principal T-bundle, for a split torus T, over a +perfect field k. Consequently, one can easily see that for a principal T-bundle +X over a smooth variety Y , the motive M(X) is mixed Tate if M(Y ) is mixed +Tate. Using a geometric approach, in [1], the first named author and Arasteh +Rad proved that (in certain cases) the motive of a G-bundle X over a scheme +Y , for a split reductive group G, lies in the category generated by mixed Tate +motives and the motive corresponding to Y . Thus the mixed Tateness of M(Y ) +implies that M(X) is mixed Tate. In this note we prove the inverse implication. +Namely, we prove the following: +1 + +Theorem 0.1. Let X be a principal G-bundle over Y , for a split reductive +group G over a perfect field k. Assume further that either k = C or X is locally +trivial for the Zariski topology on Y . Then M(X) is mixed Tate if and only if +M(Y ) is mixed Tate. In particular when k is a finite field and M(X) is mixed +Tate, the Q-vector spaces Chi(Y ) are finite. +This is corollary 2.14 in the text. +1 +Notation and Conventions +Throughout this article we assume that k is a perfect field. We denote by Schk +(resp. Smk) the category of schemes (resp. smooth schemes) of finite type over +k. +To denote the motivic categories over k, such as +DMgm(k), DMeff +gm (k), DMeff +− (k), DMeff +− (k) ⊗ Q, etc. +and the functors +M(−) : Schk → DMeff +gm (k) +and +Mc(−) : Schk → DMeff +gm (k), +constructed by Voevodsky, we use the same notation that was introduced by +him in [7]. For the definition of the geometric motives with compact support in +positive characteristics we also refer to [3, Appendix B]. +Moreover for any object M of DMgm(k) we denote by M∗ the internal Hom- +object HomDMgm(M, Z). +Note finally that throughout this article we either assume that k admits +resolution of singularities or we consider the motivic categories after passing to +coefficients in Q. +2 +Slice filtration and motive of G-bundles +Let us first recall the definition of cellular varieties. +2 + +Definition 2.1. A variety is called cellular if it contains a cell (i.e. a vari- +ety isomorphic to an affine space) as an open subvariety such that the closed +complement is cellular. +Definition 2.2. An object M of DMgm(k) is called pure Tate motive if it is a +(finite) direct sum of copies of Z(p)[2p] for p ∈ Z. +Remark 2.3. +(a) Using localization triangle, one can easily check that the +motive associated to a smooth cellular variety is pure Tate. See also [5, +Cor. 3.6]. +(b) Let P be a parabolic subgroup of a split reductive group G. Let us recall +that the action of Borel subgroup B on the projective homogeneous variety +G/P induces a cell decomposition on it. E.g. see [6, Thm 2.1]. +Definition 2.4. We denote by TDMeff +gm (k) the thick tensor subcategory of +DMeff +gm (k), generated by Z(0) and the Tate object Z(1). An object of TDMeff +gm (k) +is called a mixed Tate motive. +TDMeff +− (k) is the localizing subcategory of +DMeff +− (k) generated by TDMeff +gm (k). +Definition 2.5. A variety X is called +(a) mixed Tate, if Mc(X) ∈ TDMeff +gm (k). +(b) stratified mixed Tate, if it admits a stratification {Xi}i∈I, such that Xi is +smooth and mixed Tate, for every i ∈ I. +Lemma 2.6. The motive Mc(X) of a stratified mixed Tate variety X is mixed +Tate. +Proof. Let {Xi}i∈I be a stratification of X such that each Xi is smooth and +mixed Tate. Consider the partial order on I which is defined as follows: +for i ̸= j : +i < j +if and only if +Xi ⊆ Xj. +One defines the length of a stratification {Xi}i∈I to be +ℓ({Xi}i∈I) := max{n; ∃ i0 < i1 < · · · < in with {ij}n +j=0 ⊆ I}. +3 + +Now we prove the lemma by induction on the length of the stratification +{Xi}i∈I of X. The base case is trivial. Assume that the length of the stratifi- +cation is n, and the lemma holds for all stratifications with length < n. Set +U := +� +j +Xj, +where j runs over all maximal elements of I with respect to the partial oreder. +Since X∖U is closed and X = U, the subset U is open and dense in X. Now the +statement follows from the induction hypothesis and the following localization +triangle +Mc(X ∖ U) → Mc(X) → Mc(U) → Mc(X ∖ U)[1]. +Let us now recall the definition of the n-th slice filtration and motivic fun- +damental invariants from [HK]. +Definition 2.7. Define the n-th slice filteration v≥nM of a motive M as the +internel Hom object Hom(Z(n), M)(n). +The functor cn(−) is a triangulated endofunctor +cn(−) : DMeff +− (k) → DMeff +− (k), +which assigns the n-th motivic fundamental invariant cn(M), to a motive M ∈ +DMeff +− (k), constructed by A. Huber and B. Kahn in [3]. Below we briefly recall +their construction. +Definition 2.8. +(a) For M in DMeff +− (k), by adjunction there is a morphism +an : ν⩾M → M (resp. f n : ν⩾nM → ν⩾n−1M), corresponding to identity +in +Hom(Hom(Z(n), M), Hom(Z(n), M)) = Hom(ν⩾nM, M), +(resp. = Hom(ν⩾nM, ν⩾n−1M)). +(b) Define ν d and cdMc(X) = CHd(X)[0], where CHd(X) denotes the +homotopy invariant Nisnevich sheaf with transfers U �→ CHd(X⊗F F(U)). +Here F(U) is the total ring of fractions of U. If X is smooth, and with +coefficients in Q, the assumption of characteristic 0 is not necessary. +(d) Let X be a smooth variety such that M(X) is a pure Tate motive. Then +there is a natural isomorphism +M(X) = +� +p +cp(M(X))(p)[2p], +5 + +with cp(M(X)) = CHp(X)∗[0]. Here +∗ +− denotes the dual of a free abelian +group. +Proof. Part a) follows from definition. For part b) see [3, lem 4.8] and for c) +see [3, Prop 1.7 and Prop 1.8]. For the last part see [3, Prop 4.10]. +Let Db(Ab) be the bounded derived category of abelian groups Ab. Denote +by Db +f(Ab) its full subcategory consisting of those objects whose cohomology +groups are finitely generated. Consider the fully faithful triangulated functor +ι : Db +f(Ab) → DMeff +gm (k), +which sends Z to Z[0] and respect tensor structures. Its essential image is the +thick tensor subcategory of DMeff +gm (k) generated by Z(0). Similarly there is +a fully faithful tensor functor ι : D−(Ab) → DMeff +− (k) from bounded above +derived category D−(Ab) to DMeff +− (k), whose essential image is generated by +Z(0). +The following proposition [3, Prop 4.6], gives a criterion for mixed Tateness +of a motive M, in terms of the associated fundamental motivic invariants cn(M). +Proposition 2.10. Assume either chark = 0 or coefficients in Q. A motive +M ∈ DMeff +gm (k) is in TDMeff +gm (k) if and only if cn(M) ∈ Db +f(Ab) for all n and +cn(M) = 0 for n large enough. If M ∈ TDMeff +− (k), then cn(M) ∈ D−(Ab). +In [3] theorem 8.8 the authors construct a relative slice filtration on a torus +bundle. They use this filtration to compute the motive associated to a split +reductive group G. Below we recall their result. +Proposition 2.11. Let T be a split torus of dimension r and X a principal T +bundle over a smooth variety Y over a field k. There is a filtration +ν⩾p+1 +Y +M(X) → ν⩾p +Y M(X) → ... → M(X) +in DMeff +gm (k), where M(X) ∼= ν⩾0M(X), ν⩾r+1M(X) = 0, together with dis- +tinguished triangles +ν⩾p+1 +Y +M(X) +� ν⩾p +Y M(X) +�❧❧❧❧❧❧❧❧❧❧❧❧❧❧ +M(Y )(p)[p] ⊗ ∧p(Ξ) +[1] +�❙❙❙❙❙❙❙❙❙❙❙❙❙❙ +(2.3) +for 0 ≤ p ≤ r. Here Ξ = Hom(Gm, T) is the cocharacter group. +6 + +Remark 2.12. Note moreover that in [1, §4] the authors construct a nested +filtration on the motive associated to a G-bundle using the theory of wonderful +compactification of reductive groups. +Theorem 2.13. Let T be a torus of dimension r. For a T-bundle X over Y , +we have the following statements: +i) If Y is stratified mixed Tate, then X is mixed Tate. +ii) Assume that Y is smooth. If X is mixed Tate, then Y is mixed Tate. +Proof. First we prove ii). For this let us first assume that r = 1, i.e. T = Gm, +and X is a Gm-bundle over Y . For the P1-bundle X ×T P1 = P, consider the +following exact triangle +Mc(X) → Mc(P) → Mc(P ∖ X) → Mc(X)[1]. +By projective bundle formula [8, Thm 15.12], we get +Mc(X) → Mc(Y ) ⊕ Mc(Y )(1)[2] → (Mc(Y ))⊕2 → Mc(X)[1]. +Applying functor cn(−) to the above triangle we get +cn(Mc(X)) → cn(Mc(Y )) ⊕ cn−1(Mc(Y )) → (cn(Mc(Y )))⊕2 → cn(Mc(X)[1]). +Note that cn(M(1)[2]) = cn−1(M), see proposition 2.9 a). For n > dim Y , we +have cn(Mc(Y )) = 0, see proposition 2.9 c), and therefore we have +cn(Mc(X)) = cn−1(Mc(Y )). +Since X is mixed Tate, by proposition 2.10 we see that cn((Mc(X)), and hence +cn−1(Mc(Y )), lie in Db +f(Ab). Now, regarding the following triangle +cn−1(Mc(X)) → cn−1(Mc(Y )) ⊕ cn−2(Mc(Y )) → (cn−1(Mc(Y )))⊕2, +and proposition 2.10, we observe that cn−2(Mc(Y )) also lies in Db +f(Ab). Fol- +lowing this way, we can argue recursively that ci(Mc(Y )) lie in Db +f(Ab) for all +i < n. +Note that the general case reduces to the case where r = 1. Namely, recall +that for a principal G-bundle G → Y , and a closed subgroup scheme H of the +7 + +reduct´ıve algebraic group G, the quotient map G → G/H ∼= G ×G (G/H) is a +principal H-bundle. Regarding this, we can view the T-bundle X as a sequence +X =: Xr → Xr−1 → · · · → X2 → X1 → X0 := Y, +such that Xi is a Gm-bundle over Xi−1 for 1 ≤ i ≤ r. +To prove i) let us first assume that Y is smooth. +In this case we have the +following sequence +ν⩾p+1 +Y +M(X) +� ν⩾p +Y M(X) +� +ν⩾2 +Y M(X) +� ν⩾1 +Y M(X) +� +� +M(X) +� +... +λp(Y, T) +[1] +�❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ +λ1(Y, T) +[1] +�❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁ +λ0(Y, T) +[1] +�❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁ +(2.4) +of exact triangles, according to proposition 2.11. Here +λp(Y, T) := M(Y )(p)[p] ⊗ Λp(Ξ), +for 0 ≤ p ≤ r. Recall that M(X) ∼= ν⩾0 +Y M(X) and ν⩾r+1 +Y +M(X) = 0. +Since ν⩾r+1 +Y +M(X) = 0 we get an isomorphism +ν⩾r +Y M(X) ∼ +−→ M(Y )(p)[p] ⊗ Λp(Ξ). +Therefore ν⩾r +Y M(X) is mixed Tate. This is because Y is smooth and mixed +Tate, and that M(Y )∗ ∼= Mc(Y )(−n)[−2n]. Note that for mixed Tate motives +M and N in DMeff +gm (k), the Hom object Hom(M, N) is mixed Tate. This fol- +lows from the fact that Hom(Z(i), Z(j)) equals Z(j − i) if i ≤ j and vanishes +elsewise. In particular we see that M is mixed Tate if the dual object M∗ is +mixed Tate, which justifies that for smooth schemes the definition 2.5. agrees +with the alternative definition using the functor M(−). +Regarding the above discussion and the diagram 2.4, we may argue recursively +that ν⩾i +Y M(X) are mixed Tate, for every i ≥ 0. +Now, assume that Y is stratified mixed Tate, with stratification {Yi}i. Then, +by the above arguments we see that Xi := X ×Y Yi are mixed Tate and hence +X is stratified mixed Tate. Thus we may conclude by lemma 2.6 +8 + +Corollary 2.14. Let G be a split reductive group over a perfect field k. Let X +be a principal G-bundle over Y . Assume further that either X is locally trivial +over Y for the Zariski topology or k = C. Then M(X) is mixed Tate if and +only if M(Y ) is mixed Tate. In particular when k is a finite field and M(X) is +mixed Tate, the Q-vector spaces CHi(Y ) are finite. +Proof. Assume that Y is mixed Tate. Then M(X) is mixed Tate by [1, Thm +0.4 c) and d)]. +Now assume that Mc(X) is mixed Tate. Let T be the maximal torus in G and +let B be a Borel subgroup containing T. Applying the above theorem to the +T-bundle X → X/T, we see that Mc(X/T) is mixed Tate. This implies that +Mc(X/B) is mixed Tate. Note that B = TU for the unipotent group U, and +thus X/T → X/B is an affne bundle. The latter is a projective homogeneous +fiberation over Y . Now the statement follows from remark 2.3 b), proposition +2.9 d), motivic Leray-Hirsch theorem [1, Thm 2.8] and [4, Thm 1.1.b)]. For the +last statement see [2, Prop 3.9]. +Acknowledgment: +The first named author warmly thanks B. Kahn, J. Ay- +oub and E. Arasteh Rad for useful comments and discussions. +She is also +grateful to Prof. Luca Barbieri Viale for steady encouragement. +References +[1] S. Habibi, E. Arasteh Rad, On the motive of fibre bundle and its applica- +tions, Analysis, geometry and number theory, 2017, Pisa : Fabrizio Serra, +2017 , 77-96 +[2] S. Habibi, E. Arasteh Rad, Motivic Remarks On The Moduli Stacks +Of global G-Shtukas And Their Local Models, +preprint available at +https://arxiv.org/abs/1912.09968 +[3] A. Huber, B. Kahn, The slice filtration and mixed Tate motives. Compositio +Mathematica, 142(4), 907-936. doi:10.1112/S0010437X06002107 +[4] J. Iyer, Absolute Chow-K¨unneth decomposition for rational homogeneous +bundles and for log homogeneous varieties. Michigan Math. J., 60(1):79–91 +(2011). +9 + +[5] B. Kahn, Motivic cohomology of smooth geometrically cellular varieties, +in Algebraic K-theory, Seattle, 1997, Proceedings of Symposia in Pure +Mathematics, vol. 67 (American Mathematical Society, Providence, RI, +1999), 149–174. +[6] B. K¨ock, Chow motif and higher Chow theory of G/P, Manuscripta Math. +70 (1991), 363–372, +[7] V. Voevodsky, Triangulated categories of motives over a field, in Cycles, +transfers and motivic cohomology theories, Annals of Mathematics Studies, +vol. 143 (Princeton University Press, Princeton, NJ, 2000), 188–238. +[8] C. Mazza, V. Voevodsky, C. A. Weibel. Lecture notes on motivic cohomol- +ogy, Clay mathematics monographs, v.2. (2006). +Somayeh Habibi, Faculty of Mathematics, Tehran Polytechnic Univ., 424 Hafez +Ave., Tehran 15914, Iran, email: somayeh.habibi@aut.ac.ir +Farhad Rahmati, Faculty of Mathematics, Tehran Polytechnic Univ., 424 Hafez +Ave., Tehran 15914, Iran, email: frahmati@aut.ac.ir +10 + diff --git a/vdE5T4oBgHgl3EQfMA4V/content/tmp_files/load_file.txt b/vdE5T4oBgHgl3EQfMA4V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8f696d70848ec740d3cfc0c4f7ad781a8774ae9 --- /dev/null +++ b/vdE5T4oBgHgl3EQfMA4V/content/tmp_files/load_file.txt @@ -0,0 +1,264 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf,len=263 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='05477v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='AG] 13 Jan 2023 A remark on a result of Huber and Kahn Somayeh Habibi and Farhad Rahmati January 16, 2023 Abstract A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Huber and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Kahn construct a relative slice filtration on a prin- cipal T-bundle X → Y for smooth Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' As a consequence of their result, one can observe that the mixed Tateness of the motive M(Y ), associated to the smooth scheme Y , implies that the motive M(X) is mixed Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' In this note we prove the inverse implication for a principal G-bundle, for a split reductive group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Mathematics Subject Classification (2000): 14C25 Keywords: Voevodsky motives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' mixed Tate motives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' slice filtration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' G- bundles Introduction In [3], A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Huber and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Kahn establish their theory of slice filtration for motives inside the Voevodsky’s motivic categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' As an application, they construct a relative slice filtrarion on a principal T-bundle, for a split torus T, over a perfect field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Consequently, one can easily see that for a principal T-bundle X over a smooth variety Y , the motive M(X) is mixed Tate if M(Y ) is mixed Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Using a geometric approach, in [1], the first named author and Arasteh Rad proved that (in certain cases) the motive of a G-bundle X over a scheme Y , for a split reductive group G, lies in the category generated by mixed Tate motives and the motive corresponding to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Thus the mixed Tateness of M(Y ) implies that M(X) is mixed Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' In this note we prove the inverse implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Namely, we prove the following: 1 Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Let X be a principal G-bundle over Y , for a split reductive group G over a perfect field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Assume further that either k = C or X is locally trivial for the Zariski topology on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Then M(X) is mixed Tate if and only if M(Y ) is mixed Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' In particular when k is a finite field and M(X) is mixed Tate, the Q-vector spaces Chi(Y ) are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' This is corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='14 in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' 1 Notation and Conventions Throughout this article we assume that k is a perfect field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' We denote by Schk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Smk) the category of schemes (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' smooth schemes) of finite type over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' To denote the motivic categories over k, such as DMgm(k), DMeff gm (k), DMeff − (k), DMeff − (k) ⊗ Q, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' and the functors M(−) : Schk → DMeff gm (k) and Mc(−) : Schk → DMeff gm (k), constructed by Voevodsky, we use the same notation that was introduced by him in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' For the definition of the geometric motives with compact support in positive characteristics we also refer to [3, Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Moreover for any object M of DMgm(k) we denote by M∗ the internal Hom- object HomDMgm(M, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Note finally that throughout this article we either assume that k admits resolution of singularities or we consider the motivic categories after passing to coefficients in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' 2 Slice filtration and motive of G-bundles Let us first recall the definition of cellular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' 2 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' A variety is called cellular if it contains a cell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' a vari- ety isomorphic to an affine space) as an open subvariety such that the closed complement is cellular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' An object M of DMgm(k) is called pure Tate motive if it is a (finite) direct sum of copies of Z(p)[2p] for p ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' (a) Using localization triangle, one can easily check that the motive associated to a smooth cellular variety is pure Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' See also [5, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' (b) Let P be a parabolic subgroup of a split reductive group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Let us recall that the action of Borel subgroup B on the projective homogeneous variety G/P induces a cell decomposition on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' see [6, Thm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' We denote by TDMeff gm (k) the thick tensor subcategory of DMeff gm (k), generated by Z(0) and the Tate object Z(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' An object of TDMeff gm (k) is called a mixed Tate motive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' TDMeff − (k) is the localizing subcategory of DMeff − (k) generated by TDMeff gm (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' A variety X is called (a) mixed Tate, if Mc(X) ∈ TDMeff gm (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' (b) stratified mixed Tate, if it admits a stratification {Xi}i∈I, such that Xi is smooth and mixed Tate, for every i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' The motive Mc(X) of a stratified mixed Tate variety X is mixed Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Let {Xi}i∈I be a stratification of X such that each Xi is smooth and mixed Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Consider the partial order on I which is defined as follows: for i ̸= j : i < j if and only if Xi ⊆ Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' One defines the length of a stratification {Xi}i∈I to be ℓ({Xi}i∈I) := max{n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' ∃ i0 < i1 < · · · < in with {ij}n j=0 ⊆ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' 3 Now we prove the lemma by induction on the length of the stratification {Xi}i∈I of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' The base case is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Assume that the length of the stratifi- cation is n, and the lemma holds for all stratifications with length < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Set U := � j Xj, where j runs over all maximal elements of I with respect to the partial oreder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Since X∖U is closed and X = U, the subset U is open and dense in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Now the statement follows from the induction hypothesis and the following localization triangle Mc(X ∖ U) → Mc(X) → Mc(U) → Mc(X ∖ U)[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Let us now recall the definition of the n-th slice filtration and motivic fun- damental invariants from [HK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Define the n-th slice filteration v≥nM of a motive M as the internel Hom object Hom(Z(n), M)(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' The functor cn(−) is a triangulated endofunctor cn(−) : DMeff − (k) → DMeff − (k), which assigns the n-th motivic fundamental invariant cn(M), to a motive M ∈ DMeff − (k), constructed by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Huber and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Kahn in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Below we briefly recall their construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' (a) For M in DMeff − (k), by adjunction there is a morphism an : ν⩾M → M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' f n : ν⩾nM → ν⩾n−1M), corresponding to identity in Hom(Hom(Z(n), M), Hom(Z(n), M)) = Hom(ν⩾nM, M), (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' = Hom(ν⩾nM, ν⩾n−1M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE5T4oBgHgl3EQfMA4V/content/2301.05477v1.pdf'} +page_content=' (b) Define ν0.98, +strel(’square’, 8)); +(3) + +0 +5 +10 +15 +20 +25 +0 +1 +2 +3 +Kodak image number +PSNRCA-FSE − PSNRFSE in dB +bs 16 +bs 8 +bs 4 +0 +20 +40 +60 +80 +100 +0 +0.5 +1 +1.5 +Tecnick image number +PSNRCA-FSE − PSNRFSE in dB +bs 16 +bs 8 +bs 4 +Figure 4. +Difference PSNR in dB of the reconstructed pixels between the centroid adaption (CA-FSE) and the unmodified weighting function (FSE) for a +dense loss pattern, Left: Kodak, Right: Tecnick. +sparse loss pattern=imdilate(rand(img size)>0.95, +strel(’square’, 8)); +(4) +Visual examples for the two loss patterns are shown in Fig. 1. Lost +pixels are colored in black. The dense loss pattern in (3) causes a +loss of about 28% of pixels and the sparse loss pattern in (4) about +4%, respectively. The challenging patterns are very similar to the lost +areas that occur in different applications. We choose the dense pattern +to evaluate the algorithms on larger lost areas. +For both loss pattern types, the simulations were repeated for 10 +different loss patterns to obtain significant results and to avoid special +effects from a single specific pattern. For each loss pattern, the PSNR +is computed on the reconstructed pixels only. Then, for each loss +pattern, the average is computed for the 10 different loss patterns. +Fig. 4 shows the results for the Kodak and the Tecnick database +for the dense loss pattern. Whenever the difference is positive, CA- +FSE obtains a better reconstruction result. The dotted lines show +the average result for each test set. The corresponding values to +the dotted lines are listed as diff in the first two lines of Tab. II +together with the absolute PSNR values. In the lower part, Tab. II +contains the corresponding averaged results of the simulation using +the sparse loss pattern. Further, both tables contain results from +four recent reference methods from the literature, namely based on +kernel regression (CKR) and (SKR) [5], total variation (TV) [6], and +constraint lagrangian shrinkage (CSALSA) [7]. +Generally, all PSNR values increase by about 3 dB for the sparse +loss pattern as can be seen by comparing the upper and the lower +part of Tab. II. The sparse loss pattern causes less and above all +smaller lost areas. From the latter, all methods can profit. Comparing +the achieved gains of CA-FSE to the unmodified FSE, the gains +also increase for less and smaller lost areas. The weighting function +w [m, n] is centered on the centroid of all lost pixels within the +block b. When there are less lost areas, the case of more than one +distinct lost area within b occurs less often. When there is only one +lost area within b, the w [m, n] can be centered exactly on the centroid +of this one lost area. +The behavior of the results is similar for all datasets with exception +of ARRI, where CA-FSE obtains an additional gain of about 1 dB +for the dense loss pattern compared to the other databases. For the +sparse loss pattern and bs 16, a remarkable gain of 6.1 dB is obtained +compared to FSE. We further investigated this case and found that +images from the ARRI database have a small black border. Slightly +wrong reconstruction values on this border can cause a very big +loss in PSNR. Compared to the other databases, the extreme large +gains of CA-FSE and CSALSA for ARRI mostly come from border +effects. To further analyze the performance disregarding the border, +we additionally evaluated the reconstruction quality, omitting a border +of 16 pixels. Excluding the image border, the gain of 6.1 dB shrinks +to 1.8 dB which is in the same range compared to the other databases. +The centroid adaption does not only lead to a better reconstruction +of lost areas at image borders but also can improve the performance +within the image. +The achieved reconstruction gain grows with increasing size of the +block b. The reason is that the centroid of lost pixels is computed +within the block b. So, for a small block size, the center of the +weighting function can only be moved in a small range. Nevertheless, +the influence on the reconstruction result is remarkable keeping in +mind that for bs 4 the center can move by a maximum 1.5 · +√ +2 ≈ +2.1 pixels for the extreme case that exactly only one pixel located +in one of the corners of b is lost. With a larger block b, the center +can move to a larger extend. This explains the increasing influence +on the results with increasing block size. +Fig. 5 shows reconstruction results for visual comparison. When +pixels are lost in smooth image regions, an adaption of the weight- +ing function has no real influence on the reconstruction result. In +structured regions, and especially when lost areas occur at edges, the +proposed centroid adaption of the weighting function is advantageous. +The improved reconstruction performance of CA-FSE at the image +border can be seen best for the gray border at the top of the images. +V. CONCLUSION +In this paper, we consider the challenging reconstruction of arbi- +trarily shaped lost areas in images. We propose CA-FSE, a centroid +adaption of the weighting function of FSE, to address the arbitrary +shape of lost areas. Over our large test set, CA-FSE consistently +improves the reconstruction quality. The treatment of lost areas +at image borders is improved. With the proposed adaption, the +reconstruction performance of FSE can be improved by 1.29 dB of +PSNR on average for arbitrarily shaped lost image areas. . +Further work aims at optimizing the reconstruction, i.e., only one +distinct lost area within the currently considered block is recon- +structed at once. We also aim at an adaption of the shape of the +weighting function. + +Table II +AVERAGED PSNR VALUES IN DB USING THE DENSE LOSS PATTERN (3) IN THE UPPER PART AND THE SPARSE LOSS PATTERN (4) IN THE LOWER PART. +CKR[5] +SKR[5] +TV[6] +CSALSA +bs 4 +bs 8 +bs 16 +[7] +FSE[8] +CA-FSE +FSE[8] +CA-FSE +FSE[8] +CA-FSE +proposed +diff +proposed +diff +proposed +diff +Dense Loss Pattern +Kodak +21.952 +21.476 +23.264 +23.856 +25.419 +25.764 ++ 0.345 +24.998 +25.447 ++ 0.449 +23.895 +24.856 ++ 0.961 +Tecnick +23.667 +23.058 +25.394 +25.258 +28.096 +28.165 ++ 0.069 +27.529 +27.733 ++ 0.204 +26.570 +27.215 ++ 0.645 +Middleburry +28.564 +27.524 +31.354 +31.227 +34.912 +35.025 ++ 0.113 +34.225 +34.481 ++ 0.256 +33.086 +33.874 ++ 0.788 +Arri +26.582 +26.151 +26.966 +28.610 +31.189 +32.357 ++ 1.168 +30.497 +31.528 ++ 1.031 +28.775 +30.543 ++ 1.768 +Sparse Loss Pattern +Kodak +24.602 +25.931 +25.209 +26.219 +28.105 +28.782 ++ 0.677 +27.480 +28.586 ++ 1.106 +25.055 +27.866 ++ 2.811 +Tecnick +27.618 +29.552 +28.851 +28.674 +31.901 +32.162 ++ 0.261 +31.265 +31.932 ++ 0.667 +29.224 +31.144 ++ 1.920 +Middleburry +33.979 +36.285 +35.359 +35.278 +39.545 +39.913 ++ 0.368 +38.777 +39.634 ++ 0.857 +36.305 +38.667 ++ 2.362 +Arri +29.051 +31.332 +28.601 +33.086 +34.875 +37.648 ++ 2.773 +33.919 +37.206 ++ 3.287 +29.815 +35.936 ++ 6.121 +Original +CKR[5] +TV[6] +FSE[8] bs8 +Lost +SKR[5] +CSALSA[7] +CA-FSE bs8 +proposed +Figure 5. +Visual reconstruction results for a detail of image 15 of the Kodak +database using the dense loss pattern. The lost areas are black in the lower +left image. +ACKNOWLEDGMENT +The authors would like to thank Eduard Sch¨on and Nils Genser for +their valuable assistance. Further, we gratefully acknowledge that this +work has been supported by the Deutsche Forschungsgemeinschaft +(DFG) under contract number KA 926/4-2. +REFERENCES +[1] C. Fehn, “Depth-Image-Based Rendering (DIBR), Compression, and +Transmission for a New Approach on 3D-TV,” San Jose, CA, USA, +Jan. 2004, pp. 93–104. +[2] U. Kim and M. Sunwoo, “New Frame Rate Up-Conversion Algorithms +With Low Computational Complexity,” IEEE Trans. on Circuits and +Systems for Video Technology, vol. 24, no. 3, pp. 384–393, Mar. 2014. +[3] N. Bozinovic, J. Konrad, W. Zhao, and C. Vazquez, “On the Importance +of Motion Invertibility in MCTF/DWT Video Coding,” in Proc. IEEE +Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), +Philadelphia, PA, USA, Mar. 2005, pp. 49–52. +[4] W. Schnurrer, +J. Seiler, +and A. Kaup, “Improving Block-Based +Compensated Wavelet Lifting by Reconstructing Unconnected Pixels,” +in Proc. IEEE Int. Symposium on Signals, Circuits and Systems +(ISSCS), Iasi, Romania, Jul. 2013, pp. 1–4. +[5] H. Takeda, S. Farsiu, and P. Milanfar, “Kernel Regression for Image +Processing and Reconstruction,” IEEE Trans. on Image Processing, +vol. 16, no. 2, pp. 349–366, Feb. 2007. +[6] J. Dahl, P. Hansen, S. Jensen, and T. Jensen, “Algorithms and Software +for Total Variation Image Reconstruction via First-order Methods,” +Numerical Algorithms, vol. 53, no. 1, pp. 67–92, 2010. +[7] M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “An Augmented +Lagrangian Approach to the Constrained Optimization Formulation of +Imaging Inverse Problems,” IEEE Trans. on Image Processing, vol. 20, +no. 3, pp. 681–695, Mar. 2011. +[8] J. Seiler and A. Kaup, “Complex-Valued Frequency Selective Extrap- +olation for Fast Image and Video Signal Extrapolation,” IEEE Signal +Processing Letters, vol. 17, no. 11, pp. 949–952, Nov. 2010. +[9] ——, “Optimized and Parallelized Processing Order for Improved +Frequency Selective Signal Extrapolation,” in Proc. European Signal +Processing Conference (EUSIPCO), Barcelona, Spain, Aug. 2011, pp. +269–273. +[10] N. Asuni and A. Giachetti, “Testimages: A Large-scale Archive for +Testing Visual Devices and Basic Image Processing Algorithms,” STAG +- Smart Tools & Apps for Graphics Conference, Sep. 2014. +[11] Kodak test images. http://r0k.us/graphics/kodak/. +[12] S. Andriani, H. Brendel, T. Seybold, and J. Goldstone, “Beyond the +Kodak Image Set: A New Reference Set of Color Image Sequences,” +in Proc. IEEE Int. Conf. on Image Processing (ICIP), Melbourne, VIC, +Australia, Sep. 2013, pp. 2289–2293. +[13] D. Scharstein and C. Pal, “Learning Conditional Random Fields for +Stereo,” in IEEE Computer Society Conf. on Computer Vision and +Pattern Recognition (CVPR), Minneapolis, MN, USA, Jun. 2007, pp. +1–8. +[14] H. Hirschm¨uller and D. Scharstein, “Evaluation of Cost Functions for +Stereo Matching,” in IEEE Computer Society Conf. on Computer Vision +and Pattern Recognition (CVPR), Minneapolis, MN, USA, Jun. 2007, +pp. 1–8. +[15] D. Scharstein, H. Hirschm¨uller, Y. Kitajima, G. Krathwohl, N. Neˇsi´c, +X. Wang, and P. Westling, “High-Resolution Stereo Datasets with +Subpixel-Accurate Ground Truth,” in German Conf. on Pattern Recog- +nition (GCPR), M¨unster, Germany, Sep. 2014, pp. 31–42. + diff --git a/y9E4T4oBgHgl3EQfAAvD/content/tmp_files/load_file.txt b/y9E4T4oBgHgl3EQfAAvD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54fc8c14505c2aa8c542dde4ebc00bcc88582009 --- /dev/null +++ b/y9E4T4oBgHgl3EQfAAvD/content/tmp_files/load_file.txt @@ -0,0 +1,392 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf,len=391 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='04840v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='IV] 12 Jan 2023 Centroid Adapted Frequency Selective Extrapolation for Reconstruction of Lost Image Areas Wolfgang Schnurrer, Markus Jonscher, J¨urgen Seiler, Thomas Richter, Michel B¨atz, and Andr´e Kaup Multimedia Communications and Signal Processing Friedrich-Alexander Universit¨at Erlangen-N¨urnberg (FAU) Cauerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 7, 91058 Erlangen, Germany Email: { schnurrer, jonscher, seiler, richter, baetz, kaup } @lnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='de Abstract—Lost image areas with different size and arbitrary shape can occur in many scenarios such as error-prone communication, depth-based image rendering or motion compensated wavelet lifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The goal of image reconstruction is to restore these lost image areas as close to the original as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Frequency selective extrapolation is a block-based method for efficiently reconstructing lost areas in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' So far, the actual shape of the lost area is not considered directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' We propose a centroid adaption to enhance the existing frequency selective extrapolation algorithm that takes the shape of lost areas into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' To enlarge the test set for evaluation we further propose a method to generate arbitrarily shaped lost areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' On our large test set, we obtain an average reconstruction gain of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='29 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Index Terms—Image Reconstruction, Signal Extrapolation, Error Con- cealment, Wavelet Transform I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' INTRODUCTION Image reconstruction aims at restoring lost areas in images as close as possible to the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' There are several applications where lost areas of arbitrary shape can occur and need to be reconstructed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=', when distortions like scratches or dust are to be removed from scanned images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In multiview scenarios, lost areas can occur espe- cially at object boundaries, when an intermediate view is computed by depth-image based rendering [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In motion compensated frame rate up conversion [2], lost areas can occur in predicted intermediate frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' A very similar pattern of lost areas occurs when the block- based motion compensation is inverted, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=', in the update step of compensated wavelet lifting, unconnected pixels can occur [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In [4], the reconstruction of these unconnected pixels was shown to be advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In all of these applications, the different size and the arbitrary shape of the occurring lost areas is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Several methods exist for reconstructing lost image areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In [5], classical kernel regression (CKR) is extended by nonlinear kernel adaption to obtain steering kernel regression (SKR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' [6] proposes a framework based on total variation (TV) that can be used for re- constructing larger lost areas in images, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=', to remove text overlays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In [7], the constraint split augmented Lagrangian shrinkage algorithm (CSALSA) is proposed and used for reconstructing images with lost pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The frequency selective extrapolation (FSE) generates a model in the frequency domain based on the available pixels [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In natural images, pixels closer to each other have a higher correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The correlation reduces with increasing distance of the pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' When some pixels are lost, support pixels closer to these lost Original Dense loss pattern Sparse loss pattern Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Detail of an image, arbitrarily shaped loss patterns (black is lost), and images to be reconstructed pixels should have a higher influence on the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' FSE is a block-based method which uses an isotropic weighting function to control the influence of the support pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' So far, mostly block losses were considered where the currently considered block was completely lost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=', the weighting function is centered w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' the currently considered block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In this paper, we focus on the reconstruction of images with arbitrarily shaped lost areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' We propose a centroid adaption of FSE (CA-FSE) to address the arbitrary shape of lost areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' To evaluate the performance, we propose a method to simulate the arbitrarily shaped lost areas occurring in the above mentioned scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 1 shows an example image with dense and sparse loss pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' By applying different loss patterns to different images, the performance of the reconstruction methods is evaluated and compared on a very large test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In Section II, we briefly review the FSE algorithm and introduce our proposed centroid adaption in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Simulation results and discussion follow in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' FREQUENCY SELECTIVE EXTRAPOLATION Frequency selective extrapolation (FSE) [8] is a block-based it- erative method for reconstructing lost areas in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' With the optimized processing order [9], the size of lost areas is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The more available support pixels a block has, the earlier it is processed and bigger lost areas are processed from the border to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' However, in contrast to the size, the shape of the lost areas is not considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' For every block, FSE generates a model g [m, n] = � k∈K ˆckϕk [m, n] PSfrag replacements Pixel categories of L: originally known A already reconstructed R Bo: lost, outside of b Bi: lost, inside of b L = A ∪ R ∪ Bi ∪ Bo m n b d d PSfrag replacements Pixel categories of L: originally known A already reconstructed R Bo: lost, outside of b Bi: lost, inside of b L = A ∪ R ∪ Bi ∪ Bo m n b d Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Composition of extrapolation area L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The corresponding weighting functions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' for the unknown pixels based on the available pixels in the support area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' A weighted superposition of 2-D Fourier basis functions ϕk is generated where the set K contains the indexes of all basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In every iteration, the influence ˆck of the basis function ϕk is increased that reduces the approximation error the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 2 shows the composition of the extrapolation area L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The area L has a size of M ×N pixels and consists of the currently considered block b in the center, surrounded by a support area of width d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The pixels within L come from three categories, namely originally known pixels A, already reconstructed pixels R and lost pixels Bi ∪Bo that have not been reconstructed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Thereby, Bi contains lost pixels within the currently considered block b and Bo contains lost pixels outside of b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Lost pixels within the currently considered block b are reconstructed by an inverse transform of the model g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' For a more detailed description of FSE together with pseudo code, please refer to [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' PROPOSED CENTROID ADAPTION To control the influence of the support pixels during the model generation, a weighting function w [m, n] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The influence decreases with increasing distance to the center of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The weighting function is computed to w [m, n]= \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ρ √ (m− ¯ m)2+(n−¯n)2 for [m, n]∈A δρ √ (m− ¯ m)2+(n−¯n)2 for [m, n]∈R 0 for [m, n]∈Bi∪Bo (1) where [ ¯m, ¯n] is the center of the block b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' On the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 3, an example is shown with w [m, n] centered in b corresponding to the areas named in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The decay is controlled by ρ and the weight of already reconstructed pixels R is attenuated by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' So far, mostly block losses were considered where b was com- pletely lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Especially for arbitrarily shaped lost areas, the case occurs quite often where only a part of the reconstruction area is actually lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The weighting function w [m, n] controls the influence of the support pixels on the model generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Support pixels closer to the lost area shall have more influence than pixels farther away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' So far, the weighting function does not consider the case that only a part of the pixels within b can be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' To adapt the model generation to the arbitrary shape of the lost area, we propose to center the weighting function on the centroid of the lost pixels Bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=', within b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The proposed weighting function is computed by Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Example for moving the center (marked by a green star) of the weighting function w [m, n] to the centroid of the lost pixels (black) within the currently considered block b (red box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The different kinds of pixels are labeled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Left: w [m, n] centered in the currently considered block b, Right: w [m, n] centered on the centroid of the lost pixels inside the currently considered b w [m, n]= \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ρ √ (m−mc)2+(n−nc)2 for [m, n]∈A δρ √ (m−mc)2+(n−nc)2 for [m, n]∈R 0 for [m, n]∈Bi∪Bo (2) where the centroid [mc, nc] of the lost pixels Bi within the block b is computed by mc = 1 |Bi| � ∀m∈Bi m nc = 1 |Bi| � ∀n∈Bi n On the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 3, an example for centering w [m, n] on the centroid of the lost pixels within b is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' This leads to the intended weighting of the pixels with respect to the distance of the actual lost area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In the following, the centroid adapted FSE is called CA-FSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' SIMULATION RESULTS AND ANALYSIS For evaluation, we used the images from the two image databases Tecnick [10], consisting of 100 images of size 1200 × 1200, and Kodak [11], consisting of 24 images of size 512×768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' To extend our test set, we took the first image of each sequence of the ARRI database [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' We further took views 0 from Illumination 1 and Exposure 1 of the Middleburry multiview databases from 2005, 2006, and 2014 [13], [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The luminance of each image was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' We used three typical parameter sets for FSE and CA-FSE, called ‘bs 4’, ‘bs 8’, and ‘bs 16’ listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Table I FSE PARAMETER SETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' FSE parameter set denoted as bs 4 bs 8 bs 16 size of block b 4 × 4 8 × 8 16 × 16 border width d around b 14 12 16 FFT size 32 32 64 decay parameter for w [m, n] ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='7 attenuated weight for R δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='5 orthogonality correction γ γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='5 For the simulation, the loss patterns are generated using the following commands in MATLAB dense loss pattern=imdilate(rand(img size)>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='98, strel(’square’, 8));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' (3) 0 5 10 15 20 25 0 1 2 3 Kodak image number PSNRCA-FSE − PSNRFSE in dB bs 16 bs 8 bs 4 0 20 40 60 80 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='5 Tecnick image number PSNRCA-FSE − PSNRFSE in dB bs 16 bs 8 bs 4 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Difference PSNR in dB of the reconstructed pixels between the centroid adaption (CA-FSE) and the unmodified weighting function (FSE) for a dense loss pattern, Left: Kodak, Right: Tecnick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' sparse loss pattern=imdilate(rand(img size)>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='95, strel(’square’, 8));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' (4) Visual examples for the two loss patterns are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Lost pixels are colored in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The dense loss pattern in (3) causes a loss of about 28% of pixels and the sparse loss pattern in (4) about 4%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The challenging patterns are very similar to the lost areas that occur in different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' We choose the dense pattern to evaluate the algorithms on larger lost areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' For both loss pattern types, the simulations were repeated for 10 different loss patterns to obtain significant results and to avoid special effects from a single specific pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' For each loss pattern, the PSNR is computed on the reconstructed pixels only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Then, for each loss pattern, the average is computed for the 10 different loss patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 4 shows the results for the Kodak and the Tecnick database for the dense loss pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Whenever the difference is positive, CA- FSE obtains a better reconstruction result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The dotted lines show the average result for each test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The corresponding values to the dotted lines are listed as diff in the first two lines of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' II together with the absolute PSNR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In the lower part, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' II contains the corresponding averaged results of the simulation using the sparse loss pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Further, both tables contain results from four recent reference methods from the literature, namely based on kernel regression (CKR) and (SKR) [5], total variation (TV) [6], and constraint lagrangian shrinkage (CSALSA) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Generally, all PSNR values increase by about 3 dB for the sparse loss pattern as can be seen by comparing the upper and the lower part of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The sparse loss pattern causes less and above all smaller lost areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' From the latter, all methods can profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Comparing the achieved gains of CA-FSE to the unmodified FSE, the gains also increase for less and smaller lost areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The weighting function w [m, n] is centered on the centroid of all lost pixels within the block b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' When there are less lost areas, the case of more than one distinct lost area within b occurs less often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' When there is only one lost area within b, the w [m, n] can be centered exactly on the centroid of this one lost area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The behavior of the results is similar for all datasets with exception of ARRI, where CA-FSE obtains an additional gain of about 1 dB for the dense loss pattern compared to the other databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' For the sparse loss pattern and bs 16, a remarkable gain of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='1 dB is obtained compared to FSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' We further investigated this case and found that images from the ARRI database have a small black border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Slightly wrong reconstruction values on this border can cause a very big loss in PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Compared to the other databases, the extreme large gains of CA-FSE and CSALSA for ARRI mostly come from border effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' To further analyze the performance disregarding the border, we additionally evaluated the reconstruction quality, omitting a border of 16 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Excluding the image border, the gain of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='1 dB shrinks to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='8 dB which is in the same range compared to the other databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The centroid adaption does not only lead to a better reconstruction of lost areas at image borders but also can improve the performance within the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The achieved reconstruction gain grows with increasing size of the block b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The reason is that the centroid of lost pixels is computed within the block b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' So, for a small block size, the center of the weighting function can only be moved in a small range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Nevertheless, the influence on the reconstruction result is remarkable keeping in mind that for bs 4 the center can move by a maximum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='5 · √ 2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='1 pixels for the extreme case that exactly only one pixel located in one of the corners of b is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' With a larger block b, the center can move to a larger extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' This explains the increasing influence on the results with increasing block size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' 5 shows reconstruction results for visual comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' When pixels are lost in smooth image regions, an adaption of the weight- ing function has no real influence on the reconstruction result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' In structured regions, and especially when lost areas occur at edges, the proposed centroid adaption of the weighting function is advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The improved reconstruction performance of CA-FSE at the image border can be seen best for the gray border at the top of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' CONCLUSION In this paper, we consider the challenging reconstruction of arbi- trarily shaped lost areas in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' We propose CA-FSE, a centroid adaption of the weighting function of FSE, to address the arbitrary shape of lost areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Over our large test set, CA-FSE consistently improves the reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The treatment of lost areas at image borders is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' With the proposed adaption, the reconstruction performance of FSE can be improved by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='29 dB of PSNR on average for arbitrarily shaped lost image areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Further work aims at optimizing the reconstruction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=', only one distinct lost area within the currently considered block is recon- structed at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' We also aim at an adaption of the shape of the weighting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Table II AVERAGED PSNR VALUES IN DB USING THE DENSE LOSS PATTERN (3) IN THE UPPER PART AND THE SPARSE LOSS PATTERN (4) IN THE LOWER PART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' CKR[5] SKR[5] TV[6] CSALSA bs 4 bs 8 bs 16 [7] FSE[8] CA-FSE FSE[8] CA-FSE FSE[8] CA-FSE proposed diff proposed diff proposed diff Dense Loss Pattern Kodak 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='952 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='476 23.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='287 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='815 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='936 + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content='121 Original CKR[5] TV[6] FSE[8] bs8 Lost SKR[5] CSALSA[7] CA-FSE bs8 proposed Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Visual reconstruction results for a detail of image 15 of the Kodak database using the dense loss pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' The lost areas are black in the lower left image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors would like to thank Eduard Sch¨on and Nils Genser for their valuable assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Further, we gratefully acknowledge that this work has been supported by the Deutsche Forschungsgemeinschaft (DFG) under contract number KA 926/4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E4T4oBgHgl3EQfAAvD/content/2301.04840v1.pdf'} +page_content=' Fehn, “Depth-Image-Based 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